Complex Analysis (C.I.M.E. Summer Schools, 62)

F. Gherardelli ( E d.)

Complex Analysis Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 3-12, 1973

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-10963-8 e-ISBN: 978-3-642-10964-5 DOI:10.1007/978-3-642-10964-5 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Cremonese, Roma 1974 With kind permission of C.I.M.E.

Printed on acid-free paper

Springer.com

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.LM.E.) I Cielo - Bressanone - dal 3 al 12 Giugno 1973

«COMPLEX ANALYSIS» Coordinatore: Prof. F. Gherardelli

A. ANDREOTTI:

Nine lectures on complex analysis

J. J.

Propagations of singularities for the Cauchy» Riemann equations

177

The mixed case of the direct image theorem and its applications

281

KOHN:

YUM-TONG SID:

Pag.

»

1

CENTRO INTERNAZIONALE MATEMATICO ESTIVO HC. I. M . E. )

NINE LECTURES ON COMPLES ANALYSIS

ALDO

Corso

tenuto

a

ANDREOTTI

Bressanone

dal

3 al

12

giugno

1973

- 3-

A . Andreotti

Preface. In the spring of 1972 I h ad the opportunity to lecture at Lund University and more extensively at Amsterdam University and at the C.I.M.E. session in the. summer of 1973.on some topics of complex analysis of my choice. The subject has been chosen within the limited range of my personal knowledge and is intended for a non excessively specialized audience. We have tried th erefore not to obscure the ideas, in the attempt to obtain the most general statements, with an excess of technical details; for this reason, for instance, our main attention is devoted to complex manifolds, and we have recalled basic facts and definitions when needed. The purpose was not to overcome the listeners with admiration for the preacher but to share with hime the pleasure of inspecting some beautiful facets of this field. Indeed I was very grateful .to receive many valuable suggestions; in particular I am indebted to L. Garding, L. Hormander, F. Oort, A.J.R.M. van de Ven and e.pecially to P. de Paepe who undertook the heroic task of writing the notes. The material deals with the theory of Levi convexity and its applications, with the duality theorem of Serre and Malgrange, and with the Hans Lewy problem. The limited time at our disposal may account for some conciseness that, however we hope, will turn to the advantage of the reader. P. de Paepe has corrected several mistakes of mathematics and presentation; probably only few remained undected. San Pellegrino al Cassero,

September

1973, Aldo Andreotti.

-4 -

A. An d r eot t i

CONTENTS Chapter

I. 1.1. 1 ..2.

1.3. 1.4. Chapter

II..

Chapter III.

Chapter

Chapter

IV.

V.

5.1. 5:2.

5.3. 5.4.

Elementary theory of holomorphic convexity. Preliminaries. Hartogs domains. Open sets of holomorphy. Levi-convexity. Psuedoconcave mani folds. Preliminaries. Meromorphic functions and holomorphic line bundles. Pseudoconcave manifolds. Analytic and algebraic dependence of meromorphic functions. Algebraic fields of meromor phic functions. Properly discontinuous pseudoconc~e groups, the Siegel modular group. Preliminaries. I Pseudoconcave properly discontinuous groups. Siegel modular group. Psuedoconcavity of t he modular group. Poincare series. Projective .imbeddings of pseudoconc ave manifolds. Meausre of ~ s e ud o c o n c avi t y . The problem of projective imbe dding of pseudoconcave manifolds. Solution of the problem for O-concave manifolds. The case 0 f dimU; X ~ 3. Meromorphic functions on complex s paces. Preliminaries. Psuedoconcavity for c omplex s p aces. The Poincare problem. Relative theorems.

1

4 5 12

19 22

27

31 35

40

43 45 47 52

56 59

63 67

71 73

74 82

-5-

.~ .

Chapter

VI.

6.1. 6.2. 6.3.

6.4. Chapter

VII . 7.1. 7.2.

7.3.

Chapter VIII.

8.1. 8.2. 8.3. 8.4. Chapter

IX.

9.1. 9.2.

9.6.

Bibliography.

E.E. Levi problem. Prelimina1rl.es. E.E. Levi problem. Proof of Grauert's theorem. Characterization of projective algebraic manifolds. Kod ai.r-at s theorem. Generalizations of the Levi-problem. d-open sets of holomorphy. Proo f 0 f theorem (Grauert) d • Finiteness theorems. Applications to projective algebraic manifolds. Duality theorems on complex manifolds. Preliminaries. v Cech homology on complex manifolds. Duality between cohomology and homology. v Cech homology and the functor ElCT. Divisors and Riemann-Roch theorem. The H. Lewy problem. Preliminaries. Maijer-Vietrois sequence. Bochner theorem. Riemann-Hilbert and Cauchy problem. Cauchy problem as a vanishing theorem for cohomology. Non-val1di~ of_Poincare lemma for the complex. {R(S),d S S • Global results.

Andreotti

86 92 94

103 111 113 118 119 124

128 13Q

131

138 147

150 154 155

158 160 163

166

- 6-

A . Andreotti

I.

Chapter 1.1 of

Ele!llent ary t he ory o f hol omor ohi c c onvexi t y .

Prel imin aries. a) Let 11 be 'ill o p en s et in n co pies o f t he co mplex f ield

ions

lC

n

II:

I I

t he Cartesi an pr oduct wit h c oor di n at e funct-

Zl"",Zn' A f unction

poi nt

Z

o

it

.!l on which

J1.

f

f: .a. - lC

is c al l ed l101omorl) hic if for ev ery

t here ex i ~ts a neighborhood U( zo ) of Zo in admi t s an ab solutely c onverge nt power s er i es ex-

p ans i on f or every Here a

0),

aa...

.1 I'"

a.

.I '

Z

,'"""n

= nat ur al number s including 0. d. 1 n = zl ,zn

f = (f ••• , f ) : .Ii. ~ a;m i s s aid to be holomorphic . l, m if e ach co mnonen t f 1 ~ i ~ m, is hol omorphi c . i, The c ompos i tion of two hol omor phic maps i s ( wh ere it is de f i ned ) a hol omor ph i c map.

A map

]Rn , i = ~. b) We will write Z = x + iy with x, y 1 1 Then x = 2(z+z) y = 2i(Z-Z) where t he b ar den ot es compl ex 1 1 con jugation, and we will wri te dx = 2( d z+dz) , dy = 21 (dz-dz) l For any fu nction f : .0.-11: of cl ass C we h ave

ae

=

~=l

a! <3

Zj

d Zj +

where af aZ j

= .!(of 2 ex j

af a'Z j

=

.!(~.

of n jh z.

a=

+ .!~) 1 oYj 1 of ) a

2 oXj - I

Yj

J

dZj

.

-7 -

A. Andreotti

we define

n

af

azj

af

=

j~l

if

=

~ U j=l 0-

Zj

The following theorem est ablishes a cr i t er i on for a function 1 . ot class C to be a holomorphic function. l is holomorTheorem. A function f : .n - ; of cl ass C phic iff at every point of n f satisfies the Cauchy-Riemann equations: (1.e.

at

all

at

=

0, •.. ,

=

-

°

af aZ n for

c) Let polycil1nder

f

= 0, 1 1 j

i.e.

~

n) •

be holomorphic in a neighborhood of the closed for

then for every Z c ~, Cauchy integral formula

the interior

0

f

1

P,

~

j

~

n

we h ave the

From this tormula it follows easily (by ex pansion of the kernel of the integral in power series) that a continuous function t: Jl, - a: which is separately holomorphic in each variable is a holomorphic function (Osgood's lemma). This is even true it the condition that t is continuous is removed (Hartogs' theorem) but t his is much more difficult to prove.

-8 -

A. .Andreotti d)

We recall the following result: the set of points where a holomorphic function has a zero of infinite order is open and closed (principle of analytic continuation) .. In particular if f . is defined in.n. , if.n isconnected and if f vanishes at some point of..o. of infinite order then f is identically zero on n . e) We denote by H(n) the set of all holomorphic functions in n. It is a vector space over t. We can provide /I (Jl) with a locally convex topology defined by the family of seminorms

=

Ir ]

sup K

where K is a compact s ubset of .fL. A fundamental system of neighborhoods of the origin is then given by the sets V(K,

e) =

fr €

!I(Jl.)

for K compac t in 11 and E. > O. This topology is the topology of uniform convergence on compact subsets of n . _ If Kl C KZ e, K c, • • • is a ae quenc e of compact sets such

3

U that -in K c, K 1 for m = 1,Z, ••• , and.D:: m-=1 Ifm m+ one easily verifies th at the count able s et of seminorms defines the swne topology, Thus #(Q) isa metrizable space, one c an t ake for Lns t anc e as a di s t ance the function O

d(f,g) =

~

mel

~ ~m

<S'

Ilf-gll~ 1 +

Ilf-gll

K :n

r,

g eo HUl.).

Since continuous functions satisfying the Cauchy integral formula are necessarily holomorphic, it fo l lows th at H (U) is a complete met r i c sp ace (i.e. a Frechet sp ace) and therefore a Batre space. We - also note th at bounded sets B c: /I (il) are relatively compact. This is a consequence of

- 9-

A. Andreotti Vitali's t :leorem: functions on II.

ffv~

If

1s a seguence of ho1omorphic

s uch th at for every compact set

exists a constant

C(K)

v

C(K)

In particu1ar'the unit ball in the norm

(e r,

1.2

1.e.

[14] ).

1s re1ais a space

H(fL)

For more

~

det~ls

we

[28], [31] I [431.

Hartogs domains.

m3

Consider the subset in

Z

II II Km+1

II II ~

compact with respect to the norm refer to

~

= 1,2, •••

then we can extract from ffu~ a sUbseguence ~ful converges uniformly on any c ompact subset of n .

of Frechet-Schwartz

n

for which we have

K ~

/lfJI

K c.

=x

(coordin ates

x I .YI t ,

+ iy) T

= liz) o

where

<

< a
Bec ause

b

I

and

0 s, t c

e5 \J

fa < I z I

<. b,

0

~ t
5

< c < d.

0

fits sh ape we call T a "top hat ll • A top hat n n-1 (or Hartogs domain) in II: a: "-II: , n 2 2, is the set of all 0

=

=

points z (zl""'z . n ) in contained in a to p hat in Theorem

a:n

for which

m3 •

(1.2.1). (Hartogs).

top hat T =

! Iz 1 / j

in ~

II:

<. b ,

~2 I Z j I 2"

n

, f

I Zj/2

j~2 d

2

<

~

f

(zl'

(j~~ I z jI2)t)

be holomorphic in the

c2 S U {a

<.

I zi

<: b ,

S

n ~ 2. ext ends holo morphical1y t o the filled up top hat n j;2

I Z j)

2

2 <. d

is

5•

-10-

A. Andreotti

!:!::221.

Let the functions Cauchy integral

p e Ii,

~,

Jl

be defined by the

f(~, z2' ••• ,zn)

S -zl

lsI =b--p P

If

is large

is well-defined, continuous and holomorI zll b-l/p and j~2 Zj 12 <: d 2

~

I

phic in each variable for Therefore p.

Set

is holomorphic.

~

g

= ~,

there and

glT

on

b,

~ I zll <

" h(T)

~

h: "T

Let

of

=f

2

-

n 11:,

holomorphic).

then

g

because

I Zjl2

Moreover

is defined in

< c

2

g-t

S

~~

is independent of

T,

is holomorphic

is holomorphic in

and

T

T,

is zero

is connected. Q..E.D.

n

be a biholomorphic map onto an open set (i.e. h is invertible and h and h- l are both II:

h(T)

Then any holomorphic function on

holomorphically to

" h(T).

extends

This is a consequence of Hartogs l theorem and the fact that the composition of two holomorphic maps is a holomorphic map. It is sometimes called the "disc theorem".

We quote some "s i mpl e consequences of Hartogs l theorem."

to ~ 1

Let

n

~ Iz j/

rI

2

L

2

< r

2

and

J

Zj/2 < r2S.

f then

follows that

f

extends holomorphically to the ball

In tact we can put a top hat

punctured ball so th at cannot have an

holomorphic on the punctured ball

T

covers the origin.

a holomorphic function "isolated s ingularity"

f

in

n

in the

I n particular it ~

2

variables

nor an isolat ed zero

(since this would be an isolated singuaarity for li3

T

l/f).

Open set. of holomorphYT

a) Open sets with a smooth boundary. An open set D.. in rt _ [L has a smooth boundary if for every point Z e. 0 n o we can find a neighborhood U(zo) and a C 6' function

=

len

f:

U(z ) • lR such that o

-11-

A . Andreotti

This amounts to say that by a local diffeomorphism near

n

0 U(zo)

space. which

can be transformed into an open subset of a half

Indeed we can select a set of real

= xl

+ - +(zo)

a: n

be an open subset of

and let

zo"

on.

we will say that

t.

f

G

f

over

with a smooth boundary Given

f

€

/I (Q.)

Let.ft

Zo

Zo

if

and a holomorphic

=

f

•

a:n

be an open subs et of

We say th at..Cl.

with a smooth

is an open set of holo monphy if for

zed Il. = o

fell (rl)

of

such th at

o

every boundary point function

il () U(zo)

Thus

°S.

c,

m 2n)

(=

V(zo)

fIn. boundary.

coordinates in

is holo morphically extendable over

II (n u V( z »

Definition.

xl

be a boundary point of.n..

we can find a neighborhood function

C tJ>

is the first coordinate.

is an open subset of the halfspace Let.Q.

Zo

Ii.

-

we c an find a holomorphic

(l.

which cannot be extended holomorphically

Zoe

Examples.

1.

Every open subset 12 c..

set of holomorphy. is not 2.

extend~ble

(b,O, ••• ,0). on

a Q. 3.

over

The ball it =

holomorphy.

a:

wi th smooth boundary is an open

Indeed for every

6

»s:

f = (z-Zo)-l

a: n

is an open set of

Zoe

Tj~l IZ j 12

<:.

f = (zl - b)

Indeed

Zo

2 b -1

~

Since the unit ary gr o up

in

is not extend able over U(n)

a c t s transitively

' by holomorphic tr ansform ations, the a..s s e r t i on follows. The ' circular shell Il. =

°< a

< b,

if

Indeed for every point we can p l ace a top h at

n

c,

jt. I zjl 2

<.

b

2

$

for

is not an ODen s et of holo morphy. 2 Z of the inner bound ;ITy o~l Iz = a Jo)2 o " J= T in [ l such th at zo" T. ~

2

1a 2

-12-

A . Andreotti

b) Open sets with arbitrary boundary. By a domain we mean an open connected set. Let /J c. A be two domains in a: n and let S Co #(A) be a set 0 f holomorphic functions in A. We sa:y th at ~" is an S - completion 0 t /J. if

/\

i.e. if every f e S extends holomorphically to 4 . Note that by the principle of analytic continuation the extension of f to f £ #(A) is unique. For instance for a top h at T, '" T is an )/(T)-completion of !

(nn :2).

Definition. Let n. be an open set in of holomorphY if: for every domain Il c, fl

b. is cont sd.ned in

.a. ."

a: n •

We say th at D..

every

is an open set

'" - completion ,1

of

Remark: " Op en sets of holomorphy with a smooth bound ary are necessarily open sets of holomorphy in the sense of this general definition. We will see later th at the converse is also true. we will refer for the moment to the definition given before for o pen sets of holomorphy with a smooth boundary as the "provisorlal definition of open sets of holomorphy". c)

Holomorphlc convexity. characterization of ope n sets of holomorphy. An open set SL in a: n is called holomorphically convex if for every compact subset K c. n.. the holomorphically convex ,.. envelope K of K in IL, de f i ned by

K= tZ".n I is also c omp ac t , '

If(z)/

~

IIfll K

for every

t «

#((1.)

5,

- 13-

A . Andreotti Theorem

(1.3.1).

(1)

convex if for every divergent exists an

f

!lUI.)

4.

sup

!:!:£.2.!. convex.

Condition

is ho10morphical1y

X,;5 c (l.

seguence ~

I

=

D

implies th at

+ '" ("condition

.n.

K is comp act and

Indeed, if

divergent sequence

[xvS

But then for every

f ~ /I(U) ,

~

n

in

n.

K

£xv}

t

I f(X) I

DII).

is ho1omorphic all y is not we can f ind a C

K •

11 f

~

II K

be ho10morphical1y convex.

D holds.

show th at condition

c:
which

D.

contr adicts condition Conversely let

a: n

e,

such th at

/f(X,,)

V

.D.

An open set

Of t his f act

~e

We want to will give two

proofs. 1st proof: sup

'"

~ xv~

By absurdity;

xvl c.

~

gent sequence If(X)1 <

Q.

suppose th at there eXists a diver-

such th at for every

t ~

II (ft)

By passing to a subsequence we may assume

tP.

contained in a connected component of 12..

generality .we may t hus assume A =

1 fe J.I (Q.)

Then 1I(n.)

Now A theorem

I

sup v

.fl

/f(X )

"

'" =m~l

is a closed subset 0 f

connected.

I

= - A)

(A

thus

Set

~ 1 ~ .'

m A. 1-1 (.0.) •

Thus by the Baire category

must contain an interior point.

A

and symmetric

Without lo ss of

But

A

is convex

A must contain a neighborhood of

the origin say V(K,

( for a comp act that (1)

z) K

c,

=[f n.

.. 1I(n.) and some l.

>

0).

We may assume as well

K has non-empty interior. By thiS We in .('L •

mean a sequence

[x ,,~

with no a ccumulation point

-14-

Ii Andreotti

Now for every fore ror every

f

f

il 0,

<1

JI(Q)

sup

In particular, replacing

r

I fI(x)1

sup

!r(x

sup

I f(X Il ) J s

v

)1 v

~,

!

~

there-

IIrJl K •

by

sup -V

i.e"

~~

[r(xv)1

v

rEA,

f ~ #(Q),

we get

IIfll\K

-f.

1

So

(f}i

!lfll K •

This shows that

forevery

fE-lim).

Inorphic convexity 2

nd

proof: such that

Let ~ XlJ~

[Xv5

0

f

Hence

1/ K

tXv~CK.

This 'contradictstheholo-

Jl •

Select a sequence

1~ S

of compact subsets of

be a divergent sequence in..Q.

Repl acing

by subsequences we may a ssume th at X

Since

I)f

X

m

1

K

m

= Km

m E. Km+l

•

we c an find

Choose positive int egers

.A-

m

for

m = 1,2 , •••

~ ~ lI(fIJ

such that

succ essively so th at

~ K

S

m

and

- 1 5-

A. Andreotti

Now

f

£

s

..Am

D..

every compact subset of some 1hus

Km

converges uniformly on every

~

and thus on

as any such set is contained in

Km' f

is a holomorphic function in LL.

inequality we derive that sup

Therefore

v

Theore.

I f(x

(1.3.2).

\J

/f(X

) J

s

+

If J1

rI'

I>

m.

•

(Cartan - Thullen).

is an open set of holomorphY ifi

f£22!.

m)

But from the last

n

a

An open set

mn

G

is holomorphicallY convex.

is holo morphically convex, then condition

holds, thus clearly Jl

is an open set of holomorphy.

Conversely suppose that want to prove that fL

D

n.

is an open set of holomorphy.

is holomorphically c onvex.

the case, then there exists a compact s ubset

K

Co

We

If t his is not

" K

fL such that

is not compact.

IlZjJl

Because for each coordinate function we h ave ~

K is bounded. sequence in ..Q

Let

c;" 2 xvJ C K

denote the polycilindrical norm in p

= polycil1ndric al

Certainly p >

o.

If P

holomorphically convex. set of points in f.l

s

00

(la.

m n

, and let

di s t an c e of then

Jl

Let K

.n. = m n

We may assume p c

and

Z

._

1/ = sup

an."

which is cl early
Let K'

whose polycilindrical distance from

1-P. Then K' is a compact subset of 11.. For every 4 of non-negative integers an d every f ~ #(Q)

~

= IIZj/l~,

be a divergent

x" ~ Zo ~

such that

K

be the K

is

n - tuple

A . Andreotti

1

at

a

D f •

1I(f!.) ( 1 )

therefore for an.y

For

z

~

K

x e K"

we have by the Cauchy formula

an.d therefore

1

(2)

(~) From (1) a point

and

x

~

(2)

1~1

it follows that the Taylor seri es of

f

at

c

P/4S.

1\

K 1 at

t1

D f ( x ) (z_x)d

is majorized by the series

1\ r II K'

)

;ii~ I

d.

and therefore is abs ol ut el y convergent in Q(x) Now for v Let A

sufficiently large

Q(x,,)

f1z-x

1\

contains the p ointz

be the connected component of Q(x.)

)in

=~

Il

n.

we set

containing

D

et

=

o

•

x 1J.

-17-

A. Andreotti Then

Q(x'll)

A.

II(U)/tJ -completion of

is an

ss it contains the point

Zoe

Therefore

But

Q(X iI

.a-

¢

)

Jl cannot be an open

set of holomorphy. Remark. If .Q

n II: with a smooth b oundary. is an open set of holomorphy then by the Cartan-Thullen

theorem J1 condi tion

.n.

Let

be an open set in

is holomorphically convex and therefore s atisfies

D.

Hence .R

visorial sense.

is an open set

0

.

f ho lomorphy in the pro.

For open sets with a s mooth bound ary the pro-

visorial and the general definition of open "set of holomorphy are eqUivalent.

1 :4

Levi (1) - convexity; a)

a

Let.a.

C ~ function.

expansion of

II:

be an open subset of

and let

~

:.Q ~ lR be

a e D.. we consider t he Taylor

At a point

f;

n

with obvious notations for the partial d er i v a-

tives, we h ave,

~ (z) '" ~ (a) +

r. d. 4(a)(za-act)

+

[,;:4(a)(~-8.ct)

+

+ t~~j34(a)(zd-ad)(z~-a~) + tZ 9 --t>ca)(z -8. )(z -8. )

etp

+6 Because

have

~ ~( a) d-

l)

.

a:

do

fl

q(a)(z -a )(z -8.) a a; ~ 13

d.p

and because

'" ~ d.-«. a)

d

~,

~

+

= ta
o( IIz-a

/13)

is r eal-valued, we must

f (a) '" a--4J( a) ; df3

In particular the quadratic form

Eugenio Elia Levi,

(3

aci~- k. a)

= Cl-

c{p

4(a) •

-18-

A. Andreotti

is hermitian; it is c alled the Levi-form of A biholomorphic change of coordinates near L(~) with a linear change of variables a v

ata. a acts on

~

J(a)v

~

where J( a) is the Jacobian matriX of t he change of vari ables a t a. It follows th at the number of positive and the number of negative eigenvalues of the Levi-form at a do~s not depend on the :hoice of local coordinates. Remark. lln which

If (dp)a ~ 0 we can perform a change of coordinates a is at the origin and in which the new ~ - coordinate

1a ~ -0 Ha)(z -a ) + d dd.

Then

tL'

0d ,,40 ( a )( z -a Hz -af.l.). r--

del.

~Y

takes the following Taylor exp ansion:

~

~

(z)

= 4(0)

+ 2 Re

~

+ L(Oo(z) + 0«1 z 11 3 ) •

b) Let us assume th at (d~)a ~ 0 and, for simplicity of not ations th at a is at t he origin. Set U

Then au plane to

=~:z

=u - U

E.a./

4>(z) <

~(o)

=

i s s mooth near a 0 U at the ori gin is given by

and the r e al tangent

E This plane contains the equ ation

(n-l) - dimensional complex plane with

This is called t he analyt i c t an gent plane to will be denot ed by Ta(aU).

cU

at

a

and

-19-

A. Andreotti

Consider the Levi- form

0

f

~

at a restricted to

Ta ( au) ,

-.. We obtain in this way a hermitian form in n-l variables and again we realize that the number of positive and negative eigenvalues is 1ndependen~ of the choice of local holomorphic coordinates. Suppose now th at U is defined in a neighborhood of another C t9 function t with ( dy-) a # 0 U

=zz€

alt(z)(

a

by

t(a)S.

By subtracting constants from 0 where"t.> 0 we mus t h ave h( a).> O. Now

a •H =a (h

ar + ah.r)

= hoJt

+

ah.at + Jh.;;>'F

+d

ah.Y

and therefore

This sho ws th at the signat ure (i.e. the number of positive and neg ative eigenvalues) of the Levi-form r estrict ed t o t he analytic t ange nt plane to d U at a i s inde pendent also of the choice of the defining function ~ f or U near a.

- 20 -

1I.Andr~o1ti

Proposition (1.4.1). ~ U be an open subset of IC n with a smooth boundary. At any point a e aU the Levi-fchrm of any defining function for a U restri cted to the analytic tangent plane to. oU !i a has a signature which is independent of local holomorphic coordinates and of the choice of the defining function. Let pea) (q(a» be the number of strictly positive (strictly negative) eigenvalues of L(4')aIT ( d U) . These are a biholomorphic invariants of the triple (U, a u, a). Note that we must have pea) + q(a) S n - 1 • As an exercise we can show th at there is an analytic disc of dimension p

(i.e. the biholomorphic image of the unit ball

DP .. l t c mil

I ~ It i I 2 <

in;P)

1

such that

'\ ( 0) .. a

Analogously there is an analytic disc q such that /S(O) 6"'

(Dq)

=

t$

:

Dq ~

a: n

of dimension

a

-1 aJ c=.

U

Indeed we can moose coordinates at the origin such that

with

- 21-

A. Andreotti

.A. where the j's are ) 0 • Therefore near 0, for a > 0

o (lz 2 J2

sufficiently small, if

Izp+ 112<.

+ ••• +

j

£.

~. Zp+2

and

= ..• •

zn = 0

>

then Hz) 0 • . This proves the first statement.

The second one is proved with

a similar argument. c)

Theorem

(1. 4. 2).

(36]) •

(E. E. Levi

set of holomorphY with a smooth boundary.

.Q be an open

~

Then the Levi-form at

each boundary point restricted to the analytic tangent plane is ~ositive semidefinite.

Assume, if possible, that

~.

oe<3D.,

eigenvalue at the point for

-en

with

oj>

= o.

(0)

+

L(<$)oIT (an.)

o

has a negative

being a defining function

By suitable choice of the holomorphic

aoordinates we may write near

0

n

, (z) = 2 Re zl (1 + j;l :B':l.rst restrict There exists

to

~

e

,i.e.

D

r

C

Q..

for

zl

such that for

>0

lR 3 (l ~ Xl s, 0 ~ ,

Therefore:

nr =lIm

e

(zl

= Xl

= 0, (\ z II

+ iYl)'

z3

= •••

=zn=OS.

2e on the region

<.

we have

SUfficiently s mall,

4>

<

0

on the discs

-22-

A . Andreotti

Also if

£.

is sufficiently small,

~('

Zl ,= 0,

z

Hence there exists

=2" ~

A

c

)z21

A

LJ

B

= 1J.

8,

I zl l2 + I z3/ 2

0 <: 'l..£. 8,

+ ••• +

Q.

Iz)2 c ~5C:: n

the origin

I zll2 + i Z3/ 2 A is an /I em IA 0

In.,

a,

such th at

and let

By the disc-the!?rem cont~ns

<.

'1,

and there eXists

Let

l zzl c e , z3 = ••• = zn = OSC:: a , 0<:'0<:' e , such that

~

+ ••• +

..

-completion 0 r J. But is not an open set of

thus Q

"4

holomorphy. It is natural to ask if the above necessary condition for an open set.Q in a: n with a smooth bound ary to be an open set a: holomorphy is also sufficient (Levi-problem). The answer is n affirmative for open sets in a: but not for open sets on c omop:L.ex manifolds.

We will r (turn later to this question.

Exercises. I.'

Prove that every convex domain in

a: n

is a domain of

holomorphy. 2.

Suppose that Q.

~

• aa

at

a

lo~al

has a smooth bound ary and th at a t a point

the Levi-form restricted to the analytic tangent plane to d

n.

is strictly positive.

holomorphic coordinates at

elementary convex at

a.

a

Prove th at we can choose such that JQ

is locally

-23-

A. Andreotti

Hint: we can replace the defining function ~ by an increasing convex func tion 0 f ~ so th at L(V)a is strictly positive (for instance take e Cf -wit h c » 0, see I 23J, p. 263) and then use the remark in a).

t

t =

3.

Under the same assumption of the previous exercise, prove that there is a fundamental system of neighborhoods B(a) of a which are domains of holomorphy such that B( a) (l Q is an open set of holomorphy.

Hint: in the above specified local coordinates take for B(a) any small coordinate ball with center in a, then apply the first exercise. The material of this chapter is covered in all standard books on complex analysis as [ 28], [ 31].

-24-

A. Andreotti

Pseudoconc ave

Chapter II. 2.1

In ani folds.

Preliminaries •

• ) Presheaves. A presheaf on a topological space X is a contravariant functor from the category of open subsets U of X to the category of abelian groups i.e. for every U an abelian group 5(U) is given and for every inclusion of open sets V C U a homomorphism r UV

5(U)

~

5(V)

is given such that for every chain of inclusions Wc.Vc=.U of open subsets of X we have V U r W0 r V = A presheaf

;)

=

5(U)

r UW

is called a

~

if for every

open set n c:.. X- and every open covering ru. = ~ Ui1 iG I the following sequence iS lexact

o where

e

~

0

f

5(.Q)

is defined by

= and where cS

(l

r U (f) i

f

E

5 (0..)

is de fi ned by

Example: ~ = fHomcont (U, D:), r Uv 5, where Homcont (U, lC) denotes the space of continuous functions on U with values in lC and where r UV are the natural restriction maps, is a presheaf and also a sheaf ..

-25-

A . Andreotti

In a similar way one defines sheaves of rings and also sheaves of modules over a she af of rings. b) a ~ ~ over X of abelian gro ups is the data of a topological space a .continuous surjective map ~: ~ ~

T ,

such that tt is a lac al homeomorphism

d)

s = s(f)

an open neighborhood phism of

~)

s

1. e. every point

for each p oint

or

x eo X,

w! s

s uch th at

onto an open subset of

#

.L

rz:: cJ

has

is ~ homeomor-

X

x = 1/"-l(x)

has the structure

of an abelian group in such a way that the map

~

l(

1J'

X

~

(1)

t

given by is continuous ..

(r,

Given a stack set

Uc X

of abelian groups, for every open

we can consider the abe l i an group

r( n, 'f) of all

ir, X)

=l

B

"Bections"

U

restriction map

U 7

:

Tis

continuous, tr over

s

U.

r (V, 'Jl

7

r V:

If

B = i d e ntity on

0

uS

V e, U , t he n atural is defined and one

obtains in this way a presh eaf which is also a Bheaf.

3 =t 5(U)

Conversely, given a presheaf as s oc i a t e t o it a e t ack We set for every

(1', rr, x

= lim

(1) the "fibered p r od uc t " '!.><.X

j "

U~X

ter

or

lying above the diagonal

1I.f1: 'j.T

one can

as followB. 5(U) ,

is a cl .a a of equivalence of couples

:r. r

S

x.. X

~

1i

X)

U ; r V

7

Xx.X

= (1r ,, 17' ) - 1

(/.1).

;

i.e. (U, r)

an element of with

is d e f i n ed as the part of

b.

of

Xx X

by p ro j e c t i on

X

- 2 6-

A . Andr eotti

x e U,

f e- S(U)

under t he r el ation (U1' f l)

if th ere ex ists

U " x, 3 r

Ul

U

3

( U2 ' f 2)

'U

U

3

= r

fl

U2

The equivalence cl ass in or :r x o! and it is called the germ of t

We then define If we t ake on form

cr -

_

s uch t hat

c "i () U2

U C'r' x~xj x

f2

U

3

(U;

is den ot ed by

x,

at and

f)

11

by

T1 (

r;-?

:J x) = x ,

~ as ~ basis for open sets the s ets of the f or all f E S(U), we obt ain, as one verifies,

a stack of abeli an groups

('r, fi",

X).

st arting in t his construction with a sheaf, constructing the corresponding st ac k and t hen the corresponding s heaf of sections we ge t back the original s he af. We t hus h ave a one~to-one correspondence between sh e aves of abeli an groups and st acks of a be lian groups. Although this could ge nerate some confusion it is cu stomary to "r ep r es ent a s he a! by the associated s t ack (see for i ns t an c e [25] and 30] or [18J).

r

c) Meromorphic functions. Let now X be a complex man i f ol d and let e- be t he she a! 0 f germs of holomorphic f unctions on X. For every open set U c X it i s defined by t he s p ace II ( U) and the natural restriction maps. The sp ac e /I (U) i s a r ing. Let D(U) be the aubae t of I-I(U) of di vis or s of zero, i. e . D(U) is the set of those holomorphic functi ons on U v anishing on some c onn ected c omponent of U. Let cr(U) be the quotient r ing of //(U) wi t h respect to D (u) i.e. ~(U) i s the set of quotients ~ with f e- II(U), g e- #(U) - [.J(U) with obvious i dent ific ations: ! g

fl

= gl

iff

fg '

=

f Ig

-27 -

A . Andreotti

It

Vc U U r V: 1/ (U)

1s an inclusion of open sets, the restriction map

~ ..r(V)

sends #(U) - O(U)

into #(V) -

0 (V)

and

thus induces a homomorphism of rings U

t$ (U) -i" e;t (V) ,

r V:

We obtain in this way a presheat.

The corresponding sheaf;?/( is

called the sheaf of germs of meromorphic functions on ':it(X) :: [(X,?11

on

is called the ring. of meromorphic functions on

~ (X)

Note that

X.

X. the ring

?t(X)

E:.

but' q?(X)

may be actually

amaller than %(X). Example: thus

Take

the Riemann sphere.

X:: F (D:),

l while

4(X) :: a:

/(X)

is isomorphic to the field of all

rationali'functions in one variable If

X

./t (X)

is connected then

Then #.(X) :: I:

t, '/t'(X) ::: lC(t).

6? (X)

and

In the sequel we will always assume that

X

are fields. is a connected

manifold.

2.2.

a)

Mermorphic functions and holomorphic line bundles. Holomorphic line bundles.

by a holomorphic line bundle on where

F

is a complex manifold,

Let X

X be a complex manifold;

we mean a triple

TT: F -." X

(F, 71', X)

a holomorphic

surjective map such that i) ii)

ff

is of maximal rank

field q)

x e X ,7-1(X)

for every

is isomorphic to the complex

in such a was th at

II:

the map F

given by

"x

F

(u, v)

-;>

F

-;>

v+v

is ho1omorphic

the map

13)

D:xF-;.F

given by

(A, v)-;:>Av

is ho1omorphic.

Given two ho1omorphic line bundles

X a mOrphism f : F L)

-7

E

(or bundle map)

such that

ff:: W 0

f

(F, IT, X),

(E,,,,-', X)

is a ho10morphic map

over

-28-

A. Andre.atii.

1i) is

for every

x IS X

the induced map

f

a:-linear.

A holomorphic line bundle (F, it is isomorphic to the bundle

ff,

X)

(X

x:

n--l(x) ~u:-l(x)'

is said to be trivial if prX ' X).

~ a:~

Every holomorphic line bundle is locally trivial (as it follows from the implicit function theorem). Therefore there exist an open covering 'U = U 0 f X and biholomorphic maps i

such that

j

fr0'f

-1 i

(x, y) = x,

we have two trivializations of ~i

.. -1 ( x,

0 ~ j

On F

U

i

() U j

and thus

v ) = ( x,

gij(X)

v)

with

(1)

is holomorphic and never zero

On Ui n Uj n Uk we must have gij gjk = gik (consistency con ditions). The collection of f unctions ~ gij ~ are called t he transition fu nctions of t he bundl e F (rel at ive to th e l oc al trivi aliz ations 4i). (2)

Conversely, gi ven on an open covering 2( = 1Ui $ a system of transition f uncti ons (1) s atisfying the consi s t ency c ondi t i ons (2) one can construct a holomorphic line bundl e wi t h l oc al trivializations on the s ets U having t he given s yat.e ms as a i sys t em of tr ansition fUllctions. Two holomorphic line bundl es gi ve n on t he s mae covering U = U1 by. tr ansit i on f unctio ns ~ gi jS , f f i j are isomorphic i f th ere exist holomorphi c maps Ai U ~ lC* s uch th at i

5

on

-29-

A. Andreotti

Given a holomorphic line bundle t he s p ace of holomorphic sections

l

( f, zr , X)

we c an consider

= i dX S• In terms of local trivializations of F on the covering a holomorphic section is given by a collection

r (Xi

F)

=l s

: X ~ F

ho LomorphLc , tro s

5

~

= Ui

of holomorphic functions such th at (cf. b)

Given two holomorphic sections

51

30 ).

= ~ sl1~

and

So = f 80i1

of the bundle F, if 8 is not i dentically ze r o on any open 0 s et we can construct a meromorphic function on

given locally by

X,

In this way we e an obtain all meromorphic f unctions of /t'( x)

indeed we have Proposition (2.2.1). Every mero morphic func tion m ~ complex manifold X is the quotient of two holomorphic sections of an appropri ate holomorphic line bundle on X.

f!22!. For every point V such th at m\ V

=~

with

x

~

p, q

6

X we can find a nei ghbor hood

!I(v) ,

q

Ei

)/(V) - D(V).

Since. the ring (7 x is a unique f actorization domain (c r, [28]), if we take V s Ufficien tly small we may ass ume th ~t the germs and llx of p are c oprime and germs Py and qy co prime (cf. [29],

Px

llx

and q at x are coprime. But if Px and if V is sUfficiently s mall, then also the of p and q at any point y € V are

[48]).

-30-

A. Andreotti Let

fV(xi)] i€I

Then on

V(X

be a covering of

=~j

qi

i.e.

By the Euclid lemma

Piqj = Pjqi

Pj

must divide

Pi

and

must divide

Pj

i.e.

Pi'

Pj

Pi = gij

z ero.

with

gij

a unit in

gij

qi

Moreover on

V(X

i)

on

i

H(V )

(l

j

= (l

V(x

gij

vex j)

qj

n V(Xk )

= This shows th ,t the coll ection functions of a

1 Pi!

coll ection

r qi ~

CI.

1

gij5

line bunJle

holo mor ~h ic

gives

~.

we must have

holo mor phic s e c t i on giv es

Do

is a s e t of transition

F

over

"i

of

m Let

51

F

the collectand the of

with

F

So

as r equired.

So

(F, 7r, X)

on a covering

X,

holomorphic s e c tion

not i de ntically zero on ill1y open set.

c)

H(V )

n V(X j) is holomorphic and never i) It follows then th at we also have

this means th at

So

with such neighborhoods.

i)

Pi

ion

X

be a holo morphic line b un dl,e on

U = ~ u~ i"I

of

X

X

by tr an s it ion functi ons

One c an c onsider t "e Itl-th tensor p owo r' of which i s gi v e n by t h e t ran s it ion fun e tions

Fit,

(~, 'ill' X)

t gij) 1 '2

•

gi ve n

f

gil>.

-31 -

.A. .Al1dreott i

We can then consi der t h e grad ed ring

rl)

=

/!.(X, F) l~ reX, of the holomorphic sections of the different te nsor po wers of F O (F trivi al bundle). Note th at i f s € t E (X , ~)

=

then

st

then

J(X.

£

field of

flex,

reX,

rex, rl),

r~).

F)

If

X

is an integral domain and one can consider the

quotieits

=f..!1 So

F)

r

is connec ted, as we always assume,

rl)

sl' so£ [(X,

- f or some

1,

so';

01.

We have in particul ar Theorem

~ (X, F) c

<9

= tJ (X,

(X)

(2.2.2).

~(X),

trivial bundle).

For every holomorphic liBe bundle

F

~(X, F) is an algebraically closed sub field of::Jf (X) ~.

h c ;{(X)

Let

satisfies an equation h where

ki

E.

t? (X,

be al gebraic over v

Q(X, F).

v- l

h

k_ +~

Let

ki

Z

dO

~

0i O.

si t i

~th

h

rl)

rex,

li F );

we·.obtain &n ell..u~t1on

=

,)

[(X, for a suitable 1 (1 i~1 1 i) and where After multiplication by d~-1 the above equqtion can £

At each point

x

e,

X

dOh

This shows that

over B x; since holomorphic at x. 60

IE

t) x

reX, rl)

=O.

satisfies an equation with holomorphic

coefficients and with the coefficient of the

Hence

F) i .e.

si' ti E

be written as follows: '"' ~-1 J-1 (doh) + dl(dOh) + •• • + dO dy

to one .

(X ccnnec t .ed) ,

+ • • • + k v -"' O

multiplying the above equation by i~ t i ~ v-l dOh + 0 h + • • • + 0v;: 0 1 where

the field

dOh

highest power equal

is meromorphic at

is integrally closed and also

doh'"

c$(X, F).

r

E

x

and integral

dOh .. r must be

I'(x,

rl),

thus

A.Andreotti

=

thus

a h

...Q... a

O

6

q (X,

F).

2.3 Pseudoconcave manifolds. A connected complex manifold X is called pseudoconcave if we can find a non-empty open subs et Y X with the following properties i) Y is relatively compact in X, Y e c, X. ii) aY = I-Y is smooth and the Levi form of a Y restricted to the analytic tangent plane has at least one negative eigenvalue at each point of d y(l) • In particular for any point Z o € 8Y there is an analytic disc of dimension ~ 1 which is tangent at Zo to d Y and is con:;t ained in Y except for the point Zoe Examples. 1) Every compact connected manifold is pseudoconcave (take y = X then ~Y = ~ thus condition ii) is void). 2) Let Z be a compact connected manifold of dimE Z 2 2. Let fal, ••• , ~}

be a finite subset of

Z.

Then X = Z - r~,

... ,am~

is pseudoconcave (take for Y the complement of a set of disjoint coordinate balls centered at the points ~). 3) Not every pseudoconcave manifold is compactibiable (i.e. isomorphic to an open subset of a compact manifold). For instance if we take P2(a:) - '{O}~ a:2 3 and i f zIt Zz are the holomorphic coordinates on a:2 , we can consider the

to

exterior form

and define a function (1)

it is

f

to be holomorphic if it satisfi es the

As usual the defining function for ay is chosen so that < 0 on Y and > 0 outside of Y.

-33-

A Andreotti

differential equation df A l' e,

=

0

In this way we define a complex structure on

~2

lO~

(which agrees with the natural one if e 0). One can show that this complex structure can be extended to JP2 ( £ ) - [0 S, that if £.# 0 is small i t provides P2(1I:) -fO~ with a pe eudoconc ave structure and that it is not compactifiable (cf. section 4.4 and (101).'

=

-

Remark: Every holomorphic function on a pseudoconcave manifold is constant. be holomorphic and non-constant on X and let such that If(z ) I = sup I fl. By the maximum modulus o y principle, zo~ () Y. If D is a I-dimensional disc tangent to 3Y at Zo and except Zo contained in Y then 11'1 D has a maximum on an interior point of D. Thus l' is constant on D and thus there is an interior point zl of Y such that !f(zl)l = I fez ) I sup 11'1. This is a contradiction. o ! In fact let. l'

Zo .. 't

=

In particular a pseudoconcave manifold X (not reduced to a single point) cannot be isomorphic to any locally closed submanifold of numeric al space lI:N (oth erwise th ere will be a polynomial on lI:N inducing on X a non-constant holomorphic fun ction) • More generally one can prove the fol lowing Theorem (2.3.1). For any holomorphic line bundle pseudoconc ave mani fold X we have dim~

(2.3.2).

f(X, F)

Let F pseudoconcave manifold X. ~

I<:

~

~

&.

be a holomorphic line bundl e over a There exists a finite number of points

!.!:. X and an i nte ger

h = h(F) s uch that if ['(X, F) vanishes at each point ai of order L h lli!!.

~I""

s

<

F

s= O.

-34-

A . Andreotti

f!22!.

Let Y be as 1n the definition of pseudoconcave man1folds. For every point x Q , we can choose a coordinate polycilinder Px' coordinates P1' 1 .. 1, ••• , n, with center x and of radius r x such that

1)

1s tr1vial

FI~

I

l p1( Y) - Pi ( x ) / = r x , l~i.s.nSc:.Y where U is the coordinate patch on which P1 are coordin ates. Th1s 1s poss1ble in view of the pseudoconcavity of Y. Let Px' be the concentr1c polyci11n~er to P with -1 x radius rxe • We can select a f1n1te number of po1nts ~, ••• , ~ such that 111) u p~ i» , 1 Let F be given by trans1t1on functions : 11)

.s(P~)=~YE:U

f1 j

15 ~ () l5aj

~

11:-

a nd set

II Ff I = sup

I f 1 j I .. e tJ •

sup

15 n P

1,j

~

aj

Z

-1 we must have ~ O. i j .. f j 1 Now choose an 1nteger h with h > fJ, for inst ance where [}J] denotes the integral part of ')J •

Note th at since

f

h

=[flJ +

1

Let s ~ reX, F), vanishing a t the points ~ 0 f order Z h. The section s is given b y holomorphic functi ons si : P~ ~a:. We- set M

There eXi sts

a

point

= sup 1

z0

f.

lSi

0

sO; a (Zo)

1

)

,

for s ome

I

ai ' 0

0

= M

s uch that

-35-

A . Andreotti

S(Pa )

(indeed Since

zo

€

jo ~ i o

Y

Pa ).

is the Silov boundary of

i

there eXists a

P'

aj

and we will have

i

containin g

zo.

Certainly

0

Therefore

By

Schwarz's lemma

where

Pi

the functions H

Since ;t -h < 0

being the coordinates on

s: II F II

we must have

H

-h H •

=0

rex,

F) .... "7"

a k

•

" a

s

up to order

h - 1

s e

definit ion.

X

r (X,

~

~,

is an in-

The righthand space is a n+h ~ (dimension ~ k( h)}.

be pseudoconcave and

Then f(X, F)

the Taylor expan-

F)

at each poi nt

finite-dimensional vector space over Let

i

h

jective map by the previous lemma.

Remark.

s ; O.

m.~

which associates to each section sion of

which implies

(2.3.1).

Proof or theorem '!he natural map

=f

H e -h

Hence

r(y, F)

Y

~

X

as in the

is injective.

Using

- 30-

A . Andr eotti

Hartogs' theorem and t he p s eud oconc avi ty of Y we c an construct .... an open neighborhood Y of Y such th at th e re striction map

y

y

~

r Y :r(y, r) ~ fey, F) i s an i s omorphi s m. Now r Y i s a c omp act map for t he Frechet topology of J (Y, F) and fey . F). ~us the Frec het sp ace f (Y. F) is l oc al ly co mp act and t herefore f inite-dimensional. This wo uld give a mo r e direct p r oo f of theorem (2. 3.1). , However t he p revious proof has the mer i t to gi ve an estimate for the dimension 0 f ( X. F) which will be

.r

useful i n the sequel.

2.4.

Analytic and algebraic d epend enc e of mer omorp hi c functions. Let X be a connected complex manifold. Let f 1 .... , f k
!!!lit wherever t hi s i s d efi ned . In oth2r words f l ••••• f are analyt i cal ly d ependent i f at any k point whe r e e ac h one o f t hese f unc t i ons i s holomorphic the Jacobian .J

aUp .•.• f k ) 1'\ ( ) o zl.···. zn

with resp ect t o a s ys tem zl ' · · •• zn

f loc al ho l omor phi c coo r di nat es. h 8.S r ank -< k ,

The mer omo rphi c f unctio ns fl ' • • •• f k are s aid to be alge br aicall y depend en t i f th ere ex i sts a non- id entically zero polynomi al p(x l •• ••• x k ) i n k variabl es and with co mpl ex co effic i ents s uc h th at wherever i t i s d e fi ned . Algebr aic dependence i mnli es anal yti c dep end en c e .

I n f act if

k > n = dima: . X th ere is nothing t o prove. As s ume Wi t hout 10 5s 0 f gen er alit y we may al s o as s ume th at

are al ge brai cally i nde pendent. Let p ( x l • ••• • xk ) polynomial j1 0 of mi ni mal degr e e in x s uch t hat k

p ( f l' .... f k ) = O.

k.s...n . fl' ••• ,fk _ l be a

- 3 7-

A. Andreotti

Differenti atin g this

But UX

a

k

i ~e ntity

we get

thus we get a non-trivial line ar r elation

(f) ~ 0,

between the differentials d f i in an open dense subset of X. ibis implies th At dfl /I ••• A dfk ; 0 wherever defined 'on X. The converse, of this statement (except for k = 1) is not true in general. For instance the functions

=

Zs

=

fs(x) e , s l,2,3, •• ~ in ~(=) are all al gebraically :lmdependent while any two of them are analytically dep endent . The converse is however true for pseudoconc ave manifolds; we have in fact the following Theorem (2.4.1). Let S be a pseudoconc ave mani f ol d . !! f1' ••• ' f k , f E ?(X) are.. analytically dependent then they are also algebraically dependent (i.e. on pseudoconc ave mani f ol ds : ~ al y t i c depe. dence algebraic dependence).

=

Proof. ' It is not restrictive to assume th at f l, ••• , f are k analytically independent. Otherwise r epl ace f ••• , f by a l, k maximal subset of f , ••• , f , f of an a ~ytically independe nt l k functions and tWte for f one of the remaining functions. ibere exists a holomorphic line bundle F on X and hol omorphi c sections such that

Indeed for each

fi

t here exists a holomorphic line bundle

and holomorphic sections t(i) 1

= :tIT t 0

t(i) with 1

t(i) ~ 0

0

Fi

such xhat

- 38 -

lJ. . .An d r eotti

F = Fl

Taking

th en

••• F

k

k

= i1J"l

So

(1) t(i) si = to • • • 1

Moreover

t~k)€

t(i) Eo r ( X, F) 0

r (X,

and

So J! O.

F)

and we have

Y

We can choose a covering of polycilinders P ~ p' ai

such that

i) ii) 11i)

1v)

Fl p

a

1

~

by coprdinate concentric i ~ N, as in the lem ma (2.3.2)

~

i s trivial i

S(15

ai

u p' ~

)

Co

.=:>

Y

Y

at e ach point

a

1

t he f unctions

f l, •• • , f k

are

f l - fl(ai) = e, 1 , •• • , f k - fk(~)= ~ k can be t aken among a s et of local holomorphic coordin ates ; (1 )

holomorphic and

(i)

This can be done by s mall tr anslations i n t he co ordinate patches of the polyc11inders P a .:;J P'a as condi tions i), ii), iii) i 1 are not effected by these transl ations and as t he s e t of points where condition iv) c annot be satisfi ed has an ope n dense complement : Also t he r e exists a holomorphic l i ne bundle G on X and holomorphic sections a 0' 1 e r (X • G) wi th 0- 0 _ 0 such that f

we may ass ume t h at v)

vi)

GJ Pa f

is trivi al i

is holomorphic at each p oi nt

ai •

-39-

A. Andreotti

As in the lemma

1/~/1

= ek

IIG 1\ = e

we define

(2.3.2)

=F

~

(where

Consider a generic polynomial in in each one of the variables in

x k +l

variabl es of d e gr ee

xk

xl"'"

and of degree

r

s

s; q i So. r

1 $. i

for

••• ,

d

So k

1 k xl ••• x k k' ex k+l

and

0

sk'

0'

l)€

f(X,

~r

over

S

• G

)

k~

x k+l

s, d k+l s, e ,

be the correspondin g homogeneous polynomial. These polynomials form a vector sp ace W(r, s) dimension (r+l)k(s+l). Now note th at 1T(sO' sp.'"

d


c1

~l'

0

k+l

'

=£ c... where

(k times»

• •• F

c,)

Let

t

of

•

The theorem · will be proved if we s how th at Ker e # fOs, where E. is the above natural line ar map W(r,s) ~ [(X, ~r.Gs). Let

h

be the smallest integer > kr}J + s w.

a ssociates to

(1(sO, •••• sk' 6

to order h-l of t he function a gives a line 3 I' map : i

0,

0-

1

)

The map which

th e Taylor exp ansion up

P(f l, •••• f k , f)

at each point

N

lm

where

/1(.~

e-

8-

is t he maxi mal i de al of t he loc al r ing

a:zt;ii) •...• S ~i) 5 S (i). By Le.nma

of convergent po wer s eri es in t he vari ables t his map is i n j ec t i ve .

-40-

A.Andreotti

Now the target apace has a dimension

6=N

r

k r Jl + Bwl + l+It) k

= N «(krH + :c..lJ

+ l+k ) ( [krp + awl + 1 + It - 1)

k-I

N«(krp + aw) + 2)1t Nkkpkrk + lower order terms in

So If we select

a

such that

s+l) Nkltl then, if

r

t

is sufficiently large, we get

dima; W( r, s) :> dima: Im l!. 2.5.

r.

and there fore

Ker

£,

.;.

o.

Algebraic fields of meromorphic functions.

a) By an algebraic field of transcendence degree d we mean a finite algebraic extension of the field D:(t ••• , t d) l, of all rational functions in d variables. Since the ground field D: is of characteristic zero, any extension of this kind is primitive so is of the form ~(tl"'" t d ' Q) with Q algebraic over D:(t ••• , t d). Let P(t l, ... , t d, t) = 0 be l, the minimal equation for t = Q over D:(tl, ••• , t d). Chasing denominators and dividing off any factor in the variables t ••• , t d only, we may assume th at P ia a polynomial in all l, the variables and th at it is irreducible. If V is the d al g ebraic, v :u-iety defined in a: +l where t ••• , t d, t are l, are coordin ates by the equ ation P(t l, ••• , t d, t) = 0 t then V is an irreducible variety and the field a:(tl, ... ,t d, is isomorphic to the field of r ational functions on V. Moreover

d

= di ma: V.

Q)

-41-

.A• Andreotti

We want to prove the

followin~

Theorem (2.5.1). On a pseudoconcave manifold X of complex dimension n, the field /f(X) of all meromorphic functions is an algebraic field of transcendence degree d s.. n , That the transcendence degree of ?z: (X) cannot exceed dimlf tollowsalready from the fact th at on pseudoconcave manifolds algebraic and analytic dependence are the same (theorem (2.4.1». The remaining part of the theorem is a consequence of Proposition (2.5.2). ~ X be a pseudoconcave manifold and III t1' •• " t k E Jt'(X) be algebraically independent. ~ exists an integer v = v(t1' •••• f k) such th at any f E ~(X) which is algebraically dependent on t l, ••• , f , satisfies a k non-trivial equation over 11:( f l, ••• , f k) 0 f degree s V • ~. We tollow the proof of theorem (2.4.1). First we tind a holomorphic line bun dle F and holomorphic sections So _.0, sl' •• " sk of F such that

f

si

i =So

Secondly we find coordinate polycilinders P ai such that

i)

F

restricted to a neighborhood of

U

pI

~

is ~

pI

~

• 1

~

i

~

is trivial

ii) 11i)

~

z»

Y

~

(i)

k

are holomorphic and can be taken among a set of local holomorphic coordinates.

Nt

-42-.

A. Andrec tti

Thirdly, since the conditions i), ii), iii), iv) remain within its coordin ate p atch, valid by small translations of P a i we may de~ermine for each ~ a small clos ed neighborhood v(~) so th at no matter how we translate the center a.... of P a on i

a point of V(a the above four conditions remain valid. i) Let Q be the union of the translates of P a just considered i i and let us compute II F II with respect to the covering

lQ~ l~i~N

.'

Finally, from the proof of theorem (2.2.1) we realize that there exists a holomorphic line bundle G and two holomorphic sections ""0" 0,0"1 of the following condition v)

Gl~

G such that

f

="-1d t1

and satisfying

is trivial.

We set and we choose Then v depends only on f 1 , •• " f k de fine, with respect to the covering

II Gil = e and we choose the centers also

f

is holomorphic,

of

P

but not on

l ~~ ~

in

f.

We also

t

V(~)

so that at

1 ~ i ~ No'

We can now proceed as in the proof of theorem (2.4.1) and we realize that if r is sufficiently large then f s atisfi es a non-trivial equation over 11:( f 1, ••• , f k) of degree ~ v. Proof of theorem (2.5.1). Let fIt ••• , f k be a maximal set of algebraically independent meromorphic functions. Let f e ~(X) be so chosen th at its degree over lI:(f1, ••• , f is k) maximal. This is possible by virtue of proposition (2.5.2). We claim that

A. Andreotti Clearly find

Q

E

1I:(f1"'"

f

k, such th at

5t (X)

fk ,

lI:(fl"'"

f)

r,

C

?( (X).

2 [lI:(fl''''' f k,

=

: 1I:(f1''''' f k)

Q)

f k, Q) : 1I:(f1''''' f k, f». [II:( ri , .•• , f k, f) :

11:( fl"'"

f k )1 •

1I:(f1"'" But the second factor of this product equals ct; first factor equals

1.

This means that

h

a:( f

£

and thus our contention is or ov e d . Theorem (2.5.3).

!N(a:)

l' : X -

point of

X.

IN(a:)

Let

X

Im ')

~

Y

Let

l,

• •• , f

k,

f)

n = d i maf at some

is contained in an i rreducible alge-

of the s ame dimensions than

Y

therefore the

be a pseudoconcave manifold and let

be a holomornhic map of rank

braic variety ~.

we can

h) = lI:(f l,· .. , f k , Q).

Then c(

h ~ '1«X) , .I

Let

X.

be the smallest algebraic subvariety of

containing

(X).

Certainly

Y

exists and is i rreduc-

ible, it is defined by the homogeneous prime ideal

~Y = ?p where

a:[zo"'"

Let

!?. (y) f ~

II: [Zo"'"

~1

r

zN]

pOl =

°S

denotes the gr ad e d ring of homo geneous

!N~~?'

polynomials on element

Eo

be the field

I(Y)

0

f r ational fu nctions on

= ~q

f

pq' - p'q (,

This s hows th at

mero morphic function on

with

q t

eous polynomials

rf\.

Y.

Any

is represented as a quoti ent of t wo homo gen-

X.

~Y' f OJ

If

f

= ~q = £: q

is a well d e f i n e d

we h ave t herefore d e f i ned

necessarily inj ective, ho momorphism

t* : R(y) ~ .::{(X).

then

a,

-44-

A...An.d.r.e.otti

Now

di mlCY = tr ansende nce d e gre ~ of

de gr ee 0 f ~( X) s dimi L as s ump t i on ab out t he r-ank

0

R (y) _

t .r-anecend enc e

But di mlC Y i.. dima: I(X ) = 'iim~ f the map T".

In part i cu lar every co nnected complex submanifol d of a pr o j ec tive al gebraic variety (Chow - t heorem).

1.

b) Let

Excercises. A be a pu r e- dimens i onal

non-sin g ul "~

by t he

IN(~)

is

al ge brai c s ub-

v ariety of lPn(lC). Let ~ = di mlCA . Prove t h J.t A h as a f undament al system of neighborhoods V(A) in lPn(~) with a smooth bound ary at which the Levi-form r estricted t o t he analy t i c tangent pl ane has at least a n e g ~t ive eigenvalues. Let B be a connected co mplex submanifold of V( A) with di ma:B + a 1 n+l. Prove th at B is contain ed in an irred uci bl e al ge br ai c subvariety of

lPn(lC)

of the s ame dimensi on then

B.

(e

r.

(1 51).

Prove th at any pse udoconc ave co mplex Lie group is a complex torus ([ 6]).

2.

3.

Let K be the canonical bun dl e o f t he ps eu doc on c ave mani fol d X ; d e fi ne the "c anonic al dimen s i on of X" as t he tr anscendence de gr ee of lj (X :, K ) . Prove t h at 0 s, c an di m X So d im~ . by ex amples t h at any value i n t h at r an ge i s pe r mit t ed .

Prove

The proofs given in t his ch apt er are i ns pi r ed by an i dea of Serre ([46]); t he method o f exposi t ion f ol lo ws ver y clo se l y an impr oved versi on given by Si eg el ([ 501) f or th e c as e o f a compact mani f ol d . For th e ps eud oc oncave c as e t hey were gi ven 1'; 'rst i n rl1 and f31.

-45-

A Andreotti

Properly discontinuous pseudoconcave groups:

Chapter III.

the Siegel modular group. 3.1. Preliminaries. a) The notions developed in the previous chapter can be slightly generalized with respect both of the notion of manifold and the notion of pseudoconcavity. Let us consider first the following situation;

r

connected manifold and

r

We will say th at compact set

Aut(X)

c

X

is a complex

a group of automorphisms of

is properlY discontinuous on

K c X

X

if for every

the set

fr~ flyKnK"~J is a finite set.

K

In particular taking for

have that the isotropy group of

t, is a finite group.

X

=fYcC!yx

o

~

y e tJ(x ) o

I y

€

U(x

a point

f xos we

o = xoS

0

rx

For any point X £ X there exa st s a o comp act neighborhood U( x ) such that o

II

o

-invariant relatively

for s ome

o)

'(f.

r ,

then rt e

rx

In fact let us choose a coordin a te p atch around

v(n,

for

E.

>

X o an d let and sufficiently small, d e n ot e the coordinate

0

X.

b~l with center x and radius S • o Set Set:) = lY E , V(e..) n v et) '" ~ • Choose e. s mall enough s uc h th at V(E.)

o

rl

o

for c. " [. o ' S(£.)::/

a

1e

fx • o

Then

e,

If for any

r , 'f

,J.

~

rx

is f i n it e if

S(E)

wi th

-y

o

0 <.

E:.

Y xo

fore th ere exi sts for

0 <..

U(x o) =

s:

~

c 1.

V(ll)·

= x

0

n

O ~L C ~ 0

x

n

0 <' (, 1 <

Certainly o

th ere exi sts

S ee,) • - ;> X

and thus 'Y t:: 1" x'

[ 1 > 0,

Go.

S( ~) ~fx

e. < Go '

Therefore th ere ex t a t ,s a s e quen c e c ontinui ty

is r elatively co mpact

~ <

[.0 '

o

with

-> x By n o• a c ontradiction. Th ere-

YX

s uc h that

I t is t hen enough to t ake

S( L) =.lx

o

•

-46-

A. Andreotti It follows then th at the equivalence relation ·

R

=2(x,

Y)

X

G

»;

I x =J y

X

Y ~ I' :5

for some

Z = x/r is·a X/r denote the n atural projection,

is closed and therefore t he quotient space Hausdorff space. then ' l'U(x o) p ( r U(x o »

p : X

Let

~

is a neighborhood or the orbit

= U(xo)/tx

o

is ho 1 omorphic i f f

of

X

and

o

0

If we stipulate th at a function Z

rx

• ~

a:

for U P - l ( U)

is ho 1 omorphic on

P

0

f: U

open in we can

extend to this new t ype of spaces the notions considered 'i n the previous chapter.

This type of space is sometimes called a

generalized complex manifold (or

V-manifold).

';f (Z) of mero morphic Z and of (locally txrvial) holomorphic line

In particular we can talk about the field

fun ctions on

Z.

bundles on Z

Every st ~tement about the generaiized mani f ol d

can be restated in terms of Not e th at i f

I'

By a

ma~i fo l d .

[' -automorphic holomorphic line bundle over

(F,", X)

we mean a holo:norp hic l ine bun dle liftin g of the ac t Lon of of

F,

r .

has no fixed p Oi n t s ( l ) X/I h as actually the

struct ure of a complex b)

an d the Broup

X

i.e.

for every

given such th at

L)

r

11)

if

0

rr

Y

For inst ance if

r

Y

€.

= rr Pi =T 2Yl t hen

over

X

with a

to a gr oup of Bund l e maps

f

a bundle map P

P,.

=P, 2

0

F

X

over

we

X

0

y :

F ~ F

is

Pl'

is th e trivial bundle

1

X)( lI:,

we will h ave

P, (x, v) = (-t X , Pr(x)v), where Py ( x)

(1)

is a never v anishing holomorp hic fun c:tion.

By this we mean t h at if "ye.1

that T X

1

o

= Xo

t hen

a nd

Y is t he i de ntity.

X

oE X

are such

Thus

- 47-

A. Andreotti 5 p 1 can be identified wi t h a collection of holomorphic ~ 'Y functions p

y : X

~

a:*

satisfying the consistency relation Pr2"Y"1(X)

= Py/'flX) PYl(X).

A system of functions of this sort is called a system of factors n we of automorphY. For example if X is an open subset of a: obtain a system of factors of automorphy considering the jacobian determinants

U& = det (ClTX)

PT

for every 'T

Eo

r

•

Given two l' -automorphic holomorphic line bundles (F,11, X,fp~)

and · (G,CAl, X,lo'"yS), a bundle map of the first

bundle .i n t o the second will be a bu ndle map

=

~:

F

-~

~0 PT ~ yO ~ for every y c ['. Fbr instance, given a r -automorphic line bundle by a factors of automorphy ~ PY (xH,

G

such that

s~stem

of

t his will be isomorphic to the

trivial bundle (which corresponds to factors of automorphy identically equal to

1)

if and only if there exists a never vanish-

ing holomorphic function

PI

(x)

=~ (x)

cjl : X ~

a:*

such that

p (yx)-l(trivial ~actor of automorphy)~l)

Given a (locally trivial) ho'l ouo rphf,c line bundle (f', rt ',Z) on

Z

its reciprocal image on

bundle

(F, ft, X, p

ZPrJ)

X

is a f-automorphic line

with the property th at

(x ) = 1 for every "'( eo f x r o o

One can verify th at this.property characterizes the reci procal images on

(1)

X of locally trivial holomorphic line bundles on

Thus the classes of facto~s of automorphY~P"'Y(X)~

Z.

modulo

~ PT (x>.? "'" l6r (x>! 1f£~ p r -ldy(X) is a trivial factor of auto morphy, correspond to classes of isomor-

the eqUivalence relation

phiBms of ['-automorphic line bundles F

is the trivial bundle.

~F,"'; X,~

pyH

in which

- 48 -

A. An d r e otti

r -automorphic line bundle ( F, TT, X, tP rJ.5 r -invariant sections fo F: r F) = [e (. rex, F) I s(,x) = Py( .x ) sex) for every

c)

Given a

we

can consider the sp ace of

r (X,

Gi ven two [' -invari ant sec tions So _ 0, sl sl is a [' -invariant meromorphic function on

So

element of

p

0r

F,

X

* )t(Z).

r -automorphic

[-invariant sections of it:

and

The proof is straightforward and is thus repeat the

consid~rations

p

*it(Z)

holomorphic line bundle and two sO; 0,

sl

such that

omitted.

sl m = -.

So

We can also

of section 2.2 c) in this more general

case. 3.2. a)

psuedoconcave properly discontinuous groups. For practical reasons it is convenient to generalize the

notion of p s e ud oc on c avi t y as follows. n Let J1 be an open set in a: , and z point. if we zo:

z

= Zo

(t l, t

h o od

L) ii)

We will say th at fin d a complex

~ an

2)

+ ~t l + a 2t 2; variable in a: 2

V of

i n E,

Zo

V (151-2, {z

L(q)z

Let now

r

<:

o

Eo

V

Zo

to

o

d fl. = .'1

-

51. a boundary

is a pseudoconcave boundary point

2-dimensional line ar sp ace

~,a2 ~

a: n ,

c"

and a

E

t hrough

lin early independent,

function


on a ne ighbor-

real-valued an d such th at

I 4>(z)<

¢(zo)

5

o.

be a prop erly di s c on t i n uo us gro up of a ut omo rp hi s ms

of a connected complex nanifold

X.

We will say th at C'

is a

pseudoconcave group of automorphis ms if we can fin d a non-empty open subset Jl L) ii)

C

X

such th at

fL is rel atively co mp ac t in for ~very po int

Zo 6 .::> l~

ei th er an interior p oi nt ary po i n t o f

v~ •

X

the orbit 0

f

rz

o

x.

i.e. an

Conversely for every [-invariant meromorphic function m we can find a

x ~

th en

contains

...;'1.. or a p s e udoc o nc av e bound-

-49-

.A. Andreotti

=

Clearly for r identity we obtain a generali zation of th e notion of pseudoconcave manifold given i n the pr evi ous chapter. It is not a di fficult ex er c i s e to c arr y over to this more general case all the theory developed t here. In p art i cul ar we obtain t he theorem (3): be a connected complex cave group of automorphisms , of X. [-invariant meromorphic functions on of transcendence degr ee d ~ dim~. Note th at 7( (Xl ::: p * ~( z )

wh ere

Z

= X/f.

Suppose for instance that there exists a I' -automorphic line bundle F on X such th at the quotient fie ld ~ ( X , F) of t he graded ring j{(X, F/

= k~O

['(X,

~/

has transcendence de gree equal to th at 0 f 1< (X) r (for instance equal the dimension of X) t hen it follows, since (X, F/' is algebraically Closed in j( ( X ) L' th at we must have

C!

5i (X, Fl = ~( X)r. b) The notion of pseudoconcave prop erly discontinuous group of automorphisms is stable by "commensurability"; two subgroups I' I' t 2 of Aut (X) are called commensurable if ['I (\ i'2 is of finite index in

rI

and

r 2'

Proposition (3.2.1). If r 1 and [2 iU'e commensurable and if one of them is properlY discontinuous and pse udoconcave so i. the other. Let r 1 be properly discontinuous, then G = as a subgroup of f 1 is properly discontinuous. Set r 2 = G~ U ••• o G~ and let K be compact in X. I f the set lye r 2 I y K () K I- rJ is infinite, th ere exi sts an index ~.

A. And reotti

i with l~. i s k and infinitely many g'a in G such that gaiK n K #~. But then for infinitely many g's in G g(K U aiK) n (K lJ aiK) # ¢, which is a contradiction sinCE K U aiK

is c ompact.

l'

Su"pose th at G = ['1

th at

condi tions

f

Set

Then

3.3

r2

is pseudoconcave. is pseudoconc ave.

an d

ii)

It is enough to show Let

n c.

specified above in

X

satisfy the

a)

r = r l'

for

1 = ~ G u ••• o ~ G.

n, = ail n.

r=

for

(1

i)

1

u ••• u

ail

n.

veri fies th e same condi t 'ions

G.

Siegel modular group.

We will ap ply th e abov e c on sider ations to the particul ar case of the Si egel up per half pl ane and Si egel modular group Let

n

and l e t ??'( (n , 0:) = Hom(o:n,a: n) matrices wi th coolllplex entries. The

be an int eger,

be the s e t of

(r3]).

nAn

n > 0,

ge nerali zed upper h alf pl an e i s the set Hn = ~Z

rt c«,

G

t

0:)

Z = Z,

Im Z >

°.

This s et c an " be i de ntifi ed with an open s ubset of

o:tn(n+l).

H is t he usual P oincar~ u p per h alf plane . l Consider t he simplect ic group S (n , lR) Le. t h e s et of linear

Note t h at

n

a ut.o mcz-oh t.sms of form

i~l

lR

><

dx i " dyi'

n

p

lR

which le ave invari ant t he exteri or n n (x, y) E:. lR x lR •

in matrix no tat i on Sp(n, lR) =

°

J = (-I We c an l et

I 0)'

~

g

00

1f!(2n, lR )

I

t gJ g =

JS

where

'»r { (zn , lR) = Hom(lR 2n ,lR 2n

i.

s p ( n , m) op e r-a t e on t he ge n e rali zed up p e r hal f

plan e as follo Vis

-si ,

A _ Andreotti

Proposition (3.3.1). automorphism 0 f Hn •

This oQeration i s a well J e fi ned

E!:2£.!. matrix. follows:

denot e the n x 11 identity Hn can be reformul at ed as

Let Z € Hn and let The conditions defining (t z , I)

= 0,

J(Z) I

I

i(t

z, I)

J ( Iz)

o.

Set (U) V

Then, since

g

=

(A

=

( Z)

B)

C D

I

(AZ + B)

cz

+ D

is symplectic, we must h ave

i ( to, til)

(~) >

J t

vU) >

0 ,

°.

This shows th at V must be non-singular, otherwise f or a vector w # 0 vw = thus it;(t uV tVW)w = 0 which is impossible. Then we can consider the matrix

°

(Zl) I

i

-1

(U V I

=

)

From .the first condition we derive t 0 i.e. ( Zl' I) J(Z1 l)

=

t

z1

= Zl

.

From the second we derive i(t z1 t I)

J(Z11)

>

O

i.e.

ti( Zl - Zl)

»

0

.

This shows th lt Zl = (AZ + B)(CZ + D)-l is well de f i ned and represents a point of Hn• Since the transformation is invertible it gives an aubemorphf.em of Hn• By the map Z ~ (Z - i1)(Z + i1)-l the generali zed upper half plane is mapped into the generalized unit disc ZZ < I which is a bounded domain in II:t n ( n+1 ) and Sp(n, lR) appears as a group of automorphisms of a bounded- dom ain. Consider the di screte subgroup Sp(n, ~) of Sp(n, lR) of tho se matrices with integer entrices. By the above rem ar~ it follows th at Sp(n, lZ) acts in a properly discontinuous way on H n

-52-

A. Andreotti The transform ations act as identity on group

Sp( n , lZ)

g

=+

I

are the only tr an sformations which

H and a r e cont ained in Sp(n, ~(l). The n (or more precisely t he group Sp (N, ~) / ~ .:!:IS)

is called the modular group of rank

n,

n = 1

For

we ge t

the usual modul ar group in one v ari able.

3.4 Pseudoconcavity Theorem

(3.4.1).

of the modular group.

If

n

L 2 the modular group of rank

n

~

pseudoconcave properly discontinuous group of automorphisms

~

Hn •

E!:221.

(q;)

Every psoi ti ve de fini te sy:nmetric matrix

Yean be

written in a unique way in the follow ing J acobi normal form: Y

= t wo w

with

D

=

w=

Wi

j\

·1

l et {/ ~

For a positive r eal number fJ, ~

[1..o J be the open subset of

Z = X + iY (,.D..fJ if

defined by

i)

where

ii)

for

i

c j

:.11i ) Note that if c(p)

>

X + iY

° such th at

to

D.IJ- '

th en there ex i sts a constant

Y) c(}1)I

(c(~) = min ( ~,

L». n

t-'

is a "fundamental If I-l ~ Po is sifficiently large then .1l ~ open set" for the modular group l' = Sp(n, lZ), Le. (1)

Z

If (AZ + B)(CZ + D)-l = Z

= ~ 11

AZ

= ZA.

A

= fJI

Z c: H then taking n, 0, A = D and moreover

=C = and therefore tJ =.:!: 1.

we get th at necessarily Thus

for e very B

-53-

A . Andreotti

rfltJ- =

i)

Hn

11) ~Y
{\ Cl..}J-

,. f1

is a finite set.

For the proof of t his st atement we refer to

(481

and

L49].

Notice th at from this statement the proper discontinuity of

r

could be ded uc e d by a more direct argument than the general one usee be fore. ()

We fix the parameter

once and for all.

will sas- th at a transformation

We

is a transformation at

infinity if

n

F=y -1 (Q,...)0

I-'

contains a divergent sequence of points in

Hn

(i.e. a sequence

Hn ) . In view of the definition il ~ this is equivalent to sas- that

with no accumulation point in of

sup

Z ~ F

=d.

d (Z)

n

Note th at the s ame is true f or

v

F = t (F) = IIp (\ Y( OtJ) •

We set v(Z) = y-l

A di r ec t c alcul ation s hows t h dt e ach component -1

vi = d n

of th e form wij's.

u

where
i

Z = X + 1Y

Therefore i f

vi

of

v(Z)

i s a polynomial in the

describes

F,

II v(Z) II

0",

and mo r e ov e r

takes

a rbitr ari ly s mall val ues. Set

II v(Z) 1/-1

r(Z) =

UJ ·

y w(Z)

so th at

where

~)f~

r(Z) =

v(Z) w(Z) = n v( Z) /I

Il w( Z) II

= 1.

= (; ~ is a t r an s f o r ma t i on at in fi. nity, then in t h e matrix C t h E- l ast ro w and c ol umn are zero.

Le mma.

If

Ind eed if

I

Z

~

F,

t here exi s t s

v

Z

is

G

0 r'

s uc h t h at

Z

='( -l~Z

-54-

A. Andreotti

y

=

\I

Z(CZ + D)

(1) Multiplying (1) p arts we obtain

on the right by

cr ~ }

=

Lr(~)

(XCX +

AZ + R • w(Z)

XD -

and equating the real

AX - B)

w(Z).

F. the right hand side represents a vector of bounded norm. Since Z" Qt.L there exists a constant v c > 0, independent of Y and x, such that

When u(Z)

Z describes

(YX,

= tyx

where (y, x) Therefore

denotes the euclidian scalar product;

Ollorr ll/2 s, (Y 0r] lr(z)J i.e.'

11

0

q)11

x) 2 e IIxl/ 2

J' C[r J) c

tr(z)

1I C ( ;

u(Z)/1

( r(Z) which is bounded as Z describes F. the last column of C is zero. v As Z describes F, Z describes

This is possible only if Set now

v_l

v

v(Z) = Y

Setting

and proceed as before.

[1] and

v

V'

Y w(Z),

=

v

sup

v

v

Z e. F

r(Z)

]]/

r?Z)

r(Z)

~II

s,

l/u(z)lI

v

r(Z) =

=o ;

we get

-:5'5-

A. An dr eotti Multi flying

(1)

t

on the left by

r(~»

(0, ••• ,0,

v

w(Z)

d

= tw(~)

C Y

we get C X +" *D - AX - B).

Arguing as before we obtain th at the l ast row of

C

mus t be zero.

As a consequence of this lemma we get that, for a trans form a-

Y = (~ ~)

tion

at infinity, det (CZ + D)

the last row and column of

(1)

Hn

Consider on

Z + X + iY.

the function "k ( Z)

A direct calculation shows

B = (CA D»

Sp(n,2Z)

= det

for every

th ~t

det(CZ + D)-2

Moreover one can pr-ove the following property for every

Zo e H , n

att ained at some point If we set

f (Z).

Y where

we h ave

= k(Z)

k(yZ)

is independent of

Z.

sup

Z c

Zl"

IZ o

k( Z)

c, d

(c f. ( 491) :

and the maximum is

a p o l'Zo'

k("!'Z) ,

sUP.

1.,]'

we obtain a r-invariant function on

Hn ..

Notice that the sets Bc .~Z ..

for any real

.QjJ-1

c,

+(Z)< c

are rel atively compact in

0,..

is contained in the set ~ Z ~

I

H In fact Bc n• J det Y < c] which is contained

in a set of the form ~ Z e at' d n c do. .' Choose c)' large enough to ensure th at for ,-

°

T with

~

(T) () T

=2Z .. a,..

~ ~

be done since the set

-y l' ••• , Ys for

Z

E

T

l

~

2y€f/YIl,..,,,.n

be the set

0

=val, sup ••• ,

rt ~~

This can

is finite.

f these trans formations s o that,

we have ~ (Z) 1

r

(Z) > c every '"'( Eo is a transformation at infinity. J

s

k('(" Z).' Y

Let

- 5 6-

A . Andr eotti € I" such t~ at y (z) €. Q tJ ei t h er IC ( r( Z» ~ c and thus --( (Z) € T so th at -y 1 i for so me i or else key (Z» s c and therefore k('!( Z) c; ¢ (Z).

I~ fact for y

=

we now take for.a. the set Bc +l ' We W.-U1t to show th ,lt for any point Zo € aQ. t here exists in r Zo either an int erior point of.a. or a ps eudoco nc ave bound ary point of fl.. Let p : Hn ~ Hn Ir be the natural pr oj ec t i on on t he quot i ent

=

;space. Since 4' is'£ -invariant, Q p -1 pea). Therefore a point Z € a Q which is no t equivalent to an interior point o of il must be on th e "surface" l. ~ = 0+1 S. For such a point we must have k(Zo) = k(Yy Zo) for those v's, 1 s.v s, s, for which k(l' y Zo) = HZ o) e-i .. In particular for t hese Y 's we must have

=

(C Zo +D)} =1, "'{\,oJ

/det

=

(~ ~).

Moreover det (C Zo + D) is indepe ndent of the l ast row and column of Zo. Consider th~ linear space L(Zo) of complex dimension n, through Zo' defined by the equations o

Zd/3 • Zd~

for

Since on that space

~IL(Z~) I,)

1

~ c(

S n-l,

/det (C Z + D)/

1

\3

~

~

=1

n-l,

Z s ymmetric .

we mus t have

• k(Z) ;

Therefore in a neighbrohood V of Zo Q 0 L(Zo) can be described as the set

on

ZZ ,

L(Zo) the set V J k(Z) <: c+1S.

The proof of the theorem will be complete if we show th at the function - log k(Z) is strongly ·plurisubfarmonic in Hn (and thus the same is true for its restriction to L(Zo»' TO see this we remark th at by the way det Y transforms und er the action of Sp(n, m) the form K(Z)dv (where

dv

= (det

y)-n+l)dv ,

is the euclide an volume element), is invariant.

-57-

A. Andreotti Since Sp(n, m) acts transitively on H th at form, up to n a positive factor must be the invariant Bergman form (c f. [51]). Therefore that quadratic hermitian form -(n+l)

a.~ log k(Z)

Hn

is the Bergman metric of Remark: Let

m

r =Sp(n,

of

be an integer.

every

m

rm

m,

rm

D

Let

a)

m =~. g £f and ['

r

Aut (D)

= det

f

holomorphic in

series of weight

If

1

r

k

compac t sets.

~

2,

and if

-automorphic line bundle D

we can consider the

L. y,,1

f(.." z) I

det (

is properly discontinuous, if

value and if

we

..

Aut (D)

k ll

13 k( f; z) =

Aut (D)

1

1Z.

6

is a system of factors of automorphy for

and thus it defines a

For any

Therefore for

for "'( e

~m

(~~~»)

is any aubgr-oup of

l py
then the set

r

are commensurable.

be a bounded domain in Py (z)

c,

(mod m)s ..

series.

set If

g ': I

is properly discontinuous and pseudoconcave.

Poincare"

3.5.

rm

One can consider the subgroup

defined by

~

I' For every

and thus it is positive definite.

f

~.

. / "Poincare

*f )

k

is bounded in absolute

then this series converges uniformly on

Indeed the convergence of

series reduced to

th ~t

the convergence of the series

L

Y6i:.

det

on compact Sets. Let

Q

ec,

D

be a polycilinder of radius

any polycilinder of radius det

*f d z

f<.

R

and let

R

cont ai.ncd in

P

Q. Since

is ho Lo.noz-phLc we h ave at the center

x

0

of

P

be

-58 -

A. Andreotti

dv z=X

= vol yep)

o

~

vol!'(Q) •

Therefore if V is the concentric polycilinder of R-r, for any point y " V we have 2

d.t~

(c

= (vol

Q of radius

P) -1 •

vol Q)"

z=y

s be the number of transformations Y Q () Q ~ We have for s If. V

Let

e,

,;,1 :rWj2

.,

y IE r

such that

c c

z s; vor (Q) t.

c s

vol (Q) since in

u

vo11(Q)

vol (D), s times.

any point is covered at most

ye.f

This proves the absolute uniform convergence of the given series in V. It follows then the absolute uniform convergence of that series on any comp act subset of D. I

Now if 9 k ( Z) is a convergent Poincare series of weight it satisfies the functional equ ation for every In f act ~k(rz)

=6.l: L ~

=6,; =

r

f(~1

f(oyz)

PI ( z ) -k

=P I

(~) yz

z)

( z)

-k

s;

ae]

(mfll d z fUt z )

9 k(z).

)

r~· L •

k k

(~ yz)

WzrL ) d z

)

k

k

k,

- 59-

A. Andreotti

This shows that any such Poincar: series repres ents an i nvari ant section of the -automorphic line bundle F- k•

n -2 r One is led . therefore to consider the ring JL(D, F ) and its quotient field Cl (D, F- 2) r • Disposing of the freedom one has in the choice of the function f one can show (e r, (191) that the field (D, F- 2 ) £ contains m analytically independent mero~orphic functions.

a

b) We can now apply the above consideration to Hn and to the modular group as Hn is iSGmorphic to a bounded domain of ~m, m = tn(n+l). We then deduce the following conclusion.

l2£ H n

n 1 the field of r -automorphic meromorphic functions on coincides with the field ~ (Hn, F-2~r. In particular

(q) (d)

the transcendence degree of th at field equals

tn(n+l);

every r -invariant meromorphic function can be written as the quotient of two Poincare series of the same weight.

Remark. We have reached the above conclusion using the fact that ~ admits a bounded model. Using instead the usual unbounded model of Hn one can develop a simular aJ:gument replacing the factor of automorphy P1 by 0r(Z)

= det

(CZ + D)

Y = (~ ~) ~

Sp(n, ~)

One is then led to the theory of Eisenstein series:

l~r

Z

f(r )

aet tCZ+D)k

and reaches a similar conclusion but the proof of convergence of Etsenstein series is more difficult (c r, 491).

r

result proved in this chapter for the modular group was extended by A. Borel ([171) to arithmetic groups acting on irreducible bounded domains of dimension 2 ~ (cf. section 4.1 example 5). ~e

- 60-

A. Andreotti

Also one can show as in H. Cartan ([19]) th at the quotient sp ac e x/r has the at r-uctur-e of an analytic space and is p seudoconcave if r . is in the sense of section 3.2.

-01-

A. Andreotti Chapter IV. 4.1.

Projective imbeddings of bs eudoconc ave manifolds.(l)

Measure of pseudoconc avity.

Let U be an open subset 0 f a: n and ~ : U - lR be a C d function. We will s as th at ~ is strongly q-pseudoc onvex at the point Zo e U if the Levi-form a)

L(~)z

o

=.~

a2 di dZa.3Z~

Zo

uu c:l

e

(where zl' ••• ' zn are local holomor?hic coordinates at zo) has at least n-q pos i t i ve eigenvalues. In particular a strongly o-pseudoconvex function is a strongly plurisubharmonic function. A s we have seen before (l./f) this notion is independent of the choice of local coordinates, and could also be formul ated as follows: there exists a (n-q)-complex-dimensional plane E through Zo such that ~IE is strongly plurisubharmonic. Remark. For-practical reqsons it is sometimes convenient to release the assumption th at 4> is C 1, ••• ,t} 1 such th at ~ = sup (~l' ••• '~l); (ii) there exists an (n-q)-dimensional plane E through Zo

such that

i E'

1

~

i

~

1,

subharmonic in a neighborhood of

is strongly pluriZo in

E.

Let X be a connected complex manifold of complex dimension We will sas th at X is q-pseudoconcave if a Cd function is given such that ~ : X +lR

b) n.

(1)

In this chapter the notions of analytic set, complex space, normal complex sp ace, Stein space will occasionally be used although our main concern are complex manifolds. For the basic definitions we refer t o the foll owing Chapter 5.

- 62-

A . Andeeotti

(i)

for every

c

>

Ln]'

q, the

sets

I

Bc = f x e X ~ (x) > c are relatively compact in X ; (ii)

there exists a compact subset Zo ~ :i - K

every point

S K of X such that at

~ is strongly q-pseudcconvex ,

Remarks.

1.'

For all

boundary

> inf.p, = Bc - Bc

c

aBc

0,

except for a set of measure

is smooth (By Sard's theorem).

the The

Levi-form restricted to the analytic tangent plane at Zo

G'

dBc

to aBc'

eigenvalues.

c

<:

inf , K

Th erefore if

has at least

n-q-l

~

n-q-l

positive

these manifolds are a

1

special c ase of the pseudoconcave manifolds studied in chapter 1 .

X,

For this reason when we speak of a q-pseudoconcave manifold n

is not comp act, we will assume that , Q.

X

if ~

2.

~

q So n-2 , hence

Indeed only the eigenvalues of the Levi-form in the

direction of the level surfaces of

~

do have a geometric

meaning; taking a rapidly increasing convex function of

the


remainin g eigenvalue can be forced to be positive on .any given compact set. 2.

The notion of

q-pseudoconc ave manifold could be generalized

in the sense of the remark made in

a), above.

EXamples.

1.

Every comp act connected manifold

any 2.

LetZ

dimension t hen J

=

X

z\i} , ing at

K = X,

Take

q,

1

is

q-pseudoconcave for

= 1.

be a connected c ompact co mplex manifold of complex n

z

j

So n ,

i -t

s, Xi

~

X

2.

Let ~ xl' ••• , x k ~ be a finite subset 0 f Xl to • • , x k S is O-pseudoconc ave. Indeed if ar e loc al complex coordin ates a t

we can select

e. >

0

xi'

Z,

vanish-

such th at the c oordin ate balls

A. Andreotti

B i

!!

f l

¥ \I z(i) I2 j

<: Eo

j:l

~ )

t

are relatively compactkin their coordinate p atch and are disjoint. Take K Z - U Bi and for ~ a C tP function such i=l

=

3. Now generally let Z be compact connected mani f ol d of complex dimension n 2 2. Let Y be a complex submanifold of

=

Z with dim~ Y q ~ n-2. Then X Z - Y is q-pseudoconcave. Indeed, at every point y € Y we can choose a coordinate

=

neighborhood

U(y)

U(1)

Let ~ U =

U ••• ' znU such that zl'

with coordinates

j=~+ll Z~ {2

n

.

Y

=l Z

€

Z~+l

U

= ... = z~ = o~.

Y wi t h a finite number

Cover

Ul'· •• , Ul of these neighborhoods and let . Uo = Z - Y and U 4 o = 1- Let f p i~ b¢ a c cP partition of unit y subord in ate to the covering ~ UiS O~i~l'

Set ~ =

t1

Pi

l Ui 'f

•

t

Then 4 > 0 on X and for E../ 0 J> ;, E ~ is rel at ively co mpact in X. Moreover ~t e ach poi nt of Y L(q) has n-iq positive eigenvalues. Therefore if E.. > 0 is s Uf fi ci ent l y small on X -? 4> ~ c. 5 L(~) h as n-sq positive ei genv alues. 4 . Let H be the gener ali zed upper h ~uf pl ane with n L 2, n and let r 'o e the modular group. From <3. 4,0:)) it follows th ~t th ere is an integer 1 > 0 such th at f or ev ery point Zo

€

H n

the isotropy gr oup I' z

o

h as an order

S. 1-

- 94 -

A . Andreotti

pbe a prime number with n > 1. Then the gr oup = g «I' r g!! I (mod p) cannot h ave fixed point s ( l ) • Therep fore X = Hnll'p is a complex manifold; From the proof of theorem 3.4.1 it foll ows th at X is q-pseudoconcave with

Let

r

r

= tn(n-l) = dim1

Hn_1 • Here the concavity is to be understood in the gener al sense of remark 2 above. q

5. More generally let D be an irreducible symmetric bounded domain in lC n with n ~ 2. Let be an arithm etic . group 0 f automorphisms of D without fixed points. A result of A. Borel shows th at X = D/r is q-pseudoconc ave for some q, o ~ q ~ n-2 17] ).

.c

«(

4.2

The problem of projective imbe dding of pseudoconc ave manifolds.

a) We have already remarked th at if X is a pseudoconc ave mani f ol d and T : X -+ P N(lC) a one-to-one ho1o morphic map of X onto i (X), then j (X) is contained in an irreduci ble algebraic variety Y of the same dimens ion than X (theorem (2.5.3». I f y.1; Y is the normali zation of Y (c r , ch ap t er 5), then the map ) : X ~ Y f actors through ~. Mor eover Y is again a projective algebraic variety. Thus X is i s omorphi c to an open subset of the projective algebraic variety Y.

.

.

(1)

Indeed if A 6'£ has a fixed 1 p 1 • e must have A = I. Now A = I + 1 1 ( 1) pB + (2 1) p2 B2 + +

1

point, for some thus 1 P 1B 1 =

pB 1

1

1

~l

o.

Dividing this relation by p we see that e ach entry b i j of B must be divisible by p. Thus A I + p2B1• Arguing in the 2 same .~ we s ee th at each entry of B1 i s devisible by p , thus A = I + P4B2• And so on. It follows th at we must h ave B = 0 and thus A = I.

=

- 65-

A. Andreotti

The problem of projective imbedding of speudoconcave manifolds can be loosely formulated as follows: to find some useful criterion to ensure that a pseudoconcave manifold X is isomorphic to an open set of some projective algebraic variety. We will then say that X admits a projective imbedding.

b)

If X is compact such criterion is provided by the theorem of Kodaira [33J

k!l X be a compact complex manifold, the .quivalent conditionsl X admits a projective imbedding. X carries a K8hler metric whose aasociated exterior form has integral periods.' There eXists over X a holomorphic line bundle F!!... X whose total space F i8 O-pseudoconcave. There eXist over X a holomorphic line bundle F ~ tM~t the graded ring A(X, F) ; l~ .f (X, separates points and gives local coordinates everywhere on X. (4.2.1).

Theorem

!QllowlDs ·~e

A. B.

C.

D.

Fl)

=

Let n d1m~; to say that ~(X, F) separates points and gives local coordinates everywhere on X means the following: (a-:) given x # y X there exist an integer 1, 1 > 0, and sO' sl e r(x,~) such that SO(y) ]

det

#0

~(y)

re ) 8

0

for every

, ••• , s

n

G

x

L(X, Fl)

X there exists an integer such that

(f (_l)i si d So /\ •••

A

A

•••

(I

1 > 0

dsn ) x

and

# o.

's l So In ot~er words the meromorphic function takes (or So sl different "values" at x and y while the meromorphic functions sl sri So , ••• , So (if so(x) # 0, otherwise renumber th e sections s) provide local coord inates at

x.

- 6-£-

A. Andreotti

We will prove here only the implications D~ A. im~lications will be proved in chapter VI.

E!:22.!.

D ~ A. For any two points x # y 1 = l(x" y) and two sections q, det Replacing ~ and assume th at c{ (y) Therefore also

[

« ( X) .

q(y)

i5 (x)

(3(y)

X we can find an int eger (3 Eo

J

l)

r:'(X, F

s uch t hat

# 0

by convenient linear combin ations, we may = 0 and thus d (x) ~(y) ~ o.

(3

=(3 (x)

det

Set

The other

~ 0

•

l) Al = ~ (x,y)-" X x X I Y (
By the above remark for any int eger k> 0 Akl c AI . Now for each 1 Al is an analytic s ubset of X x X cont aining the diagonal I:>. of X)( X. If Al ~ '" then for s ome

#

k> 0 : Akl Al• Since a decreasing sequence of analytic subsets of a compact manifold must be st ation ary th ere exists an 1 o > 0 such th at Al = A. Therefore o

1

the sections of s: (X, F 0) do sep ar ate any couple of points x # y i n X. Simil arly by a simpler argument one proves that 1 0 can be cho sen such th at also 1 (11) the sections 0 f r (X, F 0) h ave no common zer os ,

(i)

(iii)

they give local coordinates everywh ere on

X.

- 67-

A. Andreotti 1

{sO"," sk} be a basis of I (X, F 0) f : X ~ F (II:) given by k

Let map

x

~

and consider t he

(sO(x), ••• , sk(x»,

Thi s map is holomorphic by (ii), is one-to-one (by(i» and biholomorphic (by( iii) ) • There f orer is an isomorphism 0 f X onto ~(X) which is a projective algebraic manifold by theorem (2.5.3). It is enough to prove th at implication for X = Fn(m). Consider on Fn(m) the line bundle of the hyperplane sections. this is given by

A ~ D.

as we can write

Z

~). zi

It is then

immedi ate to verify th at this is a holomorphic line bundle A

section on

Fn

is given by

(zO'""

zn) ~ (zO'""

where

F.

Zn+l) separates

points and gives local coordin ates everywhere. Remark. If X is not compact the proof of the implic ation D ~ A bre aks down, al though one h as reason able conditions to ensure th at condition D is fulfilled. For instance: n if X = H/r is a quotient of a bounded demain H c. m by a properly di s co n t i nuous group (without fixed poi nt s ) then one can show directly, by means of Poinc are series, that condition D holds;

- 6 8-

A. Andreotti

X· is a complex manifold and on

if

F

holomorphic line bundle such that F 8 K- l "positive and complete",

there exists a

X

is "positive"

and

K denoting the canonical

X.

bundle of

[13J."

We refer for these st atements to

We thus formulate the problem of projective imbedding in . the followin g mann er: ~

o

be g-pseudoconcave manifold with

X

~ q S"

n-2,

n = dima: X. Assume th at there exists on X a holomorphic line

.fl..

bundle

separat es points an d gi v es local

F such th at (X, F) coordinates everywhere on X.

Does

X

admi t a projective

imbedding?

4.3 Solution of the probl e m for (c r , [101 , (121). ( 4.3.1).

'lheorem ~"JP o s e

Let

X

O- pseudoconc ave man ifold.

~

X

th "lt t here exists on

th a t J2 (X ~ F) everywhere on X.

O-pseudoconc ave .man i f o l ds

a holomorphic line

b ~md l e

F

poi n t s an d ..... gives l ocal co o r din ates ;;.s_ep~ar=-a:::.t=-e;;.;s;;....o-=:=:.;;......;.;;,;;.;;.;.. .;..;;.;;;...;;;.:..;;.=;....;:;..;;.;:;.===-;;-.

Proof

(following an i dea of

( u:)

=2x Bc

o

admits a proj ective imbe dding.

We c hoose .; X

I~

Co

lP,

~

( x) .> Co ~

H. Gr auert). inf ~

X

l)

c

0

d

i nf

K

such t h at

h as a smooth bo undary .

is comp act, we can choose an

the s e c t i ons of I (X, F

<

1

Bec ause

su fficient ly l arge so t h a t

h ave no c ommon zeros, sep ar ate p o i n t s

and gi v e loc ;J1 coordin at es e v e r ywhe r e on

Bc

o

•

This is don e by

the same argume nt d e v e l op e d i n the p r evi o us s e cti on . «(3)

Let

N+l

b asis of I (X, F

= dimlC [ l).

1

(X , F )

an d l et

We consid er t he map ;-

:

'be a

A . Andreotti

defined by

xl-? (sO(x), •••• sN(X»'

and one-to-one. (1)

Z

is

X

Bc o

Zo e A ' o

~

Since

I-'

A

f

Let

n

= dimt

: D

~

15

<.

such th at 1

t = O.

Ao •

1, D

X.

Consider th e subset of 6

ih

~ i o' • • • • inS

i t is a dis c r e t e s e t by

non-constant and

This is a contradiction.

dS c,

This is an analytic subset of

i

Ao

X where

~

fl (D ) c

A=~x~XI (~(_l)h for all

A

O-dimensional ;

is a subharmonic function on

having a max i mum at ()

must be

I

= zo' 0

the same

has a maximum a t some point o is of positive dimension, we can find a

I tl f

is contained in

Then d A

1.'

one-to-one holomorphic map

Then

)

A has an i rreducible component

Assume if possible th at of dimension

of

Co

o-pseudoconc ave any analytic subset

X-

contained in

X

(B

X.

Bec ause

(T)

~

By theorem (2.5.3)

an irreducible algebraic variety dimension than

of

This map is holomorphic

(y),

1\

i

•••

1\

0

~0I

••• ,

dS

N '§

ih

X : A.·•• ,..

dS

i

n

)x

=0

•

B

Thus co thus at mos t count able. Obviously

X

c ont ained in

X -

extends to a holomorphic map l'

which i -s of r-anx

n

a l l p o Ln ts over whi ch o f r an k

X-A"';> Z

at; e ac h po i n t .

')

We a gr e e t o d ele t e from

may b e ex ten:ied by a ho Lomo r-p hLc map

n,

(1)

We can als o as s ume

t •••

t

A

-70-

A. Andreotti We will call this set

(e)

A

again.

A is a finite set then one com-

If we can show that

pletes the proof as follows.

1

multiple of n

It is knowg that ~ ~:

[52], X

1

by a convenient

so that the corresponding new map is of rank

A.

at the points of

(cr.

We can replace

Z*

Let

Z

Ie.-.>

Z

~

be the normalization of

is also a projective algebraic variety Since

(531) ~

X is normal the new map Z* of Z by a

Z factors through the normalization

holomorphic map

N*

Since T

is of rank

n

everywhere, and since ",*

normal is locally irreducible, I therefore an isomorphism.(l) (~)

Let

"

T

the closure

0

being

must be one-to-one and

A is a

We are thus reduced to show that ~e

*

Z

f the graph

T

,.,

of 1"

in

~inite set.

X)I.

lIN( an.'

The set T must be an analytic set as it is t he irreducible component containing T of the analytic set

l

(x , t)

X ><.IN(D:) I

6

si(X) t j - Sj(x) t i

=0

for

o x i, j ~ N •

Let

"'* ~

X-A ~ Z *

normalization graph of

t"'* l'

•

be the factorization of 1'" t hrou gh the "* be t he closure of the Z*w _ Z of Z. Let T

This is also an analytic set as it is c ontained

in the analytic s e t ~.

n than T. projections.

(I

. -1 .. w) (T)

J'

~*

Let ex: : T

-7

Now let

G A

a

X,

which has th e s ame dimension A*

l?> : T

and set

be a coo r dLn ate b all centered at a other point of A.

U

(1)

is of r an k

n

)( (X) =

-:;>

*

Z

"'* ., ( a)

be the n atural =

t3d -1 (

a).

Let

and not cont aining any

everywh ere and gen e ral l y one-to-one as

a:

Z.

-71-

A. Andreotti

er·

Now

:z,

must be one-to-one as

be one-to-one as it is one-to-one on d.

(3/

-1 ·

mu",t also be one-to-one. Z·

op~ ;bl!)thus a neighborhood in

.A

:.fI..,

-;>

n,

We · cla1ll that each component

-1

(U-a)

must

(U)

and since

=

Therefore 11 (3o.-leU) is of ~( a) . Since Jl, is

N.

of 7' (a)

B

-1

~

= d. \3-1 : a - r· ( a) ~ U

normal, the holomorphic map A holollorphic map

pI

is normal; so

extends to a

which is not of co-

cimension 1 must be an irreducible component of the singular set ot Z·. In tact if B contains a non-singular point b e Z· and if

is of codimension 2. 2,

B

then A

must have a non-

vanishing Jacobian at b. Thus A. -1 is defined near a )..-l(a) = b i.e. ~. must be a local isomorphism at a. Hence

a

tP

A.·

Now we remark that

(tj)

compact.

and

Thus for each

.-

rv·.

is a proper map because

C(

a '" A T (a)

many compact analytic subsets of

Z

•

is

is a union of finitely

=Y

Z - l (Bc )

of codimension

one and finitely many irreducible componentg of the singular set of

Z· (which must also be contained in Y.)

tV.

previous ariument it follows that if

..,1* .

a,. #.

= ..

a,. ,

a

Moreover, from the are in

2

a then "( (a,.) (l T (a ) ~ Z Z We will prove that A is a finite set if we show that

and

A

Y con-

tains at most tinitely many irreducible compact analytic subsets of codimension

1

Z•

(here we use that the singular set of

has a finite number of irreducible components). Let e:

>

0

B -.

co

Ic _

o

Then!

co-e.. given by r,

Co- c: > iff ~ -e = ~ 4 > c o- l:S.

be so small that

biholomorphically to

.

e•

is compact in

•

Z

Y

-

N*

-r

and that

extends

7

Set

(Bc _€,). 0

and we can, by the isomorphism on

Y-

extend this function by putting it equal to

c o-

transpose the function

~

YCo-e.• €,

on

Yc

Let us o- e.

and

-72-

A . Andreotti

call

t

-r: Y

~

the cont inuous f unction t hus obt ained. The f unction m, just d e fin ed, h as t he fo llowing properties

i) '( is continuous; Yc ={,/,< c ] cc:. Y

11)

iii)

on

Y-

yc. o

E-

°is

c

c

= sup 'f; Y cO' and stron gly plurisbh armonic.

for any

z,

0

Then every compact irred ucible anal utic s ubse t of Y of dimension ~ I must be contained in ~c _ E. by the same o ,'argumen t given in (y). Moreover Y is holomorphically convex as it follows from i), ii), iii) and the solution of Levi problem (cf. ch apt er VI). It then follo ws by the reduction th eory of holomorphically convex spaces (cf. [20]) th at th ere is a compact analytic subset C ~ Y of dimension I at each one of its poi n t s and such th at any irreducible compact analytic subset of Y of dimension ~l must be cont ained in C (see section 6.3 remarks ? and 4). Since C has finitely many irreducible components of codimens ion 1 the theorem is pr ov ed .

=

2 this result can be considered Remark: only if di mE X sEj.tisfactory as the only type of pseudoconc avity available is O-pseudoconcavity. 4.4.

The c ase of di mE X L 3.

If dimE X L. 3

in the pr evious theorem the assumption th at th e ring J1(X, F) sep ar at es poi nt s c an be dropped; one has in fact the following Theorem (4.4.1). Let X be a O-pseudoconcave mani f old with dim~ X ~ 3. Suppose t here exists on X a holomorphic line bundle F s uc h th at t he r ing Jf(X, F) gives l ocal coordinates everywhere on X. Then J( (X, F) do e s also separate points of X so th at X admi t s a proj ecti ve imbedding.

-7 3-

A. Andreotti

The proof of this t heorem i s r "ther c omplic :,t ed.cnu will be omitted. However t he interest of t he t heorem r es i de s i n th e f act th ctt it do es not hol l f or dima: X = 2. We will indic ate ho w to con struct <;. ccun t er- ex amd, e ( c f. [10]) based on the n ext Lemma.

Let

V be a c onnected comp act manifold. Let W ~c.. U ~einrrr open subsets of V. Let ~ be a c onn ect 2d complex manifold and let 1/:

X7V-W

be an holomorphic map malting X int o a ) -sheet ed non-r ami ned covering of VII dima: V2.2 and if V i s simple connected either A 1, or else X c a.ino t be comp actif'J ed be isomorphic to an open s e t of a c omp act (i. e • X .;:,c.;;;an=n;.;;o..:t;.-;;;.;;....:.;;;..;:.;:=:..;;,;;.:.;:;.....:.;;.-=~==~..:;.:~:.:.....:::....::.::.:::.I;:.;;::;::..:. complex s Dace).

w.

=

The re ason for the validi ty of this l emm a can be said briefly 'V as follows. If X woul d co mp actify int o X t hen on e c an s how th at the holomorphic map 11' extends to a holomorphic map ~

N

rr:X~V.

Since V is simply connected the map tr must be r amifi ed, if ..l > l, somewhere i n V. But the r ami fi c at ion s et mus t be analytic and contained in U i .e. it is a compact an aly t i c subs et of a Stein manifold. Therefore the r ~ ific at i on s et cons ists of a fini te s et of points E. Since V is s imply co nnect ed, non-singul ar, of dime nsion 2.. 2 , also V-E is s imply connected. Therefore _~l(E) is an irreducible non-ramifi ed covering of V-E. But t hi s is po s sibl e

X

only if J-

= 1.

Remark. One could also assume th at X is r cvaifi ed ov er V along an analytic set F Co V-U such th at V-F remains s i mply connected.

(1)

A Stein manifold is a hol omorphi c convex manifold on which

holomorphic functions separ at e po ints.

-74-

A. Andreotti

=

Let T ~21r be an 8lgebr cu.c torus of complex dimension two .L e f'Ln ed as the quoti ent of ~Z by the group of translations gener a t ed by the vector columns

Construction of the example.

r

of the matrix

r:

0

~2J

zn

1

z21

where

group

T

~

T

points.

Let Z

K =

with

TIT;

isolated t non-degenerate, conical

o

of fourth order in

It is known th at

~£.'

simply connected.

4'

E.

give

-U

3

and let lP3(~)


a.

is isomorphic to an points.

16,

~

close to

.0

16

Select

16

centered at the

=q,=1 \? 'U.
U

with

F3(~)

Consider a surface and han-singular.

as a non-singular surface of

lP

3(lC)

,

is

is a stein open subset of et>

0

- U

admits a non-ramifi ed 2-sheeted covering, the

16

sm8l1 neighborhoods of the

h ave been r emoved. X

1$

is di ffeomorphic to

E.

K

Moreover

UE. = 4>e..(lU

Therefore of,.

Im Z ) O.

this is a complex analytic

singulc~

lP '

U<1 in

~

singular points of

T

zZ2

conical, non-d e gene r-at e double

16

of fourth order in

~o

disjoint small balls

torus

z2l

z = Z,

A theorem of Kummer shows that

algebraic surface

11)

z12

the map which associates· to each point of the 2 This map is involutive, 7 = I, and

space of dimension

i)

t

HZ'

zll

its inverse.

has 16 fixed points.

¢~

=

z22

is a point of Siegel half plane LetT: T

Z

Let

X

16

fixed points

be t ,'is double c over-i.ng ,

We

the complex structure that makes the natural map

11" : X -<> q, e -UE:. ho1omorphic. Since 4>€. -U e is O-pseudoconcave

Moreover if

F

X

is also

O-pseudoconcave.

denotes the ho10morphic line bundle of the hyper-

p1ro1e section of ~

e

,

J~(4

7

e-Uc I F)

gives local coordinates,

-75-

A. Andreotti everywhere on

"q (X,

tr *F).

~ -u. The s ame is therefore true for t e... But th e preceedin c: Le.nma tell s us th cct X

be c ompa c t.Lf'Led , in p ar ti c ul ar

X

cannot

cannot c,d mi t a proj ective

i mbc:dd 1 n g . Remarks. 1.

We h av e constructed an exam ple of a

th ,' ,t canno t be compxc td f'Led

t

O- fJseudoconc ave mani f ol d

a c t ual l y if we let

on a small disc

A

U ~ fXtSt '£t1 of

2-dimensional msnd r oLds , all

around C,

c-

vary

D.

little

we h ave constructed a f Uinily O-pseudoconc ave

and no one of which can be compactified. 2.

Makin g use o f the r emark made a f t e r the lemma one can show

that example

3

of

c omp l ex s t r uc tur e on

(2. 3)

gi ve s al so a non-sc ornpac t.Lfd abl e

lP ( G: ) 2

-1 Or.

-76-

A. Andreotti

Chapter

5.1.

V•.

Meromorphic functions on complex spaces.

Preliminaries

n• Let Jl be an open set in a: A closed subset A c Jl. ::Is called an analytic set if A is locally the zero set of a f1l1.1te number of holomorphic functions i.e. for every Zo ~ A a)

there exi - ts a neighborhood U f , .•• , f 0; "l1(U) such that 1

of

Zo

in .....11.

and

k

On A we define as the sheaf 0A of holomorphic functions, the sheaf of germs of restrictions to A of holomorphic functions in some open set of Il.. A morphism (or holomorphic map) from the analytic set ( A, D- ) to the analytic set (B, B) A is a continuous map 1'-: A ~ B such th at

e

~ • crB,T(x)

c=.

& A, x

for every

x e A.

The notion of isomorphism is then defined and then the notion of complex space is obtained repl acing in the definition of complex manifold open sets of a numerical space and isomorphisms between them by analytic sets and isomorphisms between them. The "structural she af" will be denoted by &- • A point x £ X of a c omplex sp ace X is called a simple point if some neighborhood U of x in X is Lso a.o r phf,c to some open subset of so me a:n. Let SeX) be t he set of non-simple points of X, t his is c all ed the singular set of X. This is an analytic subset 0 f X as one can prove. A complex space X is cal.Led irreducible if X-SeX) is a conn ected m c~i fo l d ; its dimension is called t he dim ension of X. The dimension of E~ arbitr-~y c omplex s pace X is, by definition, the maximal dimension 0 fits components. If Y c:. X are complex sp aces t hen dim Y So dim X. If x E X is a point o f the co mpl ex sp ace X we Jefine dim:x:!. as info dim U( x ) when U( x ) describes a fund wnent al system of neighborhoods o f x in X. Alw ~ys

holds:

dim SeX) < dim X.

- 77 -

A . Andreotti

b) The notion of meromorphic function on a complex space is defined in the same way as for c omplex manifolds. Given a complex space X we can then define two rings of meromorphic functions: ~(X), the ring of meromorphic functions which are quotient of two global holomorphic functions on X, the second of which is not a divisor of zero.

5t(X) , the ring of all meromorphic functions (which are locally the quotient of two holomorphic functions, the second of which' is not a zero divisor). If

X is irreducible, as we will alWayS assume in the sequel,

~ (X)

and

J< (X)

are fields.

e)

c) A complex space (X, is called normal if for every x ~ X the ring cr x is integr caly closed in its quotient ring. A complex space (X.·<9-) is called weakly normal if every continuous meromorphic function f : U -'>' II: defined on an open set U G X is holomorphic on U. Given a complex space (X,~X) there exists a (normal) complex space (X, &' i) and a morphi sm /' p : X -) X which is surjective and such that for any normal complex spa.ce (Y, By) any morphism Y -->X such that the im~ of any irreducible component of Y is not contained in seX) f actors through X A

'"

/1 X

y~l

p

- X The s pace (X, C1f ) is uniquely lefined up to isomorphisms and is cxl. Led the normali z ation of (X, l9). Analogous 's t at ement hoLds r-ep Luc Lng the word " normal" by "weaklynormal". We thus obt af n the notion of wea\t··normalization space of (X, 6 ) . A sp ace X and its we.:"k normalization are homeomor phi c . A no r -na'l sp ace is we''';';:!y normal.

-7~-

A-Andreotti

Examples.

t

= (x, y)~ 0: 2 I x 3 = y2 ~ is neither normal nor weakly-nermal as the function ~ is continuous on X and integral over the local ring of X at the origin as it satisfies

1.

x

The space

the equation

2

(l ) _ x = 0 x

X.

on

Its normalization and we:ik normalization is the complex space 0: with 0: ~ X defined by the map t ~ (t 2, t 3 ) . 2.

The space

as

~

X =

I

(x, y) <: 0: 2 I x 2_ y2+y 3 ,= 0"5 is not normal

is integral over the local ring at the origin but not

(~)

holomorphic; it satisfies the equation Its normalization is t .~ (t( I_t 2 ) , I_t 2). g

with

0:

However

0:

X

p -"7

2

+ y - 1 = 0

in the neighborhoods of

and assume the same value

ct

verifies th .tt any holo",orphic as a series of the form a +

at

t '" 1

~nction 0

I 1 r+s=

X.

defined by

X

is we ak l y normal.

is holomorphic and continuous ne ar the origin on

must be holomorphic

on

a

rs

and

In fact if goP

X

t = 1, t = -1.

t

= -1 One

f this type can lE written t r(1_t2)r+s, but this

is the pull back by p of the holomorphic function near the ortgin in 0: 2 ex: + ' f a x r y s r+s=l rs •

3.

Simpler examples can be obtained by t acing into consiJerat±n

reducible s paces. So for inst~ce 2 lex, y) E 0: \ xy = 0"5 is not normal {(x, Y) € 0: 2

1

(x_y2) x =

Os

b~t weakly normal;

is no t weakly normal, its weak

nor mali z at i on i s isomorphic to the pr eceding s et. As reference one can consult [29],

L1,2 ]

we ~~

normali zation.

5.2

Pseud oconcavity for cO{fi 11ex sp aces.

and

[8 ]

for the

The notion of paeud oconc av e mm i f olJ c an b e extended to c ompl ex s p aces with the fol lo\'l .'.n.:; :J e f i ni t i on .

-79-

A. Andreotti

The complex space sutset Y # f1 of L)

(X, f}) is caL.LeC1 pseudoconc ave if an open X is given such that

Y CG X ;

' i i ) For every zoc,: oy sequence of neighborhoods

------..,

(Ut) n Y)u

there exists a fundamental in X such that

is a neighborhood of

\J

Zoe

Here

(U~Y)U

v

=i Z €

Uv

I

I fez) 1~us~~ I f I

for all

f

E.

)f(U,,'> •

II

This is generalization of the notion given for complex manifolds. in fact if X is such a manifold and Y ec. X has a smooth bound~y and if V zo';; d Y the Levi-form restricted to the analytic tangent p:\,ane has one negative eigenvalue, one can construct, using the disc theorem, a fund amental sequence of coordinate polycilinders P v centered at Zo such that every holomorphic function on P v n Y extends to P. Taking Uv = P v the above condition ii) is then fulfilled. For a pseudoconcave space one h as the equivalent of theorem (2.5.1) (c r , [lJ). theorem (5.2.1). If X is a pseudoconcave sp ace of dimension n , then the field 1«X) of meromorphic functi ons on X !! an algebraic field of transcendence degree d ~ n. We remark th at fr om condition ii) it follows again th ~t every holomorphic function on a pseudoconcave space X is constant. Therefore ~ (X) = e, ~

5.3 The Poincare problem. a) If we drop t he assumption of pseudoconcavity there is no hope to obtain a st 3tement of the n 3ture of theorem (5.2.1); we h ave already remarked th at for X a:, the functions z2 e~, e ,... are algebraically independent and. therefore, the

=

- 80-

A. Andreotti

}(a:)

transcendence degree of as groun l field hopeful.

11(X)

inst€ ~d

is infinite. of

However if we take

the situ ation is more

~

First of all one h ,s the followin g useful fact

Theorem (5.3.1).

If ~

is a normal sp ace, t hen

t~(X)

;h§.

7( (X).

algebraically closed in

The proof is the same as the proof given for man d f'oLd s (theorem (2.2.2».

Moreover the previous

counter-ex~aple

disap-

pears as one has: Theorem (5.3.2). (a) I f X is a Stein so ace(l) then (b)

If X ~

(1)

& (X)

=1l(X).

is an open connected subset of a Stein manifold

Gt (X) = :r(X).

A Stein space

complex sp ace. X

(or holomorphically complete sp ace) is a (with countable topology)

satisfying the

following conditions (d )

separates points

/I(X)

exists an (ii)

f

with

H(X)

<=.

E,

such that

6'(X)

sup X . in which

ally convex. A sheaf is called coherent neighborhood

U

of

(ii)

"1

x'; y, x, y" X,

there

tXi~ c. X

I f(X i ) I

=

there exists an

0".

is satisfied is called

holomorphic~

of 19--modules on a complex space

(cf. (23J) x

if

f(X)'; f(y).

for any divergent sequence f

A space

Le.

if for each

X

x ~ X there is a

a nd an exact sequence of the form

In particular the sheats

e P,

p 21,

are all coherent.

Kernels, cokernels, im ages, coimages of homomorphisms of coherent she aves are coherent. Stein sp ace

X

the space

t

(X,

For a coherent she ,,!

'T)

generates

'J x

'!

over

on a

r:9

x

-81-

A . Andreotti

~.

(a)

Given

h

lJ = f f ~ r3 I h = ~

Locally

zero divisor.

fxh x

with

q

?((~)

€

consider the she qf

ex

lS

P

for

and

/J

S I:

!.J =1f e l = Ker ~ <9-

sh

If

!J x o = r!3 X

X

o

o

=t ~

with

s(x

E (X, c9 ) •

o.

71(

defines a section the open set Let

A

X

0

s _ 0

and

is holomorphic there eXists

(since. X

is

X

Given

-?

J'J( of the

~

is an open set of h

e}{ (X)

and 1enoting

f germs 0 f meromorphic func tions on

h : X

tJ ,

t9 (X).

Let US .tssume first th at t he s he af

h A

But, by the definition of

(a particular Stein manifold). by

J

are coherent, .it follows that

h ~

assumed irreducible), hence (b) 1

6J /q 0

is a point where

=.is

h

S

(mod q)

0

Therefore by theorem

o)';'

Thus

~

X

€.

~

tp

& /qe

and thus

is coherent.

['(X, j )

not a

q

Thus loc ally

19, q e

we h ave

X •

6

holomorphic and

q

E:

Since

x

s t a c .ced a p ac e

X.

n 11:,

h

7?( over

J1f .

be t he connect ed component of h eX) in Because n .".: ?J1 -i' a: is a local homeo morphism,

the n atural pr o j e c t i on X

a c qui r e s a

n atur ~

complex mani f ol d . sp ac e

??(

for every

compl ex structure of an

is Huu sdor ff", X .c

X

n-dimensional

This l!l u.nifold is Hausdorff as th e st ac ked

(t h eorem

By th e very meanin g of A

of

subsp ac e of the numberical s p ace

H. Cart an).

a:

N

Every complex

is a S t e i n sp .c e ,

Stein s p ace (0 f finite d Lmc a s Lon ) a :lmi t s holo mor p hi c me\:) int o a nume r Lcal, a p . .c e

A

X, t h er-e

;l.

a:N •

For a Stein

man i f o l d t h .r t m .p c an o e chosen to be an Lso n o r-ohLsm onto i t s t ma ge ,

Every

p r op e r o n e-to-one

- 82-

A . Andreotti

exists a meromorphic f unction

h I heX) =n-*h;

"h

" such th at X X the germ of

on

at each point of

germ represented by i. A Now X is the l~gest domain s pread over

hx

;0.

is the A

~n

on which h can be continued. Therefore -log & ( ~ ) , 6(x) polycilindrical A distance of x from the boundary of X is a plurisubharmonic function (c r, [31l). " is a stein manifold. We can then apply to "h the Hence X ,,1\ /\ argument developed in (a) and write h =:e. with p, q A ~

;0.

holomorphic on

" X

and

Then

tr*

is an isomorphism between result.

X

q

a" "q

and

heX) heX)

=rr * h

and since

we get the desired

(b)2 Let u& now consi der the general case, in which X is an open subset of a Stein manifold Y. By the imbedding theorem for Stein manifolds (cf. r281, [311) we may a3sume th at Y is a submanifold of s ome numerical sp ace ~N. By a theorem of Coquier-Grauert ( [ 23]) ~ o ne can find a connected neighborhood of Y in a: N and a holomorphic retr: Y, ley) = y for al l y € Y. Consider the mero morphic on ....., -lex). BY th e s peca. a; ~, c ase (b)l we can I *h f unc t a" on ".

U

find two ho Lo mor-c hf,c fu nctions p, q wi t h q ~ 0 on 1" -leX) such th at q T * h = p. Since q ~ 0 there exists a mul t i d (J index d.. € :JNl s uch th~t D q IX ~ 0 while D q\X :: 0 i f t~.1 <.

I d. I •

d

Then

D d

D

(q T * h)

= Dd p. d

al X h = D P IX

Restricting to

wi th

X

we get

d

D q X~ 0

as we wanted. The previous theorem gives the solution, in some particular c ases o f the so-c all ed Poincare problem: when, for a compl ex ~ X, do we h ::tve <9 (X) = J( X ) ? Of course t his probl em is not always solvable, for inst ance for comp act projective irreducible al.g ebr-at,c v arieti es we have iR (X) = ~ but

- 83-

A. Andreotti

}f(X) #

~

if the variety is not a point. if /I(X)

O(X) = 1«X)

One could hope th at

separates points of

X,

but this in

general is still an open question. b)

For a complex sp ace

s ep ar-a t ed!'

tR

rel at Lon

y =

which is not "holomorphically

X

on

f(X) = f(y)

iff Let

X

one is L eud natur-al.Ly to consider the equivalence

X/rte

for a ll

<E

#(X).

be the topological quoti ent space, and let

p : X 7 Y be the natural projection. iff

l'

p-l(U)

X, Y

is open in

define for

U

open in

Cy(U) ={f: U-7~

I

thus

r

A set

U c Y is open

Y is a Hausdorff sp ace.

We

p-l(U)_>~

continuous, P*f

holomorphic •

was to

We give in t his

Y

the structure of a ringed sp ace (a

.13pace with a she af of rings of germs of continuous functions) and the problem a.ri s e s to see when If t his is the c a s e, then sp ace and

(y,(9)

(Y, e)

is a complex sp ace.

is a holo morphically aep ar at ec

X 7 Y is a holomorphic surj ection.

p :

em th en consider 7( (X)

(]i. (X)

which c ontains

Moreover one

as an exte n sion of the field

and may be equal to

~ (X)

if on

p * '7t(Y) Y

the Poincar~ problem is solvable, as one o b~ously has ~(X) = p*~(Y). To decide when

(Y,<9)

has the structure of c, complex sp ace we

h ve the follovdn g sufficient criterion (of topological n ature): Proposition

(5.3.3).

comnlex space (i) (ii)

Y = X/(j( p: l K c. Y

Let

X

be an irreducible weakly-normal

If is loc ally comp clct,

-;>

Y

is semiproper,

i.e. for every comp _Jct set

there exi sts a comp "ct set

Ie' c. X

p(K') = K. Then

(Y,

e,l)

is a wealdy norlnalcomnle x sp .,ce.

s uch that

- 84 -

A. An d reotti Ex ample.

Let

be a we akly-normal, h o'Lomo rph Lc al. Ly c onvex

X

sp ace.

Th e n

p-l (K)

must b e comp -vc t , o th e rw t se we

p

is a prop er map.

sequenc e ~ xiS c p-l(K) ~p

If(X

=d'.

i)/

function on p • f

Y

Ln d e ed i f C ,1Il

an d thus f i..nd an

= X/~

wi t h

where

whose pull b a 'c by

p

is c omp act,

di ve r gent

it

f e !I( X)

sup I fJ = sup p.f p-l(K) K

But

K c: Y

s ele c t

p.f

is

f.

Y

and

is the

Since

is continuous this is impossible.

Moreover such th at

is Loc al. L y comp sc t ,

Y

p(x) = y.

X is

Since

can find a finite s e t

0

Let

f functions

f

l,.

th at the set

has a comp act closure and is contain ed in

K' .

p(K')

Then

(Y.

e)

€

p(U)

'"

f

k

x

€.

X

convex. we

in fI (X)

such

is a nei ghborhood of

y

which i s comp act.

We s atisfy thus the c onditions and therefore

y

holomorphic ~lly

st ~ted

by the pr e vious proposition

is an analytic ' sp ace whi c h is also weakly-

normal. But on

Y holomorphic f unctions separate points and holo morphic

convexity is i nhe r i te d by norm al Stein sp ace.

Proof of proposition (a: )

Y

from

X.

is a weakly

(5. 3.3).

First we remark th at. given

~

c omp act set

find fini tely many ho Lomoro hLc functions Jt(X)

(Y .0 -)

Thus

In t hi s case we t hus h ave

s uch th at: iff

f

l,

•• "

KG

X we can

f

in

k

- 85-

A. Andreotti

,12

Indeed the equivalence rel ation

is r-epr-e e e n t ed in

X" X

by the analytic set

f[ Locally

= f ~1/I(X)

dC

equ ations.

~ (xl' X2 )

\ f(x l) - f(X 2 )

X- X

f.

can be given by finitely many of its defining Since

K

~

K is compact finitely many of these

rJ2 n

defining equations suffice to define

K '" K.

(~)

Sec ondly we show the follow ing fact:

Let

X, Y

g: Y

Let

(i)

if

(11) if

= 0~ •

¢: X ~ Y

be complex spaces and let

~ ~

be continuous.

*g

g

is mer-omoz-phf,c so is


is semiproper and surjective then if

meromorphic so is

be holo morphi¢

Then ¢


is

g.

This fact is proved making use of the following remark. A

continuous function

h: X

.le f i n ed on a complex space

-;>~,

X is meromorphic iff the graph of X

~

h

is an analytic subset of

11:. (c r,

This is a p ),rticuLtr case of a general th eor-em prop.

[111,

3.12) •

In this instance if

X is weakly normal there is

~most

nothing

to prove; in general a sp ace and its weak norm ;>lization h ave the "same" meromorphic functions.

Making use of this remark,

st atement (L) then follows from the fact th at the graph of 4> * g equals

(~.id)-l (graph of g)

holomorphic an d the graph Denote by

rN

and

J'

0

f

g

the graphs

We h ave a commutative di agram

...

while

QXid: X x II: -"'Y x II:

is analytic. 0

f

~

*g

and

g

respectively.

X

r --» I'

1;r 1

fT

X~y

where

X.

rr and rr are the n atural projections and where

= (1/>)( id) 1 '::' •

1

is

- 86-

A. Andreotti Since

ff.

and

tt

are homeomorphisms

4 is semiproper. the composed map

Horeover

r

7\. is semiproper bec ause

is closed in

y"

Therefore

It.

Y " It

is semipJoper and holomor-phic. so th,t r is analytic.

By assumption

d>

* g is meromorphf.c

A theorem of Kuhlmann (cf. L40J, se also [111, a generalization of a theorem of Remmert l45) stutes th at the image of a holomorphic semipro per map between complex spaces is an analytic set. By virtue of th at theorem Em = f mus 't be analytic and therefore g must be meromorphic.

X.

(Y) Let b ~ Y and let compact closure ,.

V be a neighborhood of

Choose K G X compact such th at f 1, ••• , f k '"' II (X) such that

Let gi: Y -;> Ie sider the map

be such that

p(K)

p * gi

= V.

= fi

for

b

with a

Select

1 s; i

s,

k,

Con-

x.:

defined by X(X) = (gl(X)" •• ' gk(X».

Let

L =

(V).

Then,

being continuous, L must be compact as V is compact. Since X is gijective X must be a homeomorphism of V onto L. Therefore Z = X(V) i6 open in L and there exa s cs an oo en set k (open in lek ) such that Z = L /) fl. In particular Z fl c. le is closed in n . Set W p-1(V) and consider the composite map

=

- 87-

A . Andreotti

This is given by x ~(fl(x), ••• , fk(x», therefore J is a holomorphic map which is also semiproper as p is semiproper. By the Kuhlmann theorem ~ (w) = Z is an analytic set. " " be the weak-normalization of Z. We get a Let (Z,8) bijective map .p: V .., " (7). '\ We claim that ljr is an isomorphism of (V, e V) onto (Z, This follows directly by application of statement ($) and the fact that on a weakly normal space continuous meromorphic functions are holomorphic.

z.

5.4. Relative theorems. The previous considerations lead us naturally to the following situation. X -";> Y Let X, Y be irreducible complex spaces and let p a holomorphic surjection. This gives an injective map p* : '~(Y) ~ "Jf (X) and we want to investigate when /(X) considered as an extension of p*7f(y) is an algebraic field. For instance Y could be the separation space of X and in favorable conditions we would have IX' (y) G; (Y) ~ ci(X). In this connection one can prove the following:

=

(5.4.1). I f p: X-:>Y is a proper holomorphic map then is an algebraic field over 'p *'If( Y), 0 f transcendence degree ~ dim t X - dim t Y. Theorem IY(X)

Example. Let X be a weakly-normal holomorphically convex space. Let Y be the separation space of X. Since Y is a Stein sp ace, If (y) = ~(Y) and p* ~(Y) ~~(X). We t hus obt ain the following result as p : X - 7 Y is proper: For a weakly-normal holomorphically convex sp ace X the field It (X) is an algebraic field over q (X) of transc endence degree ~ dim X - ~imt Y, where Y is the senaration s u ace of X. t If in theorem ( 5.4.1) Y is a point we ge t back t heorem (5.2.1) for a compact sp ace. One is le ad to conj ecture an analo gue of theorem (5. 4.1) also for a pseudoconc ave s itu ation. This actually is t he case. For t his purpose we introduce th e following definition.

-8 8-

A. Andreotti

Let X, Y be irreducible complex sp aces and let f .: X -? Y be a holomorohic surjection. We s~ th "t f is a pse udoconcave map if an open, non-empty, subset f2 c, X is given with the following properties (i) (ii)

(iii)

f:n-'lY

is a proper map

for every ye. Y every irreducible component of f-l(y) intersects a for every point Zo c; d Q. =Q- Cl.. there exist two fundamental sequences of neighborhoods ~ U), VtJ Co u.,; such th at

for every

ye. f(U .. )

1

Vv ~ with

we h ave

~ (UI/ n il f (y))U nf-l(y) ::> v

V

v

n

f

-1

(y)

•

Theorem (5.4.2). ~ X, Y be irreducible complex sp aces and let X be normal. Let p: X -+ Y be a pseudoconcave map. ~ J( (X) ~ algebraic field over p. }«( Y) of transcendence degree ~

dim; X - dim; Y.

Remark. For the validity of this theorem actually one needs only to suppose th at (a) p is pseudoconcave ·over some open set A. c Y with A" fJ and such that p-l(A) is normal, (13) for every compact subset K c:: Y we can find a compact set K' c. X such th ::.t for every s « K, every irreducible component of p-l(y) intersects K'. The proof of this theorem is rather complic ated. It can be found in [111 with all the material_considered in this chapter. Remarks. 1. Theorem (5.4.2) For example the family eonstructed in

4.4,

is more general then theorem (5.4.1). V = 1Xt f , where LJ =~. t e a: I \ t I < t '-ll

remark 1, gives an example of a pseudo-

IS

- 89-

A. Andreotti

co ncave map in which no on e of t he fib ers comp actified.

Xt

can be

2.

The assumption of pseudoconc avity i n t heorem ( 5.4 . 2) c annot be relaxed. He r e is a co un t er ex .anpl,e du e to Kod aira and Kas ( c f . [111) Take count abl y many copies A of the unit disc /J. in IC, and k

•

count abl y many copies Consider the sets

M

a: •

of

II: k

= ku.2Z

!.I k "- a: k•

On M-S co nsider t he cyclic gro up f au t omorphis m

where

t

k

~

.d

w

k'

k

* . eICk

gener ated by the

Define the ac t i on o f

r

on

S

to be

t he i de ntity . One veri fi es t h ct X = Hie is a t wo-dimensional manf fol d. The :IL1ni fold X looks like A " a: * in whic h [ 0 ~ x IC * is repl aced by S (c f. ( 111 ) . We h ave a natur al holomo rphi c map p : X -;> .1 an d one h ·.s !I (X ) = p * 1-1 ( 6 ) •

Indeed, let

f " /I(X) ,

holomorphic function expansion of

f

k l (t, w) = f k(t, t 0.

0 ,n

(t ) t

-1

- nk .

f

= n=s;

fk

k ~,n(t)

f +

=

then

£

w),

d

J

Ak

a. (t) ~K, n

H(4). fo r

;<.

IC

•

is given by a

k

wn • in the Laurent

Now t

~

0,

t here f ore

This means th at i f

n ~ 0

~ ,n

has

- 90-

A. Andreotti

a zero of infinite order at

t = 0,

a... u,n = 0 i f n -F. 0 onl y t he t erm i nd epend ent

t h us

an d th erefo re in the exp ansion of f of w app e ars i.e. f <'.I/(.6). It follows then th ).t 6{ ( X) =p*&( Ll) and th at /J reduction s p ace of X. On X c onsider the functions given on !J 0 " a:~ f

g =

= w ;

r tn n e2Z

2

is t he by

n w

One verifies easily th ,;.t t hese de fi n e meromorphic f unctions on X. ni 7 Set t = e with lm » 0 since I t\
For any v alue of 7 B( i,Z) the form Zo(i) + 1 + k J' ;

h as an 1nl£inite number of zer os of 1, k G 21a

We show th .:;.t f and g are al gebrai c all y independent over ~ (X) . If, on the c ontrary, f and g were al gebraic ;.11y dependent over (X) th ere would exist an irreducible polynomial in X, Y, F(t;X,Y) '10 with coefficients th ,~t can be assumed to be holomorphic in t 6 IJ, such that F(t; f, g) ;: O. Now for so me t = t 0 -F. 0 F(t o' X, 0) F 0, but this equation

ct

f(t o' X, 0 ) = 0 should be sat i s fi ed for all v alues 2 Ri Z0 t 02k k € 2Z • This is impossible since a X= e polynomial in one v ari able h .s only f initely many z er os . The conc lusion is th at the tr ansendence d egr e e of J« X) over 4) ( X ) is 2 2, whil e i f theorem (5. 4.2) is holdin g fo r X, one should h ave a transcendence degree ~ 1.

- 91-

A An ilr e otti

Chapter 6.1

VI.

E. E.

Levi problem.

Preliminaries. a)

eech cohomology with values in a she a f ,

l{ =

Ui i ~I

be an open covering of

(f

Let

sheaf of abelian groups over a topologic al sp ace

X.

we set

Xi

for (iO'···' i q)

where I

and where we h ave set

Thus an element

....<. t 1

runs over all

l,

where

1

0

f

••• 1 ')

fi

q

Obviously'

CqCU, j- )

S

q

Cq(Z( 0·"

1

q

U· 1

(\

nUi

r,

- ;>

~

0

1

() ••• I1U

i q

is given by a coll ect1on l ' "1) •

0'·' q

1s an abeli an gr oup .

: Cq (U , 1-)

q

(q+l)-tupl es extracted from

U 1 ••• 1 = Ui q 0 0

,--:1)

be a

Let

Cq +l

We :fe ri n e

( ZI , '1 )

by the formula

=

q+l ~

j=O

U " i • • .1 j (-l)j r O U

i

••• 1 +1

q

o · • .1 q +1

where

is the n ::.tural restriction homomorphism. One verifies th at S q +1 of abe11 an .gr-o upa and maps

o

C

. <S o ( 11, "] ) - 7

0 ~q

1 1 '
= 0, b1

-?

th erefore the s eque n c e b2 -

1s a cochain comp1es. Its cohomology in d imension d e n o t e d by Hq( 2(, "'1).

>

q

will be

-92-

A. Andreotti Note th at, s ince

is a sheaf,

HO( 2{, T)

= Ker o O =l'(X,Y)

so that the above complex c an be c omp l e t ed wi t h the augment ation map

°

The groups

"?

Hq( Z(,~)

of the covering

U

are c al l ed t h e eech cohomology groups

with v alues i n "?f •

V = 2" Vj S j~ J

If

l' (X, '1) !.,.~ CO ( V , "J- )

is anoth er covering of

U

which is a refinement of refinement function By means of

J: J

-;>

= ~ Ui~ i G-I I

X

by open ,s e t s

then one can define a

with the prop erty

V j c. U-r(' j)'

one defines a homomorphism of comp lexes

J

: .~( 2!, 1) -;1>C*( z.J-, '1)

,T*

by sett1~g ( i f)

jO ••• j

= r q

Uj"{jo) •••1(j )

Vj

q

j

0·· ' q

This defines a homomorphism of cohomolo gy groups J

~ ..: v

One verifi es th at :r

H* (?(,

n«

'2.-'--

'1).

:J) -,) H* ( V,

d e oe nd s only on t he coverings

b ut not on th e c h o ice of the refinemen t f uncti on.

2{ , 2, Q-

One c an t hen

de fi n e Hq(X , '1-) =

Hq(

lim -.;>

-z( ,c-y:: )

q

I

0,

~

7.{

th e d irect limit t akc n o v er al l coverings U c all ed th e b)

(?!, Jr,

into ~

ii)

q-th eec h c oh omol ogy gr o up o f

Homomo r phi sms o f

gr o up s

L)

v

X) ,

sh e ~ves.

( 5', {~"

is a c ontinuous map

u; o ¢

X

of

X.

Thi s is

with val ues in

"5-

Given two s h e :.v e s of abelian

X)

over

d

-:; -> ,~

X

a ho momorphism

0

s uch th at

tr

f or ev e ry x
t

x

is a

f

7J

.

-93-

A ....Andr.eolli

If

~

""1

is injective

can be considered as a subsheaf of

J

and one can, then clefine a quotient she af

:1

space of

~

N

~-~ t; P('J-x) ,

iff

the topology being the quotient topology. =

'!f

as the quotient

by the equivalence relation

~

(~/'l-)x

I~,

fxl7- x

One

h~

thus

'

A sequence of sheavesand homomorphisms (1)

°-> T::" f

~> 1/ -"? °

is called exact

°~

"T' J x

e:

if for each X

1fx

->

is an exact sequence. the sequence

°

(3 x

-;>

-:>

x

~

X

the sequence

Jlx-7> °

In particul :cr if

~~

!i:

-;>

1

fl7 -;>

is a subsheaf Of'; 0

is exact.

Analogously one defines the notion of long exact sequence of sheaves. Given a sheaf homomorphism

1: j: .., 'j..,

this induces a homo-

morphism of cohomology groups

t> It:

Hq(X,

T)

-> Hq(X,,lJ •

One has the following theorem of Serre: Given an exact sequence space

X

trr'

1

d.

""I' d:.

*

HO(X'J)

*

..." H (X, J) -?'

,.,>H 2 (X, err. j ) <:t

8

of sheaves over a paracompact

then one has an exact cohomology sequence

°-> H°(X, T>

where

(1)

Hl(X,J' )

6

(3*

HO(X,J()

-:>

(3* ---;>

Hl(X,J!>

.~

~

*

is a "connecting homomorphism" which is defined in

the usual woy.

- 94-

A. A nd r eotti y

Cech cohomolo gy i s part i c ularl y useful if the topological a dmi t s a s yst em of coverings ? ( d

X.

sp ace

coverings

0

X.

f

U

i O " .i q

cofinal to all

such th a t f or an y Hr(U i

i'

o

q

T ) =0

» > O.

for al

One h 9s in fact the fo l loWing important th eorem of Leray:

T

Let space Hr(U .

J. O•

be a she af of abelLm gr o up s on the paracomp act Let 1..(

X.

i , ~) •• q

be an open covering of

=0

X

such th at

iO, ••• ,i q, ~

for all

q

and all

r > O.

Then the n atur al homomorphism q H ('2(, 7' ) -> Hq(X, "1' ) Example. Let

X.

covering o f

J. P. Ser r e ( IITh e o r em BII)

on an open set one has

i Ui S

by open s e t of ho l omo rphy .

is al s o an open set of holomorphy.

space)

"2{ =

be a c omp Lex manifold and X

U

q ~ O.

is an i s omo r p hi sm f o r all

be an open

Th en any U i

A theorem of

O

••• i

q

H. Cart an and

st~~~ ~h a:t: for any coheren t she af

":F'

of holomorp hy (or mo r e general on a Stein

Hr(U,

T)

=0

r >

if

o.

Thus coverings by

open set of holomorphy are Ler ay-coverings. c)

Ac yc l i c resolutions.

on the paracomp act

sp ace

Let X.

an exact se qu ence of she aves dO ( 2) o -) T~1o --::> l

T

vr

be a she af of abelian gr01H!s

By a r esol u t ion

~ .r-J- 2

d

of

2

- '7

the resolution is called a c y cli c i f Hq(X,

1-i )

=0

for all

CJ. )

0

and all

i 2. O.

T

we mean

- 95-

A. Andreotti De Rham theorem.

II

Gech cohomology of

X

(2)

is an acyclic resolution of

with values in "}'

".:r

~

is n aturally isomorphic

to t h e coholomogy of the complex [(X,

'10 )

i.e.

Hq (X, ar' j-)

'lJ

'1'

Ker i (X, .....:ag) __--..

- Imtf(X,

7'q-1 )

Examples. 1.

lJ-

A sheaf of a b e l ian gro ups

open set

Uc

surjective.

Hq(X,X)

=0

X

:i.:.;;::-

is called flabby if for every

the restriction map .f(X,JeJ

For a flabby sheaf for all

q> O.

j;:...

-7

r

(U,

is

one has that

Every sh eaf

~ of a be li an groups

can be considered as a subshe af of the flabby she af

t- 0

= sheaf

of germs of arbitrary (not ne&essarily continuous) sections of

'J-.

C?j-

Making use of this fact on e re alizes th at any ah e a f

ad mits a flabby (thus a c yc l i c ) resolution

2. A

A she af;:' c,

X

is call ed ~ if for every closed subset

the restriction map

For :~ soft sheaf

,X

r

(X,;:') -'> l' (A' 5~)

one h as again

Therefore soft resolutions of a she af Let ·X

be a

different~able

is s ur j e c t i v e .

Hq(X,j.) = 0

st

f o r all q > O.

are acyclic.

manifold ef dimension

n.

Let

denote the she ~f of germs of real-valued differe nti able C~ ~ r L' r +l forms of t y p e r (r = 0,1, ••• ) an d l et d : L/t """ .rr be the she df homomorphism induced by exterior di f f e r en ti dt i on . kernel

0

f the h omomo rphism

of c onstant fun c tions

IR

d:

"f 0 ~

j( 1

Since for e ach

i s a soft s he af we obt ain a se q uence

is the she af r

t he s h t af

The 0

f germs

uf

r

-9.6-

A. Andreotti

One has

d. d=O.

Moreover the sequence is an exadt sequence

(~oiricaf~ lemma): We h ave thus obtained a soft resolution of the sheaf 1& The De Rham theorem tells us that the cohomology group H* (X, lR) can be computed as the cohomology of the complex f Jt..* (X), d} of global dif f er en t i al flrms.

4. Let

Let

X be

a

a complex manifold of complex dimension

n.

JI.. r,s denote the she af of germs of complex-valued C'" dif-

ferenti able forms of type (1', s) i.e. of t hose forms ~ that can be written in a system of local holomorphic coord in ates as

Let j be the operator induced by exterior di f fe r en t iat i on with respect t o antiholomorphic coordin at es. We obt cun in t his way a sequence 0-;> ,-,'L r

J; J(r , O s, J/.r,l ~.>

Ker t. Jl.r,o-- "~r,l~ is the she af of germs of holomorphic forms of degree r. The sequence is a complex as ad = in which JLr =

°

and exact (Dolbe ault lemma). We obtain in t his way f or any I' ~ a soft resolution of the she af J7. r. In p art i cul ar for I' = )1.0 .= tJ and we ge t a s o f t resolution of t he structure she af {} of X. We h ave thus

°

°

= Ker [ f eX, J O, k ) ~r(x,JlO, k+IH ImtL'()!:.,.jt°,k-l).i [:(X,tAO,k )

Reference s:

[25J,

[30].

5

for any

k 2. 0.

-9 7-

A. An d r e otti

6.2

E.E. Levi problem.

a) Let X be ~ n-dimension al complex manifold. To clarify the exposition we will restrict our ut t ent i on to open rel atively compact subsets D of X with a smooth bound~y. We can thus assume th at D is de f i ned as follows. We h ave g1 ven a C func tion ~: X -) m wi th d ~ ;. 0 on a D and such th at D =~x 6 X b(x)< o~(l). (1

I

X = tn

As for the case

we de f i ne

D to be an open set of

holomorphy if for every zo" a D there exists an f co /I (D) whi ch is not holomorphically extendable over zoo In chapter I we h ave also proved th at if D is .S\Il op en set 0 f ho Lomo rphy t then (*) the Levi-form at each point of 0 D restricted to the analytic tangent plane to () D has no negative eigenvalues (theorem (1.4.2 )). The Levi problem consists in asking if an open set D with a smooth boundary and relatively compact in X which verifi es the 'condition (*) is a domain of holomor9hy. Vfuen no additional cond itions are pu t on the n ature of the m ~ifold X the answer to t his question is negative as the followin g example of Grauert shows. Example.

Let

n

2

2;

X=m

and let

1":

lIfn

-7

be the unit cube in

consider t he real torus 2n

/ 7Jn

X be the n atural projection.

m 2n t

Let

so th at 1'(Q) = X.

Consider the following set where

(I)

0

<. L <.

t

It is easy to verify th at if D has a s mooth bound ary aD this can be also defined by a global equation.

- 9 R-

A. Andreotti and let

-rO xl =!

E.

5 ).

lR 2n

n II: n the complex structure of II: •

lR -linear surjective map 1

inherits through )"

: lR 2n

~

~2n is a group of tr~slations, these are holomorphic

Since

for every J-.

we consider.

Th erefore

provided with a complex structure. structures is Levi-fl at Zo €

X, rel atively

is an open subset of

is compact) with a smooth boundary given by

X

Consider now an so th c,t

D

Then

D =i(Q £).

compact (as

aD

since

and thus

D,

is

D in any of these for every

~.

is identically zero.

In fact this is a local property

is a local isomorphism it is enough to verify this

7

condition on

r -l( d D) = 1s Xl = ! £ S.

the function

4>

The set

= xl :; E.

But now we c ant ak e for

q,

whic h has i dentically z er o Levi-form.

Y = r ( ~ xl = On

of real dimension

x

X,

Moreover

2n-l.

= (A, A)

In particular

is a real compact s ubmanifold of 1 -1 Let /be given by z el;n, A = ( a ) ij

D

hi ~2n

ls.jm

xl = 2 Re

fX = Os is the union o f a cone - parameter f amily of co mplex l linear subsp aces

t " lR. t

c = (0, c 2,···, c 2n) b e a r e .J. v ector in f Xl = 0 J s uch are line arl y independent over the th ~t the numbers c , c 2,.·. 2n r ation als. Since t h e mat rix 1\. h as t o s 'J.tisfy th e on l y cond i t i on Let

de t

( A,

A) .#. c f

t ~ lR

0,

we c an ch oose

f ~l z l

'I ( ICn~l)

A

+••• + ~nzn =

so th at

o}.

Then fo r ev ery

mus t be everywh e r e de nse in

Y.

Moreover

A . Andreotti y

U = tam

( ICni

1)

•

We can now show that fi'(D) = IC. Indeed, let f '" I/(D) and let z0 be so chosen in Y so th "t !f(zo)/ =maJC If I. There exists Y a t0 m such that zo'" T(a:n-l t ). ~

0

By the maximum principle for must be constant on

f

must be constant on ., (ICn - l) as to l • l) a:ni But ., (a:ni being dense in o

0

I, f must be constant on Y bec ause f is continuous. The set of zeros of a non-constant holomorphicfunction on a complex connected manifold is of real dimension ~2o-2. Therefore f must be constant on D as Y has real dimension = 20-1. Since R(D) = II: every ho1omorphic function is estendable at each boundary point of D and therefore D cannot be an open set of holomorphy. b) If we wish to obtain an affirm ative answer to the Leviproblem, one need to reinforce the Levi condition (*). For instance one could assume th at the Levi form restricted to the analytic t 'illgent pl ane is nowhere degenerate on JD. Then we have the following Theorem (6.2.1). (H. Grauert, [26]). If D satisfies cond1t~ (*) and if the Leviform restricted to the analytic tangent ~T l an e to aD is non-degenerate (thus is positive definite everywhere on JD) then D is an open set of holomorphY. When the cond itions of this theorem are s .t i.sfied we say th at has a Gt ro ngly Levi-comvex boundary. The proof of this theorem will occupy the next section. We divide the proof in several steps.

D

6.3 Proof of Grauert's t heorem. Step 1. A criterion for finite ness for cohomology. Let X be a complex manifold and l et T be the ah e af of germs of hoLomor-oh i,c sections of s o:ne hoLomo r-ohi,c VEct or bundle F.

-100-

A. An dreotti

For every coordinate p atch

U where

Flu

is trivial, we h ave

r(U, ~) ~ h(U)r

where r is the fiber dimension of F. Now leU), and . therefore #(U)r has the structure of a Frechet space. If can transpose t his structure on r(U,~). Banach's open mapping theorem ensures us th at this structure is independent of t he trivialization chosen for F over U. If' V is cIlpen and V e. U the natural restriction map rU

r(U,Oj) ~ (V,'f)

V

is continuous, and, if' V c.."- U, then r UV is a compact map by Vitali's theorem. Similar considerations could be repeated for any coherent sheaf' even if it is not locally free. Since for the sake of the proof we need only to consider the locally free case the corresponding argument is omitted.

-zo= l Wi! iE: I

Let

be a countable covering

0

f

X by coordin-

ate patches, then the ~ech cochain groups

a s a countable product of Frechet spaces have the structure of a Frechet space. Since the restriction maps are continuous, the coboundary operator 6 is continuous. Therefore in the complex COCkl',

1") ~ Cl(tJ/j) ~

C2 (ut; 'Y )

i

the spaces are Frechet spaces and the maps are continuous Le. it is a "topological complex of Freehet spaces". Lemma i)

ii) , •.,.,

(6.3.1). B

e,c-

~

A ,B

be open sets of X such that

A

r~.: Hq(A,~) ~ Hq(B,~) ~

dimE

Hq( B,<J ) <

cP.

is surjective

-101--

A. Andr eotti ~.

U:: 1U~ :!eIN

Choose a count able covering

(eC)

of

A

by open

sets of ho10morphy cont ained in coordin ate p atches on which is trivial.

We may assume th at

#,,'1

many Ui's, say BnU i In e ach of the open sets set

U i

for

U i,

such that

B!l U i 1 s, i

1

B

# '1 So

k,

i

for only finitely

k, we can take an open

k

U

Co

i::1

Z/:: ~ Vj ~ such th at zr

We now choose a countable covering of jo1omorphy of the set

lUi n

of the covering For each

Vj

let

B

F

and

j ~J

by open sets

is a refinement

B i'i.isk'

i ( j)

be such that

Vj

U~( j)'

C

We now consider t he cochain groups

(~)

n r
Cq(Z(,,,;,):: q C ( V-,

s) :: n rrt j

0'"

and the restriction map

j

q

,"5)

for

A

for

B, _

T*

which is

defined as follows: (J * f)j

0'"

j::

q

f~(jO) ••• ~(jq) I

Vj

0'"

j

for q

If we consider the Frechet structure on the cochain groups, then

*

'3"

is a comp act

map since

Vj c

A

Uj(j)Cc. Ur(j)

and since

th ere is only a finite number of groups [(U T(jo) ••• • (jq)' ~).

(y)

Consider the map

<.J : zq( z{, '1) ~ Cq-1 (V , T )

-;>

zq( Z'"-"

y)

defined by G) (<( ~{3 )

where

Z

* :: J
+ ~ .'3,

denotes t he s p ace of cocyc1es.

By the Leray t h eorem

-102-

A. Andreotti

"'F)

we have Hq('Z{, = Hq(A, ';) and Hq(~ 'T) = Hq(B,':). By the assumption the map r AB induced by '1'* among these groups is sur jective; hence tV is. Since J * is compact, ~ =iX-l"* has a closed image of finite codimension by a theorem of L. Schwartz. (1) Hence 'dima: Hq(B, '3=') <. ~. step 2. ti!Yer-Vietoris sequence. ~ X be a paracompact topological space and let Xl' X2 be two open subsets of X such that X .. Xl X2• ~ be a she af of abelian groups X. One has the following exact sequence

or

sa

0..". HO(X, Of) ~ HO(Xl' 'f) & HO(X

~ Hl(X, ~.

:n -7 H1(Xl' "f)

&

2

, :1) ~ HO(X l n x2 '''F

Hl(X 2,

or )....,.

Choose· a flabby resolution of

~

Hl(X l on

)-;>

X2!~) ~ •••

(I

X:

o -;>':f-> eO -> e,1_> ••• One has the following exact sequence for every 0 .....

rex,

Bq) ~ f(Xl' e;q) &

where ,,( a) = '\

1

f(x 2 , e q) ~ r(X l

(3 (a & b) =

°

a/x

X

1° 2

q (1

~

0:

x2 ' e q )

-.;>

0

- bl x1ox2

Taking the firect sum over all q ~ we obtain an exact sequence of complexes. Writing down the corresponding exact cohomology sequence we get the desired result. Step

3. The bumps lemma . ~ (6.3.2). ~ B be an open rel atively compact subset of the complex manifold X with a smooth, strongly Levi-convex bOundarY.

(1)

Let E, F be Frechet spaces, let u: E ~ F be a continuous linear surjection and let k: E ~ F be a compact linear map. Then u+k has a closed image of finite codimenBio~, (c r , [28J. )

-.103-

A. An dr eotti

Then there exist arbitrarily fine finite coverings 2( = ~ uj h i5.t 2! dB and for each U a sequence of open sets B ,

o s:

111)

j

t,

j S.

with smooth strongly Levi-complex boundaries such

B - B c.c. U i_l i 1

i~t

and

1

any coherent

j,

o •

Without lo ss of generality, we may assume th at on a

neighborhood such that

U=

~

=

~.

E

1

Hr ( U () Bj,"F) 0 for an;}' 1 and for all r > ~'"Y

1v)

X

.!2!:

tha~

u,

of

U

L(+)x) 0

~U.} lSi~t

balls

B

=It ,

B () U

there is a

u I~

(x) <

for every

x

G

C tP function

05,

U.

(dq)x ~

~: U ->

0

The covering

will be chosen with SUfficiently small coordinate

such that U 0 B in the coordin ates .o f 1 1 elementary · convex and thus a domain of holomorphy. U c:. U

We now select

C

p

i ~ 0

and such that ~ t?i (x) i Choose €.l, ••• ,E:. successively with t small so th at the functions

,w upp Pi

Co C

m,

for every

U

4>1 =~ -tlP l,

4>2

X,

on

>0 E

1

= Q- cl f'1 - t.2 P2 ' · · · '

have all their Levi forms

L( ~ i)X > 0

1

1!.. t,

5.

for all

€.

i

1s

with

a B.

and SUfficiently

;> 0

4>t

x

U

= 4>-E l Pl - · · · - Ct t' t

for all

x

€

U.

Define B

i

=B l x U

G

U

I

01,


for

1 c i s t ,

=

BO B. If the Gi are chosen SUfficiently small then ~Bi

will be smooth for

1

s:

i s, t ,

B (I U j will be convex in the coordinates i an 0Den set of holomorphy.

0

f

Uj

and thus

-104-

A. Andreotti

we

then have (i) (!i) (11i) (iv)

step

B = BO co Bl c, • • • Co Bt since I} =

~t on J,B O Bi-B c<:... U has support csz: Ui i_l i since 4 i-f_l =€iPi Hr (Bi (\ Uj , "F) = 0 for r ~ 0, all i,j and all coherent sheaves ~ since Bi () Uj is an open set of holomorphy, by the theorem of H. Cartan and J.P. Serre.

4. (6.3.3).

~ B a relatively compact open subset of the complex manifold X with smooth strongly Levi-convex boundary.' There exists an open set A ~ X such that

~

(i) (11)

B "=e.- A

for any coherent sheaf ~ restriction map Hr(A, ~)

r? 0, !h!. is surjective.

2!!. X and any Hr(B,~)

£.!:2.2.!.

Applying the previous lemma, taking A = Bt, it SUffices to ahow that for r > 0 the restriction maps Hr(Bi,;t)

Hr(Bi_l,:F)

are surjective for each Write B = Bi_l i sequence:

U(U

k,

1 s; i

0 B i)

i

and

So

t.

a~ply

the Mayer-Vi etoris

= 0 = Hr(B i_l

n Ui, ~), the conclusion follows from the ex actness of the sequence. As a corollary of the above considerations we get the following Theorem (6.3.4) Let B e, c; X be open with strongly Levi-conve> boundary. Then for any locally free s heaf "1 2!! X we have Since

Hr(Ui ~ Bi'~)

dima: Hr(B, 'Y ) <

&

for every

r » O.

- °105-

A. An d r e otti Note.

This theorem is true for

( 6.3.1)

~y

c oh e r e n t sh e af as lemma

c an b e carri ed o v er t o t his mor e g ener al

wi t h the

c ~we

s ame p r oo f . step 5. Solution of the Levi problem.

2x li\ "-

a divergent sequence sup

v

we can find

( c ond i t i on

~

D

f '" H(D)

such th at

of (1.3.1».

D cc. X it is not restrictive to ass ume th at the sequence

Since x

=

f(x)

v

It is e n o u ch to s ho w th ;;. t gi v en

D

converges to a boundary point z ero at

zo'

~D.

U around

Choose a coordinate p atch zl, ••• zn'

zo ':;

z

such that

with coordin ates

o

DO U =~z" U 19(Z)< 0 5

where

et>

t

wi th

It

U

U.

in

C


function with

(z) = Re f(z) + L(4)zo (z ) + O(

6

If (U) •

1f

is au fticiently smdll, then

and set d:> 1

p

in

U

=~ -!.P.

>

0

A .. D

u

L(h)

with p

2.

TaIte ~ ;> 0

on

U.

X €

U

Set

~

d4

~ 0

in

U

and

We can write

~

C .,. function and

is a

U -? lR

> 0

L(~)

II zll

3.)

= 0 s(\

0, a upp

p

D <:

e,

s o small th at

= fj.

Choose a

U, 0 ( 0 )

>

d~ l ~ 0

0,

on

OS.

¢l (x) -c

A has a smootq strongly Levi-convex bound ary an d t holomorphic on A n U. In particul n- by theorem (6.3.4) T~en

l(A, d1ma: H 9)

Write

=r

A = D

<

U

U

is

oP •

(A () U)

and apply the Mayer-Vietoris sequence

) -;> HO(D,

6)

for the sheE!-f () :

o -7

HO(A, L9

~ Hl(A, & ) -r

~

HO(A (\

u, e )

-?

HO(D () U, f) )

§..

-101;-

Consider the sequence of functions D

n U.

;lJ

A. Andreotti for

jJ

= 1, 2, •••

in

These are ho1omorphic and linearly independent over 1

~.

Since dim~ H (A,I.7) = r not all zero such th at

we can find constants c l"'"

1

~.

c r +1 <= I:

1

b(c 1 1 + ••• + c r +1 fr+l ) = O. Therefore there must exist holomorphic functions .,; hI(A h
m is a meromorphic function in A, holomorphic in D r+1 1 with principal part at Zo c i t1 For the function

Hence g

=hn

t

we thus h ave

I g(x)/ = 0".

Q..E.D

Remarks. 1. Let us call a manifold X O-nseudoconvex if there exist a C function 4 X 7 m and a compact set K <... X such that (i)

for every B

c

(ii)

=~ X

c

eo

m the set

.,

X

I

¢

(x) <

cS

is relatively compact in X at every point the Levi form L(t)x of 4> is positive definite.

x

~

X - K

A O-pseudoconvex manifold for which we can take K empty is also called O-comp1ete. It can be shown th c~ theorem (6.3.4) extends to O-pseudoconvex manifolds as follows: Theorem (6.3.5). For a O-pseudoconvex m;. O. If moreover X is O-complete then (d. (2 ] ) . for all r > O.

-101-

A . Andreotti

The proof is obtained from theorem (6.3.4) by the Mittag-Leffler procedure making use of the fact that for s;p ~ c <. c the l 2 restriction map HO(B c -? HO(B c ' 'r) 2 2 has a dense image. 2. Let D be an open relatively compact subset of ~n with a smooth boundary and such that there eXists a C ~ function 4> : a: n ~ JB wi th the properties .<;

,'1)

D

°

=t Z

€

cf I ~ (z)

°S,

<

d4> oJ on a n, L(~) LOon some neighborhood of ;}D. Then D is an open set of holomorphy. Indeed, replacing

on D. It follows then that D is an increasing tudon of open sets of holomorphy and thus a set of holomorphy. One could also apply the previous theorem of vanishing 'f or cohomology to derive the f~t that D is an open set of holomorphy.

°

°

t

Let D =f x e X I 4> (x) < Os be a strongly Levi-convex relatively compact subset of the complex manifold X. We have proved that for any coherent sheaf Hl(D,jf) is finite-dimen-

3.

°

sional. If L > is sUffiaiently small then DE; 1: x e ¢J (x) < - e f has the same property. can show that the restriction map

=

xl

Hl(D, '1') is an isomorphism, ( 2). #(D) = HO(D,e) separates In fact let us consider functions vanishing at the coherent sheaf and we have

-.>

Hl(D

e

Moreover one

,"J')

It follows in particular that points on D-De• the sheaf ~ of germs of holomorphic points x, y E D-Dg• :J is a the exact sequence II:

y

,~

0.

- 10 8 -

A. Andreotti

From t his we Qeduce the commut ative diagram with exact rows: HO(D, $-)

~ II:

x

'"

HO(D ,(}) ~

I

Since

4.

Let

=

II:

t

t3

y

°

D

C .::>

3.

be as in }<.

D

the diagonal

~(D, ~ )

-;>

Y.j,\ ..,-> Hl ( D , ;

1m 13 =

is an isomorphism,

C = ~ (x, y) .; D

Cl early

a

°

~) thus

c£ .

i s s urjective.

Consider the analytic set f(x) = f(y) A

0

l'

D " D.

fo r every

1' '' # ( D ) •

Consider the closure

S a-A of C-~. This is an analytic set. Therefore 5 n 6 is an analytic set. Any compact irreducible an alyt i c subset of .1 of di men sion ~ 1 mus t be cont ained in S 11 11. Moreover since N (D ) sep ara t es po i nts in D-D,s t he s et S n 11 mus t be comp ast. Therefore: !! D i s a strongly Levi-convex rel atively comp act open subset of a complex manifold X t here exi - ts in D a compact analytic subset A of dimens i on.L l at e ac h po i n t such th ,;.t an y i rreducible comp act an alytic s ubs et of D o f dimension ~ 1 i s containec i n A. Thi s argum ent c :~ be c arri ed throu gh for c omplex s p aces.

6.4.

Char ac teriz ati on o f pr o j ecti ve al gebrai c man i fol ds , Kodaira ' s t he orem ( 0 : J) •

As an applic at i on o f th e p revious c onsider at i ons we gi ve here a oroo f of t heorem (4. 2 ~l) o f Kod at r- a , We have alre ady given t he pro o f of t he i mpli c ati ons A < = 7 D. We will give here t he »r-oo f 0 f t he impli c at i ons A = ? B = C = ) D It is enou ;;h to pr ove t hi s for X = Pn(II: )· Let zO' . . • , Z n be homogeneous c o o r-a Ln a t e s in j'n (lI: ) and l et

A = U i

~

B.

=

Zo

zi -F

° wh ere

Yl = - t···, Yi - l = zi

we as s um e

.>..8

c oo r d i.nrt es

zi_2 zi _l , Yi = ~, Yi+l = zi+l zi zi a,

t ••• t

zn Yn = zi

-109-

A. Andr e otti Consider the hernd.tian form

dSl' =)"

A 4: m and ). > O. Then dS). =J.. and by direct calculation we get

~dS]

dsr

g

log

ziZi

where

o.J log (1 + j=lYij)

= (1 +~Yij)-2t(1 +~Yij)

on

U i

rdYjdYj - (ZYjdYj)("iYjdYj)S

~dYjdYj.

2. (1 +iYi;-2

Thus

J.~

defines a hernd.tian metric on

Pn(t).

Its exterior

form is given by the 2-form n = i3.,~ ~ log i

w),

(J'

zi zi

o

i..l ="2 a~ d

= aw).

By the w~ it is constructed d UJ),. therefore the hermitian metric Now H lZ) = lZg 2(J>n(t), in g

Fn(E).

=~z2

B

g

fg

=;.

Jg

g

= 0

thus

d w",-

=0

and

is a Kahler metric (c r, (511). is a projective line

get «J .~ l g

J

d=J.d ~

(l+yy)

g wl l • But to make t h at p e r i od an int eger (1) 4)).

0

!l(E)

the projective line

= zn = Os we

=

Therefore

A

Taking for

where

dsi

n -) log (1 + ~ YjY j •

W

>

0

2

where

Y

thus one can choose

=> C. (~)

The assumption

th at t h ere exi sts on

X

B

can b e re pl ac ed by the ass umption

a Kahler metric whos e ext erior form has

r ational period s as we c an al way s multiply th e me t r i c by a p os i tive integer to make the periods of t he corresponling exterior form integr al. Singular homo logy or cohomology b ased on d i f f e ren t iable singv

o

u1 ar simplic es will be d e no t ed by a s uffix "s", Cech c ohomology by the suffix "v" and th e cohomolo gy o f the d e Rham complex of di fferenti a b ] e forms b y the suffi x

(1)

"dR".

-119-

A. Andreotti

We then have the following commut ative diagram of isomorphisms and inclusions:

2 sH (X, ~)

.H

2(X,

JI2(X, ~)

k

<--->

JI 2 ( X,

lC)

JtI2 ( X,

t)

1"'5

d

If

l'

2""

values on

sH 2(X, lC)

-y 1'1 2 , ,, , y

sH 2 ( X, ~) modulo torsion, a

is a basis of

cohomology plass of values on

lC)

d 1'S

d.1'S

is caracterized, viaP',

and i t is in the image of

l' '( 2"'"

are rational numbers.

tion shows that; the isomorphism

~'oct'o

which associates to a differential

T

j

by its

if its

A direct calcula-

is induced by the map

f>

2-form

the singular

cochain It follows that if a closed

2-form

co has rational periods i.e.

fYi"'ii, ~ then it is represented in eech cohomology by a 2-cocycle 1Cijk~ with

c

t: ~

i jk In this argUll8nt

(~)

(modulo coboundari es with values in

a:

can be replaced by

Define the sheaf ~

monic functions on

o~ where o.(c) Let ..{ r,s of type

=c

II:

lR everywhere.

of germs of complex-valued plurihar-

X by the exact sequence of sheavea

~

It c

e

It

$' ~

11 -?>

and f.,(f Et

g)

0

=f

-

denote the sheaf of germs of (r, 8).

t).

Set for

we n ave the following

U

open in

X

g. C

fP

r

complex-valued forms (U,J1 r,s) =J/. r,8(U).

-J1.l-

A. Andreotti ~

!!ll

(6.4.1)

~

~n.

U

be a contractible domain of holomorphY

One has the exact seguence

° ...;.f(u,1-f) . . . . j{0,O(U) ~ j{l,I(U) ~Jl,2(U) • A2 , 1 ( U) , !!l!!:.! f!22!.

d = ;) +;i

is the exterior differentiation.

Obviously the composition of any two consecutive maps in

the sequence is zero. resolution

°-> a:

1"\

-T .I~

Since

°d

nl

~ .1<:..

U

is a domain of holomorphy the

d () 2 d ~..i~ -"7

is acyclic, ..Qi

denoting the sheaf of germs of holomorphic

i-forms.

U

Since

is contractible the complex of sections on U

is exact. Let fO,O (;

"q O,O(U) with CW :l.e ~ antiholomorphic in a(fo,O-gO,o)

U.

=°

We have then

=JgO,O,

fO,O _gO,O = hO,O

i.e.

fO,o

;itO,O

Hence

= gO,O

+ hO'O,

and a holomorphic function on

U.

Now

i , l ..: fl l , l ( U)

~gl,l

= 0,

such th at

Hence

° thus

h 2,o = dkl,o

a h l , o" ;;>k l , o

and

hl,O _kl,O:)k0'O gl,l

=

because

with

kl,o

3 ( h l ' O_k l , o ) tor some

°

(1,0) U

dhl,O = h 2,o

)hl,o = 7(h l,0_k1,0)

--1'l,l(U).

di,l =

thus there exists a

gl,l = jhl,O,

Now ;)- ~hl,O = Hence

with

U.

Le.

3 Jko,0.

Jol,l

form

This

= °=5i,1

hl,O

on

U

is holomorphic and closed.

holomorphic in

= 0.

r (U ,ff).

fO'O,;;

is a domain of holomorphy.

U.

Thus

Consequently

C '" tunction on

=

is holomorphic in

a sum ofm .antiholomorphic

proves exactness at J? O' O( U) . Let

thus

U.

Now

This s ho ws exactness at

-11 2-

A. An d r eotti

0')

Let

.

=i

i»

~ g

Kahler metric.

Let

coordinate balls. write

Iu1

(,<J

(g~i)

Since

dz

", 7:

"2{

be an open covering of

dw = 0

X

by

by the p r evio us lemma we can

= ~;W1

6J/ u = t

1 be purely imaginary.

u1

r

= ~ Ui~

Since

be the exterior form of the

dz",

1\

w

is hermitian,

and therefore

Then on

d.

is real, so that

~a(Wi - Wi).

".l lu = ~

We can thus assume

a Wi ' Wi

to

Uj

fI

Wi - Wj .. 21

=Pi j

Im Pij

- Pij ,

Pij : U 0 Uj ~ £ is holomorphic an d determined up to i addition by a real constant.

where On

U (\ Uj 1

Uk

(l

we must h ave

Pij + P jk + Pki Moreover Hence

= ci j k ~

c i j k - c l j k + c l1k -

! C1jk~

C

l1 j

lR.

=0

represents an element of

on ~

2

Ui

(1

Uj (\ Uk () Ul•

( 2{, lR).

v

Now the cocycle Z. Cijk~ is the eech represent~tive of the de Rham cohomology class represented by 4) (up to a sign). Indeed we have

wlu

~.i - aWj

i

..

d(iw , i)

= -aP i j

-(Pij + Pjk + Pki)

.. -dPij'

=-

c i j k'

which is the string of homomorphisms th at make explicit the isomorphism of the de Rham theorem: By the assumption that

C<)

has integral periods, by

derive that we can assume th at the real numbers integers: (1)

P1 j

+ P

jk

+ P ki

=c i j k

€

~

ci jk

(d.)

are

we

-1l~-

A. Andreotti

__ e 2 1]"i P i j

Let gij holomorphic. (~)

~cause

of

gij gjk = gik Therefore the collection

is

so th at stj (1) we get on

Ui

J stj"5

for a ho l.omor-phf,c line bundle

Uj () Uk •

I'

is a set n F ~ X.

0

f transition functions

Now where k Let and

Vi =n ~

i

-1

(U

'2::. Ui

i)

J4

II:

the fiber coordinate. j.. (z , e, )

j.'< z , E)

Br = Z(Z''S) For every

r ? 0

is comp act.

as on

F

c F

r

around t he O-section of

I ;x.(z,;) <

F.

F:

rS.

s)

t; )

= ,j . }

3 Br

log k

i

as the analytic tqngent p Lane to i

be the base coordinate

B is relatively comp act as the base space r Moreover the Levi-form of log f ez , restricted

J.'3 log f ez ,

;) z«

i

One verifies that the func t { "n

to the analytic tangent sp ace to

,) log k

-2Tl1W

defined a hermitian metric on the fibers of

Consider the tube of radius

X

= e

= k i (z ) ISil2

is a well-defined function on

Indeed

z

and let

i

d ZQ +

1 t:

i

d~

dB

th e analytic t angent plane to

L g _ dz dZ .s' ~3

d..

is given by as

He -roe the Levi-form re stricted to

B r

fhis s hows th "t t he c omplex space cons ider Lns t e ad of t h e bundle

r

1T

we can take

= 0 ,

coordin at es -alo n g th ,.t plane.

=2

reduces to

F

is strictly p os i t i v e definite: F

is

O-pseudoconvex. If we t he bundle F- 1 which is

-114-

A . Andreotti

given by transition functions [g~~f, then b, a simple verification we see that the space F-1 is O-pseudoconcave. C ,

_>

D.

Let J be a sheaf of ideals on X, for instance the sheaf of germs of holomorphic runctions vanishing on some finite set SeX.

Consider on

F

the sheaf

'}' =

n*J • ei

e F'

In

the previous instance, this is the sheaf of germs of holomorphic functions on F vanishing on the finite set of fibers n -1(S). Every

t c.

r (Vi' ':f) l' -

has a power series ex pansion

""~

- d.=O

Set

so

This gives a filtr ation of [(Vi,t) which is independent or the choice 0 l' the fiber coo rf Ln at;e ~ i' therefore Simil arly one filters the groups r ( Vi i , "1) t we get a

filtr~tion

o

v

of the Cech complex

q

c- (V",

°1 )

Which is compatible with the coboundary operator. If

~ ~ Cq(V;

1)

and

",:.

;.

one verifies th at, for given a

--~ et ~ =0

,

id.)

i o " .i q

(z)

where F- denotes the she ~f of germs of holomorphic sections of -vv the bundle F-~ Therefore we get a (split) ex act sequence (2)

0"7' C~+l

u l'/1) ~ C~
-:" CqU{,

J

ill

!-k)-7' 0

-.U 5 ~

A. And r e o tti

* _.Since the filtration of C (1 "', ';J-) is compactible with the coboundary operator, we obtain a filtr ation of cohomology

H6< V,J) = Hq< U, ~J) while from

(2)

Hi< 2!, :f) :->

..:>

H~< r;, r) .::>

•••

we get

Therefore (3)

s

k=O

q From theorem (6.3.5), i f s > 0, H 0 in (3) the left hand side is finite-dimensional. The same must be true for the right hand side of (3), t her efore

.!2!:

q.>O

ther'l'!' eXist

k >.. k O'

ko

0

Hq(X,

» 0 such th at

fll

if

J-k) = O.

Now we apply this result to the exact sequence

o

7

tJ

k

fll F1/11

~

F-k -? Q 1./1F

where ~ is the s he af of ing at two di s tinc t points given point. Writing down one realizes th at the ring separates points and gives

~

0

germs of holomorphic functions vanishor vanishing of second order at a the corresponding cohomology sequence l) Fk~ F- k) local coordin ates everywhere on X.

J« X,

=

reX,

Remark. One could apply theorem (6.3.4) instead of theorem (6.3.5) replacing in the argument F by a tube Br of radius r around the O-section of F t thU8 avoiding the use of the Mittag-Leffler argwnent to go from the first to the second of these theorems. The extension to complex spaces is given by Brauert in

[ 27J •

-116-

A. Andreotti

Chapter 7.1.

VII.

Generalizations of the Levi-problem.

d-open sets of holomornhy.

Let X be a complex manifold, let D be an open relatively compact subset of X with a smooth boundary and let d be an integer, d ~ o. Let ~ be a locally free shea! on X, for instance t9 or ~ = o,d. We repeat the considerations of the previous chapter replacing the ~pace #(D) = HO(D,b') by the cohomology group Hd(D,~). Let Zo IS cD, an element E, G Hd(D,:r) is said to be extendable over Zo if there exists a neighborhood V of Zo

cr

in

=

and an element ~ ~ Hd(D U V, "5)

X

°

such that

D

=

Rem ark & if d > the necessary and sufficient condition for an element ~ E Hd(D, to be extendable over ZoE: JD ~ that there eXists a neighborhood W 2! Zo ~ X such that

;)WnD .. o,

c; is extendable over

In fact, if hood W of W e V,

Zo

Then ~

in

X

I W=0

Zo' we can find a neighborwhich is an open set of holomorphy with

I

thus ~ WilD

= 0.

Conversely suppose there txists a neighborhood W of Zo such that ~ \ WilD :: o. We may assume that W is an open set of holomorphy. By the Mayer-Vietoris sequence we get the exact sequence : Hd(D Since ~ in

~

€

U

W, '5)

-7

Hd(W,~) = 0, Hd(D

fl

Hd(D ('\ W,

W,"}')

'J')

Hd(D, '1) ~ Hd(W,

r)

~ Hd(D () w, '3').

because d > 0, and since the im age of is zero, it follows that there eXists

I

such that ~ D

= ~.

We say that D is a d-open set of holomornhy with resp ect to the she a! if for every Zo ~ ~D we can find S E Hd(D, "f) which is not extendable over Zoe

'J"'

- 11 7 -

A . Andr eotti

=

For d 0 we re ali zed ( The or em (1. 4. 2) o f E. E. Levi) th at the bound ary of D has t o have a par-t Lcu'l ar shap e . In vi ew of th at result we ar e Le a .i t o lDnsider t he fo l l owi ng s i tu at ion: Let U be an open set in a: n and let of : U "7' m be a C ov function on

U.

We set

fl- = ~.x

€

U )

f (x)

5 I ~ (x ) = 0 S < 0

I

and we assume t Lt on S = 2x E U d ~ oJ 0 t so th at S is a smooth hyp ersurface, which consti tut es th e boun dary of fl- in U. Let us consider the Levi-form of ~ restricted t o th e analytic tangent plane to S at every poi nt Zo G S, and let us 3.Ssume that at a point Zo e S i t has p pos i tive and q neg at ive eigenv alues (p+q .s n-l) • Then one can prove the following Theorem (7.1.1 ). There exists a fund amental se quence of neighborhoods U~ of Zo ~ U s uch th at, for qny locally free ~ on U we h ave

ur

=0

s > n-p-l or

{ 0< s < q.

These neighborhoods c an be chosen t o be ope n s ets of hol omorphY. (c f. [2]). The proo f 0 f t his theorem is r ather tedious and will be omitted. As a coroll ary we get t he an alo gue of E. E. Levi's th eorem: Theorem ( 7.1.2). (E.E. Levi). If holomorphy for a loc ally free sheaf Zo

Eo

D is a d-open set of t hen at any point

'1

en

The number of negat ive eigenvalues of (*)

\

the number of positive eigenvalues of

L( ~)

ZIT

~


IT

is

~

o

L(~)z

o

Zo

Zo

n-d-l.

-118-

A. Andreotti

One can then formulate the analogue of Levi-problem for Hd(D, '1). The example of Grauert shows that condition (*) is not sufficient to 'ens ur e th at it is solvable as in the eXdmple any I/X>int 1. " OlD has a fundamental sequence of neighborhoods o U'" such that D (l V'Ii 1s an open set of holomorphy. We have thus to reinforce condition (*) assuming for instance that L( ~) T for every zo e:: ~ D is non-degenerate. An open set zo zo

I

D of this sort will be called strictly Levi d-coBvex. has the following

Then one

Theorem (7.1.3). (Grauert)d' If D is strictly Levi d-convex D is a d-open set of holomorphY for any locally free

~

~

l'

2.I! X.

7.2.

(e r, (81). The proof Proof of theorem (Grauert)d' follows the same pattern given for d o. It is based on the following remarks. (ct.) As a substitute for theorem B of H. Cartan and J.P. Serre, we need only one h,uf of the vanishing theorem (7.1.1)

=

S H ( 0U7'v,

.-) =0, 'S-

if

s >d

=

since (n-p-l) = n-(n-d-l)-l d. Here :;: is any locally free sheaf (although this part of the vanishing theorem holds for any coherent she af Y. on X). Moreo\'er one has to realize that this st atement is stable by small defoDBations of the boundary, precisely given any sufficiently small neighborhood Uv of zo ~ ~D in X which is an open set of holomorphy and given any C ~ function p : U -;> lR with p 2. 0, p(zo) > 0, supp p c..<- Uv ' we can find Co

>0

such th 21.t if we set 0<: t.. <. £0'

U)",)

4'1

=

=tX

t -£pwith €

U,J4 l ( x ) c:

o}

then we still have for

s> d •

-'119-

A. Andreotti

(~) This enables us to repeat the proof of the bumps lemma with condition iv) replaced by r i v) H ( u1 o B j , '1) = 0 tor any i, j, any locally free

sheat

-r ,

and any

r <: do'

Then the analogue of lemma (6.3.3) ness gives

and the criterion of finite-

Theor!! (7.2.1). ~ B open relativelY compact in X with a smooth boundary on which the Levi-torm restrict!d to the analytic taP,gent space has at least n-d-l positive eigenvalues. Then tor any locally tree (or coherent) sheaf "f 2!! X we have dima: Hr(B, 'r')

<:

&

if

r

» d.

(1) We now have to construct for every z (, ~D a sequence d 0 ot coho.ology classes .c:;ve. H (IV {l D,5') defined in a fixed region • n D where W is a neighborhood of Zo in X, such that i) they are linearly independent over a: i1) every non-tirvial linear combination ot them is not extendable over Zoe

We postpone the proof of this fact to the end. We remark that to achieve this purpose we will make use 0 f the assumptions that ~ is locally free and that the Levi form on the analytic tangent plane at the boundary of D is non-degenerate. (8) Let us suppose for the moment th at point (,) has been settled. Then the proot proceeds as in the case d = O. Let A be constructed as in the solution of (Levi-problem)o making a b uap on D to swallow the point Zo E: cD , A

=D

U (A () U).

We then apply Mayer-Vietoris -tHd(A, "f)

-:l> Hd(D,"J')

.,Hd+l(A, "5) -t e.'"

• Hd(A

(l

U, "f) -? Hd(D

o u, ':F) ~

-12.0 -

We consider the classes C, '"- Hd(D n V, 5') constructed in (')'). vrLet r = dim~ Hd+l (A,~) (which is finite by theorem (7.2.1». We can find constants cl' ••• ' c r+l'" a:, not all zero. such that 6(cl~ + ••• + c r+l Sr+l)

= o. hn ..

Therefore there must exist cohomology classes and h € Hd(A n u,'T) such that M U r+l hn = i~ c i 8,i + hA(JU on D () U.

d H (D,

s).

We set g =hn € Hd(D,~), then in any sufficiently small neighborhood W of Zo which is an open set of holomorphy hAoulw = 0

thus

is not extendable over

Zo because of property

ii)

of

(~).

(t) It remains to prove statement (1) : Since is locally free, in a small neighborhood W of Zo in X, f} p for some p. Thus it is enough to prove the statement for the sheaf 'J' = The proof can be based on the following

rr

'J= -::.

e.

Lemma (7.2.2). Consider in forms of type (0, n-l) i:(-l)j Zj

~n _

to}

(n ~ 2)

the differential

a. +1 j

It'd. = - - - - - - - - - - - - - - - - - - -

for ~ ~ ~n. T~ese forms are a-closed th erefore for any open 2.tl U in a:n_[o} they represent cohomology classes Hn-l(U, tt). !.

= {z

Then th e cl asses (c f.

[5]).

U contains the closed half sphere

If Eo

a;n \

t& ~

~

j=l

IZ j 12 =E, ,

Re zl

~0~

Hn-l( U, l'-'7 ) are linearly inlenendent

-121-

A. Andreotti ~.

lfd

The fact that the forms

verified by direct computation.

'1 .. L cd f cl

Let

are ~-closed can be

be a finite linear combination of those

forms with coefficients cd E E. Let us assume that for some C (9 form fJ of type (0, n_2)CL on U. If

=a r'

Yf

dZlA ••• Adz then for any holomorphic function n, we have the generalized Cauchy formula:

~.

En

SJf

Let

rJ

>

II

z c a..:..:..! a:

az a

on

(0)

be so small that the part

0

Sd

leLl

rJ .. ~2"i~; n-

w A

f

2

fl

I Zj 1

:z f.,

flA)/I"l

ZlSd

5

Re zl Z - rJ of

is contained in

S

We have

U.

S6

5

f

ft.:JA

af"=sa'f d(fwflJ.!) =i)S!

tio»,

Thus

~2ntt n- I

Idol! L ca. CJdZd.

(0).. d

B

Then

=lj=1 ~ and

A

Re( zl + d/2)

I z j l2< B

is < 0

jJ

-d/2J

n lI: •

are Runge domains in

f

+ S S f w -d

on

B.

holomorphic funct ions in f

\J '7

K

g

uniformly on

c B and

g

a

Sd

we mus t h ave

II

1.

and

A

>0

and

En.

Let

on K

Moreover B.

Therefore

v (S

We can find a sequence En K,

A Ii B

be ~ oompact subset of

containing the origin in the interior aQd let

omorphic function on

any

<:::

IN 1\

d'/4}.

is ~so a Runge domain in

as

f

d

A=~ ~/ZjI2",e., +SSO[Rez l j=1

Let

B

s!

g

be any hol-

'£ r \J l

of

such th at f ~..,. 0

- S6) =

uniformly on JS d U (S- Sd)

(S - Sd) c tc,

Therefore for

- l.22 -

A. A n dreotti

lei I

c

d.

2...-...J!a

az

= o.

(0)

a.

d = n-l.

Let us now cunsider the case local coordin ates at ~quation

h = 0

zo'

Hence all

near

oD

are zero.

CiS


By suitable choice of

Zo

is given by a local

and we may assume th at

Zo

is at the origin

and that

= Re

h

1

where of

D.

~jZi Zj

If' E.

{L I z j l 2

»0

zl + [aijzi Zj + O( II z II 3)

»0

and where

h

»

0

corresponds to the side

is sUfficiently small then

= cJ"5

{h

the closed half sphere

> o] ~~

for any 2 I Zj 1

=

lemma above, in the coordinate ball

cI with

0 < cf

a'S (\

zl

f Re

>

W of radius

contains


O. EJ

By the centered

at the origin the forms 1Va represent linearly independent cohomology classes of Hd(W, 19'), not extendable over z00 In the general case Zo

can be written as h

= Re

Z

I + LLI

d h

2 \z.1 -L

~

=0

n-l

the local equation of

aD

near

with

~~

+ 2Re (z

1

.

j)

+ dtl A ~ j

D

corresponds to

aj-z

\z

jl2

where

Aj>O,lJj>O; is positive definite and where the side of the side If

h

> O.

W is a SUfficiently small polycilindrical neighborhood of

Zo = 0

in those coordinates, then W (l

t zl

:&

•••

:&

zd+l

= 01

(l

f5

= f zo5

•

-

-123-

A. Andreott i

Consider the forms of the lemma in t h e coordinates Zl'···' zd+l' all forms defined in W. These have singularities outside of D. Moreover their classes again satiRfy property 11) of ("Y) as this is true for their re.trictions to the s )ace

f zd+2 = ••• •

zn

= o} •

(1).

This achieves the proof of statement

7.3 Finiteness theorems. In theorem (7.2.1) we emphasized only the positive eigenvalues of the Levi form. With the same procedure one can prove the following more precise theorem, «t, [2] r (7.3.1).

Theorem

k!i

D be a relatively compact open subset

of the complex manifold

X

with a 8/Dooth. strictly d-Lert-convex

'3='

2A

boundary.

Then for any locally free sheaf r(D,1=') < d' dim; H i f r ~ d.

X

we have

The group

Hd(D,~) not considered in the ~heorem, is actually

infinite - dimensional and moreover has a natural structure of a Frechet space. Theorem (7.3.1) can be considered as the "intersection" of the following general theorams of finiteness. A complex manifold X is called g~pseudoconvex if there exists on X a C d' function + r X ~ :m and a compact set K such that

i)

the sets

f X"

X

I ~ (x) c c 1

are relatively compact in

X

for every

Bc •

on X-K the Levi-form L(.) n-q positive eigenvalues.

11)

If we can choose Theor.. ~

0;'

K.

~

the manifold

m

c"

p

of

has at least

X is called

g-complete.

(7.3.2). l! X is g-pseudoconvex then for any coherent ~

X

di. Moreover if

we have

r(X,1f) < ~ E H

if

X is g-complete then for

r

> q. r

>

q

Hr(X, 'j") • O.

-124-

A. Andreotti Analogously we h ave Theorem C7.3.3). 11 X is g-pseudoconcave, for any locally tree sheaf "5" 2!! X we have it

r

< n-q-l

Due to the maximum principle there cannot be an analogue ot "q-complete" manitolds tor the pseudoconcave case. Theorem C7.3.3) is only true for locally tree sheaves. Actually the real analogue of theorem C7. 3.2) is that "for a g-pseudoconcave manifold and any coherent sheat 'S" sm X we have

.!2!: ~

r

»

q+l ,

Hr k

denotes cohomology with compact supports. II Theorem C7.3.3) is obtained from the one quoted above by duality and thus, if st ated for any coherent sheaf, the number n dime X has to be replaced by the depth of the sheaf "J" • As reference see [2], [7].

=

7.4. a)

Applications to projective algebraic manifolds (c r , We revert to the case d 0 to begin with.

=

[9]).

+ Let D be a complex manifold. Let COCD) denot e th e sp ace ot positive O-dimensional cycles i.e. the tree abeli an monoid generated by the points of D. We h ave

Pi' D}

wher~ DCk)

=D

CI ) o

fl D

D

( 2)v

denot es t he k-told symmetric product of D. For each k L I DCk) i s the quotient of the cartesi an product Dll' ••• Jt D··Ck- t i mes ) by the act ion of the s ymmetric group on k l etters permuti ng t he f actors of this c -ll't e s i an product in all pos s i bl e ways. As s uch D( k ) , and t herefore C~ CD ) . carries a nat ur al struc ture o f ~ c ompl ex sp ace.

-12 5-

A . Andreotti

Moreover we h ave a natural map given by for Note that the image of Po is a set of ho1omorphic functions on C~(D) which are additive with respect to the monoid structure of

+

Co(D).

Proposition (7. 1+. 1 ) . The sp ace (holomorphical1y complete) iff (ho1omorphical1y compl ete).

C~(o) D

is ho1omorphical1y convex is ho1o morphical1Y convex

~. If C~(D) is ho1omorp hic al1y convex (complete) t hen 0 as a connected co mponent of C~( D) has the same pr oper t y . + If D is ho1omorphic ally convex then also CO(D) is. In + fact let t Cv $ C COCO) be a divergent sequence. Let for every c = ~ niPi deg(c) = In Then deg is a ho1omorphic function on i•

C~(O)

tellin g us on which s ymmetric pr oduc t O(k) the considered cycles st ay. If deg (c y ) ~ & t here is nothing to prove. If deg (c v ) s: k , then we can extrac t a divergent seqUer.c e ~ c '5 cont ained in a given product 0 ( 1 ) (with 1 S k). \I Now Dl D~Ox ••• xD (l-times) is ho1omorphical1y convex. Let rr: 0 1 ~ D(l) be the natural pr o j ec t i on and l et ~ 1 be the l s ymmetric -gr oup acting on D • For e ac h c ~, let s 'll' .;; D1 The se quence { s" , $ must be diver be such th at 1J'(s li') = c"" 1 gent in 0 • There fore there exists a ho1o morphic fu nction g '" H(Ol) such th at su., I g( s ,) I = ry . Consi de r the y v pol ynomi al in t 11" l' 1'-1 (~ll (t - g(y x» = t . + ~( x) t + ••• + ~!(x ).

=

The co efficients ai ( x ) are holomorphic fun ctio ns which are t hu s for e ach i t her e eXi s ts a hol o ~o rp hi c function f € 17'( 0 ( 1 » with l7" * f = ,,\ . I S. i s. 1!. On t he i i

~l-inv ari ant

'-

lZ6~

A. Andneatti

sequence unbounded If D once from values of

sv,

at least one of these functions ~, mus t h ~ve acsolute value. Therefore sup !fi(sv')! =~. i~ Stein then also C~(D) is Stein. This follows at the fact that on a Stein manifold we can prescribe the a holomorphic function on any finite set.

b) Suppose now th at D is an open subset of a prcjective algebraic manifold X of dim; X = n , Given em integer d , + o !; d ~ n, we can consider the space Cd(D) of positive compact d-dimensional cycles of D i.e. the free ~belian monoid generated by irreducible compact analytic subsets of D of dimensio~

C~(D)

= fL

I

niA i

n i Eo:IN almost all zero,

Ai

compact

irreducible analytic subset of D

0

f

dim lC

Ai

=d ~

It is known that C~(D) carries a complex structure of a weakly normal complex space. Examples. 1.'

D

= X = Fn(lC),

C~_l 2.

X

(F n(lC»

= Fn(!I:)

0;_1(8)

I

D

= En

d

=

= n-1~(a:)

= ~(a:)

V lPn +2

( 2 )-1

- ~ point}

u a:(n22)-1 u

(D:) U F +

(D:) U •••

n ( 33 )-1

then

a:(n~3)-1 u ••• d

d

Let us consider the cohomology group H (D, n ). Every cohomology class S on it can be represented by a a-closed differentiaJ form 4> d,d of type (d, d) modulo a of forms of type (d, d-l). + Given C b Cd(D) we can consider (by virtue of a theorem of Lelong) the integral S(c) = ~ d,d. c

f

One verifies th at this integral is independent of the choice of the representative ~ of the class 5 ~

-12 7-

A. An d r eotti

As analogue of proposition (7.4.1) Theorem

we then h ave the

fo11aWi~g

(7.4.2).

a) The function ; ( c ) 12!: c a variable on holomorphic function so th at we get a linear map

r« :

d H (D,

C~(D)

~

'" ,C n. d )O -p H (C d(D),L7).

b) !! D is strictly d-Levi-convex then for any di ver gent sequence 1C) c C~(D) there exists a class t; E Hd(D, .nd ) such that =

In particular

C~(D)

<:0.

is holomorphically convex.

c) If D in addition is d-complete then given c l' c 2 + d d Cd(D)!!!h c l '" c 2 there exists a class !; e H (D, ..Q. ) such that

~

;(c l) '" S(c 2).

In particular

C~(D)

We limit ourself to a

is holomorphicallY complete. v~ry

brief skp.tch of the proof

(cf. T9]).

(d) First one shows that the :unction S(c) is a continuous function of c (this is the mo st di f f i cul t part). (~) Since C~(D) is weakly normal it is enough to show that

+

iB holomorphic at non-singul ar p~ints of Cd(D). This is essentially done via Morera's theorem. Loosely speaking, if h + is an analytic I-dimensional disc in Cd(D), if -y = d 4 and if ~(c)

F

rr~ Cd(D) +

is the fibered space of d-dimensional cycles over + their parameter space Cd(D), we h ave

= f : n-l(y)

h,;"'dt

17'-l(A)

t/>d,d" dt = 0

=

17 -

1

(.6)

r

d( d>

1\

dt)

=

-128-

A . An d r e otti

where t denotes the variable on ~ and c(t) = H-l(t). The last equality sign is for reason of degree as n -1(,1) is a d+l dimensional space. eY)

If

D

the class e, as we did in

is strictly d-Levi convex then one constructs using the local non-extendable cohomology classes (Levi-problem)d •

+ The separation of points on Cd(D) when D is d-complete follows by an exact sequence argument from the vanishing of Hd+\D,!L d) and the existence of a .:'i-closed (d,d) form which h as non-vanishing integral over every cycle + c e Cd(D) (this is for instance the d-th exterior power of the exterior form of a Kahler metric on X).

(8)

c) Consider, as an application, the following situation. Let f : X ~ Y be ~ proper holomorphic map between the projective al.gebr- ed,c manifold X onto the projective al gebraic v ariety Y. Let A be a compact irreducibl e analytic subset of X and B a compact irreducible analytic subset of Y such that f-l(B) A and f: X - A ~ Y - B is an isomorphism.

=

Set a = dim t A, b = d im~ B and assume th at a > b. Then there eXists a neighborhood W of C~( A) in C~(X) which is holomorphically c o~vex. In fact

f

+

+

+

+

induces a map f b (Cb(X) - Cb(A» -7 (Cb (Y) - Cb(B». The cycles B, 2B, 3B,... are isol ated points of C~(Y) thus n~lnB has a holomorphically convex ne ighborhood U in C~(Y). One then verifies th at W = C~( A) U f-~(U) has the reqUired property. Now the eXistence of a neighborhoo l W of th ~t sort is certainly garantied by the existe nce of a nei ghborhood N(A) of A in X which i s strictly b-convex, by virtue of the previous theorem.

-129-

A. Andr eotti

Chapter

VIII.

Duality theorems on complex manifolds.

8.1 Preliminaries.

v

a) Cech homology ¥d t h value in a cosheaf. As in the case of poincare duality on topological manifolds, it is better unlerstood as a duality between cohomology and homology, so in the case of complex manifolds duality should be a pairing between cohomology and homology. v For this reason we develo n here Cech homolo gy theory as a preparation for duality theorems (c f. r 7 J which we follow in this exposition). Precosheaves (cf. [18J). A precosheaf on a topological sp ace X is a covariant functor from the category of open sets U c X to abelian groups, i.e. f?r every open set U c X an abelian group [) (U) is given and if V c U are open a homomorphism iV

u

d)

(V)

~ ~

(U)

is given, such that if WcVeU are open subsets of X we have, V W W i U0 i V= i U •

A precosheaf is called a coshe af if for every open set Jl. = ~ ui 1 i' I of .Q., the following

C

and every open covering ?( sequence is exact. where

00

is defined by

and

Set &(U) = continuous functions on U with compact support in U. Define the "extension maps" as the n atural inelusions S(V) c ~(U). We obtain in this way a precosheat and also a cosheaf. Example.

X

-130-

A . Andreotti

v

Cach homology. an open covering

7...{ = l Ui~ i~I

of

= t .,)(U),

S

Given a precosheaf .

uJ

iV

and

one defines the groups

X,

cS (U:t (j""'4) .... q and the homomorphisms

by

aq-l1Jt. ""J.

=L(_l)h

i 1 O• • • q)

for We h ave

aq-l

0

q = 0

g

={gi 0"·. i q}I

for all

~ C (U, $).' q

.

and we put d -1 =

q 2. 1

We thus get a complex with differential operator of degree

o. -1

and an augmentation eX : C o(

We define

Hq(U, $)

21, S)

the q-th homology group of this complex,

=

H (0(,,$) q

If

V

=~ Vj1, jEJ

$ (X) .

""7

Ker

o

Im Q

9-1 q

is a refinement of

the refinement function

1":

J

'7'

I

U,

to each choice of

th ere is aascc Lat ed a

homomorphism

r * : Cq('Zf; £ )

-'I

Cq ('2{, ~ )

for every

which is compatible with the differential operator,

q

2:.

O.

Thus '1"*

induces a homomorphism

V u. :

'3'

Hq( V, {)' )

'7'

Hq
which, as one verifies, is independent from the choice of the refinement function.

One can then d e f i n e H (X, q

<::

c)

= lim ~

U

-1 31-

A . Andreotti

Remark.

We may not wish to use all open subsets of

1?(.

the open sets of a particular class

X

but only

This can be don e pro-

vided 7J( is stable by finite intersections and contains arbitrarily fine coverings b)

A

f

X.

J = f cS (U),

cosheaf

i UX r '

0

<5 (U)

-?>

For example let I c,

X

the obvious way.

$ = Z~ (U),

U

r k(U; I)

.s (U)

and set

On every open set

=fk ( u, I).

De fine

compacti\

in

X,

For every cosheaf of t his sort and for

every locally finite covering Hq(U, ~.

X.

of sections of l:.,

We thus obtain a flabby cosheaf on

1 Vu}.'

if

U.

be a soft sheaf on

consider the group

ly supported in

is called flabby

$ (X)

1s injective for all open sets U

u?

iV

S)

Define the sheaf i

ui

U = } U i""f hI

=0

if

O • ~ .i q

of

X,

we have

q ~ 1.

by

= as the sheaf which has for fiber at a point !U

i

O

•••1

x

the direct sum of the fibers of the sheaf

at the same point. q

We have an exact sequence of sheaves ci ci _ ci 1 q l q (*) ••• ~ ! q ~ ~q-l -7 ••• ~

where

d.

~O

o

r.

~

0

-132-

A. Andreotti The exactness tollows homology of that complex e tticient in 1" x" Now the sheaves ~ are q supports in the sequence functor r k is exact on the sequence ••• .., C (1A , S q

) ..,

trom the tact that at each point x the is the homology of a simplex with cosoft.

Taking sections with compact (*) we get an exact sequence as the so ft sheaves. We thus get exactness of

Cq_l('U,$) -7 ••• -i> C ('l{ , c3 ) O

~ t3(X)~ 0

and this proves our contention. One can prove that any fi abby cosheaf is of the sort described above. We have tor cosheaves the corresponding s t at ement to the Leray theorem: Let ~

-u.

cS

= fUi ~ i"r

be a locally finite covering of

be a cosheaf on

X with the following property

z/' = 1. Vj"5 j ~J

for every open covering

=

H (U

q

X ~

i

(\ U . n J

we h ave

V- , ~)

=0

=

for every q> 0

then the natural homomorphism

Hq(1(,J)~ Hq (x, 3 ) is an isomorphism. covering ~ of this s ort will be call ed a Leray covering for the cosheaf $. As a consequence of the Leray theorem, we mention th e folloWing fact: A

~

S>, S

Z/ =Z Ui~

i6 I

, ~II

be cosheaves on

be a Leray covering for

ku

and let de

an d ,S ~

II.

homoraornhisms U = U.1. . , .J.. 0 q are given such th at the sequence kU hU '0' II(U ) ..,. 0 0 "7 S '(U) -;> J ( U) -7

SU'ppose th at for each open set h U'

J'

X

-133 -

A. Andreotti

is exact, and compatible with the extension maps. an exact homology seguence ~ H (X,

l

SI)

-7'

.s

(8.2.1).

~

h.

H (X, l

-7

k.

S) _

h.

H (X, l

,

S II)

Then one has

d

~

k.

HO(X, I) _;> Ho(X'.') -7 Ho(X' .,) II) -? 0 Note that exact sequences do not commute in gen eral with inverse limits, thus the Leray theorem is essential to replace here holomogy on the covering U by that of X. 8.2. eech homology on complex manifolds. The following lemma is a consequence of the Hahn - Banach theorem (e r, [47J). ~

A ~ B .!,. C be a seguence of locally convex topological vector spaces over and continuous linear maps u , v !i!!!. v 0 u = o , !!ll A'

t

__u

B'

tv 4-

~

C'

be the corresponding sequence of dugl spaces and transposed maps. Then t u tv o = 0 and we have a natural algebraic homomorphism t d:

If

v

u

:;;K.;;;.e::,r""":,,,_ 1m tv

....:;. Hom cont

(~ Im u '

is a topological homomorphism then

0

{3 = b ' + t v ( C' )

£). is an isomorphism. with

b ' o u = o.

For every k £ ~:r ~ k = b + u(A) with v(b) = 0, we define = bl(b). This does not de pend on the choice of the represent ative b ' and b and thus it defines a linear map Ker v t. For every c > 0 the set Sl- b /0 BI I b l (b) I <£. 1 1m 7 u is open in B and u(A) saturated. The r e f or e o( ~) is continuous. 1&) i t lifts to a Conversely given A ~ Hom cont ( ~ 1m u' continuous line ar map J,. I Ker v -3' t. By Hahn - Banach we can /\ B .:, t. Since extend A' to a cont i nuous line ar map .A'

~ ( k)

A'

I 1m u

= 0

t hen

1· I

1m u = 0

/'

th er e f or e J..' e Ker

t

u,

- 13 4 -

A. An d r e otti

Ker

and thus it d e f i n e s an element of extension

" - '"A' AI

0

fA'

to

" then).

B

Im

- ;.,

I

defines a linear map of

topolo Gical, t hen the map of

t

v (B)

v(B)

-v

u tv ;" 0

If A

on Ke r v.

int o

into

a:

!C.

"

..v

and.A

.AI

This shows th at

i., - ~ ,

: C

.~

in

Let set

X

0

and

T

be a complex manifold.

U c.c X

lei

linear

Therefore

Ker t u Im

One then verifies that

is

v

If

= tY(}J).

rCA.)

define the same element

I

Then

is. continuous and

by Hahn - Banach can be extended to a map y and continuous.

is another

I

tv

are e ach oth ers inverse. We consider only those open

which are open sets of holomorphy.

This collection

/1( of sets if stable by finite intersections and contains arbitrarily fine coverings. Let

&

be the structure she af of

X

and l et

~ be any

coherent sheaf. For every U" /1(1 we can find a surjection f}P :; ".f -7 0(1) from which we derive (by theorem B of H. Cartan and J.P. Serre) a surjection [(UI S P ) : . r(U,j=')~. The space

r (U, f) p)

Schwartz.

has the structure of a .pace of Frechet-

One verifies that there i s a unique str ucture of a

space of Freehet-Schwartz on [(U, 1=') continuous.

of the presentation

[28]).

e p ~ 'J'~

0

:(.(U) = Hom cont

for

VC

we have chosen.

is independent (See also

U ~ 7?( we set

For any

As tr ansposed

which makes the mapd.

Moreover this structure on r(u, ~)

s

(£(U, j(), lC)

of the restriction maps

U, V, U" 7J(which are continuous)

rU

v :[ (V,ll ..,

f(V,j='),

we get extension

maps

(1)

This fact can be proved but it is not obvious

I t is obvious for sU f fi c i e nt ly small of the definition of c oherence.

(c r ,

r 22]).

U's, ' indeed it is part

A Andreotti

and therefore

z":r.(U),

i\5

is a precosheaf.

Proposition (8.2.2). For any coherent sheaf 1, 5. ~ cosheaf. Moreover for any .fl l&. 7?( and any countable locally o-r'

U

finite covering

= f ui } iEI

Hq(U~~)

=°

!llh. U

of 0.,

.!2.!:

c.

0-;-'

JJ(,

we have

q) 0.

v

The augmented Cech conplex 0-> [(fI.,rr) -+ Co(1.(,C;) ~ Cl ('2(,"J') 7

~.

...

is a complex of Frechet spaces (as V is countable) and continuous maps. By theorem B this complex is acyclic, i.e. the sequence is exact. By the lemma (8.2.1) the dual sequence is exact. But that is the sequence of the augmented homology complex.

°~ "{(0-)

Co(V, T.) ~ Cl ("2(, T.) ~ ... In particular any countable locally finite covering "Z( c;»( is a Leray covering for the dual coshe aves ~. of coherent sheaves +-

T.

8.3. Duality between cohomology and homology. This duality results by comparison of the two (I)

cq-1Ctl, 1')

S

q::;

Cq(t{, T)

S

sequ~nces

'.F)

--$

cq+l('Z(,

'2(

77{ is a countable

(II)

where:F

is a coherent sheaf and

c.

finite covering of X. In (I) the spaces are spaces of frechet-Schwartz and the maps are continuous, in (II) the spaces are strong duals of spaces of Frechet-Schwartz and the maps as transposeds of the previous ones are continuous. Moreover each sequence is the dual of the other as the spaces of Frechet-Schwartz and their strong duals are reflexive spaces.

-T3"6 -

A. Andreotti By application of the duality lemma (8.2.1)

we obtain

Theorem (8.3.1)

II

( a)

is a topologic al homomorphism then

cS q

t.) = Hom Hq+l(X , "f) !!!2.

H (X, q

Moreover

(b)

d q_1

If

cont (H Hq (X ,

'I) •

lC).

are separated. -

is a topological homomorphism then

Hq(X, ~) = Hom cont H (X, j:'.) q_ 1

Moreover

q(X,1"),

and

(Hq(X,

'1"'.),

Hq(X,"J")

lC) •

are separated.

Note th at the assumption in (a) is equivalent to the separation of Hq+l(X,;t) and that the assumption in (b) is equivalent to the sepu-ation

0

f

H (X , q_ 1

certainly ,s a t i s f i ed if H _ (X,

q l

8.4. a) sheaf

'1.)

cr.) .

Hq+l(X,'T)

or, respectively,

are finite-dimensional.

v

eech homology and the functor Let

These conditions, are

T , Jj-

Jlom f"

be sheaves of

(:f ,f)

EXT.

e-- -modules

on

X.

Then the

is defined as the she af associated to

the preshe af U -r Hom

and

y

e)u

are coherent so is the sheaf

PI om6'

(1, f).

is a family of sup norta, we set

CJ

A shea! Jj HOM(X;.,t!)

r

110m <:1" , )•

HOM ~ (U,:r, =
for any exact sequence of she aves of ~ -modules

o -r 0--;> HOM (X;:t''', ·::)

is exact.

:y;:

I

~:r ~ 'J

--rHQ!.1

" -r 0

(X:".t, ;J')-:> HOM (X; 'rl,-J)-;> 0

Le.

-137 ~

A. Andreotti

We have the following facts:

i)

an injective sheaf is flabby ii) if {j is injective, for any sheaf of &- -modules #,om ~ (j:',j') is flabby iii) for any sheaf;- of C! -modules one can find an injective

T

ft

resolution (.) 0"7;' -;> o -?> ;;:1 ~ ;;;2 ~ ••• 1. e. the sequence is exact and every ~ i' i ~ 0, is an injective sheaf. These facts follow directly from the definitions and the possibility to imbed every module in an jnjective module. Applying the functor ;Yom t7 ('~r, to the sequence (.) we get a complex of sheaves and homomorphisms o-?>;Yome- (7,f.o~ ~;(om E/T';l)~;l/om& The

q-th

(T,fl2)?

cohomology group of this complex is denoted by

t

xt

q

tY

('7';').

This is a s he a f of &- -modules and one verifies it is independent of the choice of the resolution (.). q Moreover if 7 and are coherent then E xt (j'f.) is t1 a coherent sheaf. This can be seen as follows: Let

Y-

•••

7~ 2 ~d' 1

be a resolution of

':f

7'/0 -

'Y-;.

0

(**)

be locally free she aves(l).

ApplYing

(1) On any open set of holomorphy U e X we h ave such a resolution, as any coherent sheaf on X is the quotient sheaf on U of a free · sheaf by th e theorem of eoen (cf. [ 22J). However since here we need only this resolution locally one can invoque the theorem of syzygies of Hilbert to derive the eXistence in a sUffici ently s mall neighborhood of a given point x ~ X of a finite free resolution: o -e- 6' Pd -7 61 Pd-l..",..•• ",t ) PO -,> ,} -;> 0 where d s; diIDxX (c f. [.;1> • Co-

-138-

A Andr-eotti

the functor

;fom

t9

.,~)

(

o ~ ;rom & ( ;to';; ) ?' ,;t(om(9 (,J; whose

anoth ~r complex

we g e t

"

1'9)-. /r om& .?f'P

q:-th cohomology group is again

(. xt ~

('J' 'X ),

follows by a standard spectral sequence argument. construction q>O

-;> • • •

as it

By this

G xt~(T ,~)

is a coherent sheaf: ' q is locally free we get t xt (7, t;) = 0 Y --'I l' 19 7as we can take 000=j'Xl=N2= ••• =O.

In particular if if

(~2'})

::J-

Note th at in any case

because

the sequence

o '-71fom t9 is exact as the functor

('T'ff. ) .., ;t( o~ (-r .j'0) -; ;fom& (7 ,f1) ;t(om6l C; t , . ) is left exact.

Applying to the resolution we obtain a complex whose

(*)

q-th

EXT~ (X; 'J' q = 0

For

the functor

'J', .)

,J).

~(X;

1,X) = HeM 1>(X;'J',;).

The spectral sequence of the double complex

P, K q d

~ (X;

we have analogously to the previous case EXT

and

HOM

cohomology group is denoted by

= ['4>

(X;

K ={KP,q, d}, where

110m c/ o(;p'j q»

is induced by the ma 's of the resolution

(*)

and

(**),

leads to a spectral sequence connecting the global with the local extension functors

t. )

-- 'J EXTn (X; j-

( n = p+q)

where

b)

The connection of homology ani the functor

EXT

is

established by the following (8.4.1). ~ X be a complex .anifold of pure dimension n Let fL be the sheaf of germs of holomorphic n-forms on_X.

~

n.

-1 39-

A. An d reotti

For any open set of holomorphy ~

'T sa

the suffix ~.' (<1)

k

Let

one has

X

EX~

1*(U) =.

"J'

U C~ X and for any coherent

(U;T, ~'1. n )

denoting the family of compact supports. be a locally free sheaf on

X

T

and let

U be

an open set of holomorphy in X. The sheaf can be considered as the sheaf of germs of holomorphic sections of a holomorphic vector bundle E, .~ (E). Let E* denote the dual

=

bundle of E; if E is defined by the transition functions -llI • fg i j 5 then E* is defined by the transition functions sI t gij

/i

Let r,B(E) denote the sheaf of germs of C oJ' forms of type (r,s) with values in E and let ;XU,V (E*) denote the sheaf of germs of forms with distribution coefficients, of type (u, v), with value in E*. Since U is an open set of holomorphy, the sequence

o~f(u, SeE»~ ~ (U, AO'O(E»!. (U,JiO,l(E» ~ •••

°

••• ~ [(U,Ao,n(E» ~ is an exact aequence of s paces of Frechet-Schwartz and continuous maps. By the duality lemma (8.2.1) the dual sequence is also exact. But this is the sequence ~ (U,19 (E»' ':i k(u,1C n,n(E*»! k(U,7(n,n-l(E*»!

°

••• ~rk(U,1(n,o(E*» e: 0. Now for any vector bundle E*, denoting by ~n(E*) the sheaf of germs of holomorphic n-forms with values in E*, we have in the exact sequence of she aves

°

4

Jl.n(E*} ~ it n ' O( E* ) ~ ::tn,l(E*)-r

A so ft resolution 0 f .a. n( E*) • Therefore, since E* can be any holomorphic vector bundle and hence ~n(E*) any locally free sheaf, we get:

-140-

A . An d r e ott i

for any locally free s heaf

($)

'Y

Let

o

=

H~(U,f)

and

ElCrfl

(Uj

k

~ n

r

cont (f(U,CI(E», t) i f

=0

»: )

if

P ~ n

In fact the spectral sequence converging to the term

EP'2 q

=

as

jF

q ~ 0

if

Ext~ (7 ,f..) we get

(c{)

CHt q

H~(U;

(~,;(»

is locally free.

(p+q=s). Mo r e o v e r

'2

n o

But the spectral sequence is

ElC~(U;

T::i ) = EPzo •

This proves our contention.

( Y)

T,; ) has Thus EP '2 Q = 0

liZlCT:(U;

= ;fom{; ( } ,;,) is also locally free, hence by 0 P E = 0 i f P ~ n and

= l{ (Uj .1fom,9 Cj\~ ». z degener ated so th at we h ave E

r =n.

be locally free then we h ave

:;

(U;'1,~)

&('1, Jl. n )

= j2n(E*) = 1(om

if

{ Hom

ElC~ k

J

Suppose now

th ~t

The we h av e

ElC~

( Uj

"'J'

is coherent and

T ,f. ) =

0

if

j

locally free.

p < n

and an exact sequence

o -? EXT~(U;T'I)

->

H~(U; /10m()

C.z o. f) . .,. ~(Uj Jf'om CJ (£ '1';

In fact consider the double complex has for cohomology grouns the groups For an y -0

p

».

r k(U, :t,fc>mcd" (..&p ,fq)}

EXT*k(U;

'1 'J ).

the se quence of she aves

7.Jf omt9 «6 p' /I )~ ;:{om& ()S p' ;/o ) ~;;f om£9 c.); p';

is exact as ~ ~

KP , q =

resolution of

p

~

1) -T •••

is locally fr ee and provides an inject ive .,

:/(omc: ( ,,(;P'i-).

Taking cohomolo gy with respect

to t he differe ntial co ming from th e r esolution

(- )

we get

-141-

A. Andreotti

Efiq=H~(u,;fom§~p';»=o as

ffomt9(rPp'f)

is locally free.

EX~ (U;T

It follows that

EXTn~1 (U;~'Ji) is the

H~(U;

(0)

ql-n

if

i)

= 0

if

P < n

and

l-th cohomology group of the complex:

il om~ (rP 0' 1» ~ H~(U; .;rom &(06' 1 ,f)

-7

~ ~(u;..fom61(;: l'1)

-;> •••

In particular we get the exact sequence (1)

0

~ EXT~(U; j"f)

(8)

H~(u;;fomt' (~o'f) ~ H~(U;#om& rLl

7

'1»

Consider the exact sequence of spaces of Frechet-

Schwartz obtained by applying the functor

r

to

( •• )

••• .."r(U,06l)--- f(U,et'o)~ f(U,:(>-7 o . Exactness follows from the assumption th at of holomorphy.

By the duality lemma and

U is an open set (a. )

we get an exact

sequence (2)

0

~

(0),

Comparing

1"'.(u) = Given

T

U '7' EXT~(U;

f

H~(Ut.tom6'UO,jLn»~H~(u,ifom6'(.,tl'£\.n)

T.(U)-'!-

(1)

and

y.

T.;...)

for

!!ill

E-~·q = Hp(U

?1{

EXT~(U;j,nn) =

and

0 if

U

€;J1(.

q

I- n ,

We denote this precosheaf ~

We do not need to verify i t is a cosheaf,

(8. 4.2).

c

we obtain

coherent she avee , define a precosheaf by

Lemma

2{

(2)

EXT~(U;i',S1.n)

xt~( ·T.; ).

where

and

There exists a spectral sequence

,E xt~( 1, ~».

is a locally finite covering of

X.

- 142 -

A. An dr eotti ~.

Consider the doubl e complex

K-P,q = Cp(?(,J(omk( ·~';!q».

I f we t ake coho mology with

respect to the differential coming from

(*)

we get

7, )

I E-i' Cl = C ( 2{, { xt~( ~) and then taking homology with p v . respect to the Cech differenti dl we get

2Cl = H~
IE-P

Now we remark that !fomk('f, fq)

(U)

19

= [k(u;;fom

('T, fq». 7/ om (f If. ~

Thus we have in '!/fomk( '.f q) a fl abby cosheaf as is soft, ~ being injective. Th€refore takin g first homology q v with respect to the Cech di f f er ent i al we get

.r

"E-i' q = 0 if P ~ 0

Taking now the cohomology with respect to the differential coming from (.) we get th at the spectral sequence degenerates having as total cohomology the groups

El{T~(X;1\~ We apply this lemma to

$.

=S1.n •

).

Then

[xt~( 1', S\.. n)

=0

if

q ~ n and txt:('J'I ..n.n) ='1. as i t follo ws from lemma (8.4.1). Therefore we get the following Theerem (8.4.3). ~ X be a complex manifo~d of pure dimension n and let ~ be any coherent sheaf on X. !h!a. we have Hp(X,

'1.)

~

El{~-p(X;"'f, .n,n).

The combination of this theorem (8.4.3) with theorem (~,3.l) is what is usually called the "duality theorem"; in this form it is due to Malgrange and Serre (cf. [39], [4V], [44]).

- 14 3 -

A. Andreotti

Remarks .. 1.' If j! is locally free 1.e. the sheaf of germs of holomorphic sections of a holomorphic vector bundle E, &(E), then

or

Ext ll E*

=

(r',S1

n)

=0

if

II ~ 0

being the dual bundle of ElC~-p (X;&(E),J1,n)

so that theorem (8.4.3)

and

E xt~ ('1,-St n) =

=if o.m & (oj, fl, n) =Jl.n (E* ) E.

,

Therefore

= a:-p(X, Jl,n(E*)

gives

Hp(x,B (E)*) ~ ~-p(X, .n,n(E*». In particular if this case

E is a line bundle,

E* = E- l

and we get in

~(x,9 (E).) :: a:-p(X, An(E-l»., 2. If '1" =~ (E) is locally free and i f X is a compact manifold then the theorem (8.3.1) can be applied without restrictions and we get

HP(X,6'(E» = Hom (~-p(X" j1.n(E·», I) as the cohomology groups being finite-dimensional any linear map into ~ is automatically continuous.

8.5. Divisors and Riemann-Roch theorem. a) Let X be a connected complex maaifold of complex dimension n. The sheaf of germs of never vanishing holomorphic functions as a sheaf of multiplicative groups, can be considered as a subsheaf of the following sheaves:

&.

the sheaf ~*

of germs of non identically zero meromorphic

functions, the sheaf () o· functions.'

of germs of non identically zero homolorphic

-144-

A . Andreotti

&:

While the sheaf ~* is a sheaf of ~ultiplicative groups, the sheaf is a sheaf of multiplicative .onoids. We thus get two exact sequences of sheaves (1)

°~ tJ * ~?Jf.* -> fJ ~ °

(2)

0..,. f)*~ lJ*~f) -",0

o • where the sheaves)9 and 1). of the sequences. The shea! j)

are defined by the exactness is called the sheaf of germs of (meromorphic) divisors. The sheaf ~. is called the ~ of germs of holomorphic or positive divisors. The elements of HO(X,,v) are called meromorphic divisor!! on X and the elements of HO(X,)}.) are called holomorphic or positive divisors on X. An element

D

E

HO(X,0)

(resp..

D

E

HOex,D.»

is given on a

sUfficiently fine open covering "Z/ = [Ui~i~I of X by a collection ffil of meromorphic (resp. holomorphic) functions, not identically zero, on each Ui, and such that ~j : Ui {\ U j T Ie *

s,

j

E

I

is holomorphic and never zero. The functions fgijt are transition functions of a holomorphic 1 ,(\* line bundle {DJ and represent the element
=lD~} .fD",}-l.

The study of general (i.e. meromorphic) divisors can thus be reduced to the study of holomorphic divisors.

-145-

A. Andr eott i

Consider the sp ace HO(X,CJ(D» where C)(D) is the sheaf of germs of holomorphic sections of the bundle [D]. To every s HO(X,~(D» there corresponds a holomorphic divisor, this in its turn determines s up to multiplication by a global never vanishing holomorphic function. If X i s compact (or pseunoconc ave) then the divisor of a section s ~ HO(X,C/(D» determines the section up to multiplication by a non-zero constant. The set of positive divisors corresponding to elements of HO(X, ~) ( D » ) is call ed the linear system of D and is denoted by \DI. Its elements correspond one-to-one to the point of the projective sp ace (HO(X, f) ( D» - {of) / a:* • We attribute to

I DI

dim

the dimensi on of this sp ace IDI = dima: HO(X,[5(D»

Thus

- 1 •

The problems we want to mnsider is th e followin g: given on X a divisor D compute dim IDI. Note th at if we are able to

°

establish that dim IDI 2 then we h ave proved th at the holonorphf,c line bundle £ D'5 admits a holomorphic section di fferent from the O-section. b) Let us suppose now th at X is comprc t and d im~ = 1. Then every divisor D is given by a finit , sum D = 1 niPi wi th n G 1l and Pi being the divisors as s ,' ~ i at e d t o poi nt s i of X. A divisor is positive i f f al l n are /~ . The integer i ln is c alled the degree of the divisor D. i Assume first th at D = L niPi is a pos i t i ve divi s or . In this case the bun dle l D] has cert ainly a holomorphic s ect i on s t. 0, the section corres ponding to th e divisor itself. Gi ven s we h ave a sheaf ho momorphism tS ..., [J (D) gi ven by mul t i pl i cat i on by s. The homomorphism is cert ainl y inj ective and the quotient she af is concentr ated in the points Pi and gi ven at e ach po int Pi by th e vector sp ace v(n Pi ) of dimens i on n i r epr esenti, i ng t he Taylor expans ion at Pi of holomorphic f un ~tions, trunc ated at t he orier n -1. i

- 14 6 -

A . An d r e otti In other words we h ave an exact sequence of she aves:

° ~ !J

11.

s -+ & (D) ~

V (n i ' Pi)

-7

°.

This gives the exact cohomology sequence

°

1

as

H (X,

support.

~n.

HO(X,6' )

-?

HO(X,Cl (D»-7> a:

~ Hl(X,D )

-t'

Hl(X ,6) (D»

-¥'

u V(ni 'Pi»

"=

°

~

~

°

the sheaf having

O-dimensional

Therefore we get, in particular:

dim~ HO(X,lJ(D» - dima: Hl(X, 6'(D»

= deg(D) + dima: HO(x,D) -

Hl(X,~ ).

- dima: Now: HO(X,e )

= II:

_1 I1(X,19)

as

X

is comp act thus

dima: HO(X,6J)

the space of holomorphic differentials on

r(X,C)(D»

g(X)

of

X.

= HO(X, e (K-D»

associated to the sheaf

called the canonical bundle,

where

Jt 1.

and

l K~

is the line

We thus have the following formula dim

IDj

=

K1..

is

= i(D)

dima: HO(X, [) (K-D» D.

(Riemann - Roch t heorem)

deg (D) - g(x) + i(D)

This formula can be extended to any divisor positiv~

~

The bundle

is called the speciality index of the divisor

if not

Its dimension

X.

if finite-dimensional and, by duality

v HO(X, Jt l(_D»

(1)

°

is finite-dimensional and, by duality, ~ H (X, _<1 1 ),

is e al.Ledtthe genus

~undle

=1.

maintaining for

i(D)

D

= DO

- D (f'

even

the meaning of

dima: HO(X,e (K-D». Indeed let HO(X,6>(D",».

s

denote the section, corresponding to

We have an exact sequence of sheaves

0-> e(D) -') 19(D o ) -<>

where

DO'

=Z

niPi.

Jl V(Pi,ni)

~

°

D ~ of

-147-

A. Andreotti Theref'ore dima: HO(X, e(D

o

» - dima:

H (X,l'J (DO)

= dima:

HO(X,

e (D))

-

- dima: Hl(X,{)(D)) + deg(D,..,,) Thus using for deg

Do

the formula already e st ablished we get

= dim I D \

(Do) - g(X)

-i(D) + deg (Dc")'

From this the assertion follows as we have

deg(D)

= deg

(Do) -

- deg (D ,..). c)

we add a few remarks to the Riemann-Roch theorem in

dimension one. The genus

g(X)

b

X :

1(X)

of

tl

X

2g(X).

~

1(X)

Since

of the first Betti number

=

(1) 2g(X) b (X). 1 First we show that b 07

t

eguals

From the exact sequence

a:_.>ed~&l-"O HO(X, a:) ~ a:,

HO(X,

°-r HO(X,JI.. 1) --,) H1(X,

e,

&) ~

we get an injective map

a:) •

This map associates to every holomorphic

I-form its cohomology

class as a closed form. Now of'

HO(X, J1,1) and HO(X, ji 1) can .b e considered as subspaces 1(X, H Their int~rsection is reduced to [01. In fact

er. .

if'

a e HO(, Jl 1 ) ,

g

is a

« = as-

HO(X,..ii

Eo

C ~ fun ction on Thus

'3 a g

maxi.urn principle Theref'ore

t3

1

= ,

° i ..e.

b

~

1(X)

0.., a: ~

we get an injection

and if

ft

g

is (pluri-)harmonic.

2 dima: HO(X, ~

G It

=t3 +dg

where

by reason of bidegree we get

must ve constant hence

dima: H ( X, a:)

We show now that

X,

1)

2g(X).

Jtl)

a.

i.e.

bl(X)

-7

~

2g(X).

From the exact sequence

e ~ ?1'~ °

0.., Hl ( X, a:)

By the

= 0.

l(X,t9 H ) It Hl(X,

&).

-148-

A. And reot ti

In fact HO(X, an ~ e, HO(X, (}) ~ maximum principle). Therefore dima: Hl(X, '~ ) ~ 2 dima: Hl(X,b!)

e,

HO(xdf) ~

di m

)KI

=

deg

(K)

=

D =

° we get

g(X) - 1

If' we apply the same theorem to

0)

(by the

bl(X) ~ 2g(X).

i.e.

If we apply the Riemann-Roch theorem to (2)

a:

D = K we get

2g(X) - 2.

t

To do this we have to know th at K 1 comes from a divisor. Now if g( X) ~ 1 this follows from ( 2). If g(X) then for any positive divisor D we have dim I D\ = deg (D) + i(D). But then necessarily i (D) = 0. There eXi sts on X a r ational function with a sing],e pole of first order. This function extablishes an isomorphism of X onto the Riemann sph ere and on this manifold on e verifies immediately that K = -2p, p being a point of X. In p articular for a divisor D !1!h. deg(D) > 2g-2 we must h ave i(D)

=°

= O.

As an exercise one can show now th at every compact manifold X of complex dimension one admits a porjective imbedding i.e. is projective algebraic. Indeed if .0' ..• ' St is a basis of HO(X, (D» th e map X -? Ft(l:) d e f i ned by is holomorphic everywhere. If x ~ (sO(x), ••• , St(x» ( D) > 2g t hen one verifi es by means of the Rieman n-Roch t heorem th at t he map is one-to-one and biholomorphic.

e

d) Let X be a co nnec t ed c omp act man i fo ld of dima: X = 2. Let D be a holomorphic divisor on X which will be s upp os ed o f "multi pl ic ity" on e a t e ac h point and non- s i ngul ar . We h ave now an exact s equenc e 0"";' O ~ 8 ( D ) ....,. 6) ( D) / D where

s

is a section of

-7

°

f DJ co rres pon di ng to t he di vi s or

D.

- 149 -

A. Andreotti

We get an exact cohomology sequence: 0--7 HO(X,l) ) ~ HO(X, & (D» ~ HO(D,e (D)

~ ~(x, e) j.... !?-(x, ~ ~(x, CJ) ~

2(X, H

e (D» t9

I D)

~ Hl(D, () (D) I D)

-j'

-;>

(D» -7 0.

We have dim(CI)HOCD, 6l(D)/ D) - dim£!?-(D, D (D)/ D) ::: deg

~D~ID

- genus of

(D) + 1

dim~H2(x,~ ) ::: dim£ HO(X,Jl 2) ::: Pg(X) ::: geometric genus of X dim~Hl(X,e) ::: hO,l

dim~H2(x,r:9(D» ::: dima: HO(X, S(K-D»

where tKS

denotes the

bundle corresDonding to the sheaf of holomorphic 2-forms S!.2 ::: () (K). This dimension is denoted by i(D) and called the speciality index of D. By the same argument used for dimension one we get now dim

I DI

4

deg 1 Dr)D - genus (D) + (p (X) - ho,l) - i(D) + 1. g

.

This is Castelnuovo's theorem. The l i f f er ence of the left and right side is dimll: Hl(X,8(D» which is called the "superabundance". If X is Kahler then q::: hO,l ::: hl,O ::: dima: HO(X,s&l) ::: number of linearly independent holomorp hic l-forms, and P g (X) - q ::: p a (X) is called the arithmetic genus. The inequality can be extablished without the r estrictive as s umpt i ons we h ave made on D. In particular for a multiple lD of D we get the inequ ality dim 11DI .2 1 2 deg iDS}n - (l(~-l) deg ~ D}ln + 1 genus (D) - 1+1) + ( p (X) _hO,l) - i( l D) + 1. g

If

D is positi ve and

1

large enough and pos i t i ve , then i(lDL ::: 0.

- J:? O-

A. An d r eo tti Therefore if

deg~DsjD > 0,

dim IID)

grows like

Therefore

II

t.

1

2•

°

X contains a d i vis or 0 with des OSlo> ~ trans. degree (X) 2. 'One could s how th at X i s in this

1<.

=

case a projective algebraic v ariety as it is al ways the case f or a complex surface degree

2.

(c r ,

X

[21]).

with

7f ( X)

0

f transcendence

- 151-

A . Andreotti

Chapter

The

IX.

H. Lewy problem.

9.1. Preliminaries. To simplify the exposition we restrict ourselves to the space a: n although the results will hold on any complex manifold with only formal changes of notation. Let U be an open ,... ct in a: n and let p U ~ m be a . C <>D function. We define U+ = ~ Z

E-

fz

~

u-

=

S

= (z "

U

I p(z)

uI uI

°

2.

O}

p (z) s::. OS

p(z) =O}

and we will assume d p # on S, so that S is a smooth hypersurface. On U we consider the Dolbeault complex C* (U) =

f CO,O(U) ~ CO,l(U) ~ CO,2(U)

-;>

•••

S

where CO,s(U) denotes the space of C ~ forms on U of type (O,s) and where a is the exterior differentiation with respect to antiholomorphic coordinates. Analogously we define the spaces

Co,s(Vr,

resp.

Co,s(U-),

as

the spaces of those forms of type (O,S) on fl+, resp fl-, h aving CO" coefficients with all partial deriviatives continuous up to the boundary S. In this way we obtain two similar complexes, c*(u+) and C*(U~). Define

c' O,s(U)

= ~ ~ "" CO,s(U)

I 4>

= pd. + ~ p/l (3 (3

We have

atj0s(U)c;J°,s+l(u),

;j*(U) = .lL S

U O,s(U)

E..

,d ..

CO,s-l(U)

Co,s(U),

s.

therefore

is a sUbcomplex of

C*(U)

and indeed a

di ffernetial ideal. in a similar way one defines the sUbcomplexes <1*(U:) of C*(U:).

-152-

A. Andreotti

Finally one defines the quotient complex O*(S) = tQO,O(S) by the exact sequence

o -:p j"* ( U) ~ c* ( U) -;> Q*(5) --? O. The quotient complex is denoted by Q* (S) as each one of its spaces is concentrated on S. The operator 55 is by definition induced by the operator a- on C* (U) and (J * (U). One could define the quotient complex also for the inclusions /1* + * + ' CJ (U-) c C (U-) but one obtains in this W;Jy the same complex Q* (5) as only the values on S of the coefficients of the forms considered are of importance. We thus can consider four types of cohomology groups H*(U)

the cohomology of

C* (U)

H*(U!)

the cohomology of

c*(u!)

H*(S)

the cohomology of

Q* (5).

Note tbat while the cohomology H*(U) is the cohomology of U the same is not true for H*(U!). with values in the sheaf

e

Remark. We have .{jO,O(U) =~t>o,o : U~

a:

I ~ °'°15 = °5 thus

QO,o(S)

°°

represents the space of C

°

°

u

as

In general calling the im ~ge of a form of

CO,s(U)

on

QO,s(S) the tr ace of that form on S, we can st ate th at for u E Q0 , s the condi tion u = is a nec ess ary condition for

as

°

- 15 3-

A Andreotti

-

u to be the trace on S of a form U e CO,s(u+) with
°

Thus i.e. Same argulllent :Eor Example. 2 Take U = t , on a: 2 • Let

p

zl

= Xl

+ iX 2 ,

2 2) 4 - (x + x

5 X

= 2i

z2

U-.

= x3

+ iX

4

(Z2-- Z2) - Izl l

as co ordinates 2

S is the product of the paraboloid x = Xl2 + x 22 in 4 m 3 where Xl' x 2 ' x 4 are coordinates, by the x 3-axis. 1 form a basis At each point . dZI and ap = - 2i dZ2 - ~ d~ for the (O,l)-forms. Thus we have Then

QO'O(S)

=

QO,l(S)

"" C"'(S)

QO,2(S)

=

Q*(S) -

- tC"(S)

C""(S)

=

C 6' functions on

/\

dZl

o,

Thus

i'

To compute

as'

~ C (S)

/I

dZI ~

S

° r.

by its definition, we h ave to do the following

u

- given u G C~(S) extend (in any way) to a C ~ function on a: 2 • Due to the shape of S we may assume U in:iependent of the variable X ; 4 - compute

au;

-154-

A . An dreotti

- compute

_'1J

au;

_1.1\

(1U

=

-

~

zl

dZ

+

l

»

tJ

as

N

u

aZl

dZ

z2

'\)

- "restrict" 0,1 (1: 2

c\i

'V

~

N

-

dZi.

~

(2i~p + 2iZi.dz

o z2

the form thus obtained to to get

is .independent

0

f

x

and

4

can be taken as coordin ates on In conclusion the complex on

lR

2

= xl

~

3 where

Q* (5)

+ 1X 2 '

5

l)

i.e.

compute modulo

can be taken

~,

5. is isomorphic to the complex

and

x

3

are coordinates

where

L The operator

L

=~ -

iZ l ~

oZl

0%3

is the operator of

H. Lewy,

of

r 371.

9.2.

Mayer-Vietrois sequence. A C <::P function on U is called fiat on 5 if i t vanishes on 5 with all of its partial de r i v at i v es . 5et a)

'1 0 , s ( U) = N on

5

S.

€

CO,s(U)

I

all coefficients of eP

are flat

-105-

A , Andreotti

a 1°'s(u)

We have

'j O,s+l(U)

c,

•

is another aUbcomplex of

u

~*

(U).

C (U)

The quotient complex

concentrated on coe fficients

S

and in fact a subcomplex of

• C (S)

=C

tI

•

is

(U)/'J (U)

and is obtained by restricting to

f the

0

0 , 8 ( U) '1'. (u) = JL'1 a

therefore

C rjJ forms on

S

the

U.

We have Co,a(S)

=t

!:

Eo

Co,o(S)

a:l( •••~tt.8

c" (u+)

0-7 C...(U) -p

act

l···as

for all It C· (U-) ~

(x) dZa: ••• dZ a i. a

(a'l •••O:s) •

c" (S)

.....:p

is an exact sequence

°

from which we get a cohomology sequence ° ~ HO,o(U) 7' HO,o(U+) It HO,o(U-)

-» HO,o(C· (S» ~

(1)

To connect the cohomology groups

= Ho,s(Q·(S»

HO,s(S)

°

J-/* v (U)/3=' * (U)

-?>

i.e.

*

O~tJ (U)/

o~

j'o,o(U)

.o

-:r

~. (Q)

, °(U) Let

a

~

u

€:

-"?

C* (U)/'1- * (U) -7 Q* (S)

For any choi ce of

-i>

-;>

°

0.

U and

S

the seguence

~O,l(U)

is exact.

1 o , 1 ( U)

-d O,O(U)

Assume th .\t the coefficients of .}pqllS =0.

with the groups

we may use the following exact sequence

'Y- * (U).....:pC* (S). '--i' Q* (S)

(9.2.1).

~

HO,s(C·(S»

ju

are fl at on

S.

Then

-156-

A..Andreotti

=pet 2

This means that q'l u

=f

2 Q'2.

thus

COiO(U)

thus

'

;1' Q'21 s = 0 . for some

Continuing in this

3

C(

E;

Co,O(U),

we see that

w~

u

d 0,°1'1°,°.

we have exactness at

«(3 )

E:

2

But then

= pc( 3

o: 2

a

for some

thus

= p3dy

u

must be flat on

To treat the general case we will make use

0

5

i.e.

f the

following fact Given on

S

a seguence

there exists a ok F

d pk

I5 =

C

f

f

O' func tion

d'

k

for

fk

f

l, 2 F 2!!

= 0,

2!

, U

C CI' func tions,

such that

1, 2, ..•

This can be derived as a particular case from the Whitney extension theorem. Let

Also direct proofs are available.(l)

e {jo,s(U),

f

s 21.

Then

= I'd. + ap

for some

/I (:,

and (3. Using the above remark, we can find a form CO,s-l(U) such that

r> 1

E.

131 1

5

akp

_l[

(*)

= 131 5 =0

ap k

S Thus we can write

k = 1, 2, 3, ...

for

a p" 13 1)

= (f -

f

as the coefficient of

Jp )\ 13 1

f -

satisfies the conditions (*).

= pQ'l

(1)

+

3p

/I

6

with

1

13 1

+

~u

"5p

/I ( \

vanish on

Let now

assume that the coefficients of u

f

= pGJ. S

are fl at on

a p " 13 1

while (31

u ~ ~ O t s( U)

satisfying

+

S.

and Write

(*).

In particular it follows from this lemm a th at the space

CO,O(S)

can be identified with the space

power series in

p

with

C

~

[

coefficients on

(S) ~ p } S,

of formal

-157-

A .An dne ntti

Then u - a(p(\)

Set y 1

=« i -

= peal 3~1·

- 81\).

By the assumption we get

Il=P~2+~f'/I(32 with u -

Set

-

2

o(I'f31 + tf' 132 )

r 2 =q2

--(2 = PQ'3 +

ap

- t~ e 2· /I

13 2

=p

2

satisfying (*).

~e

.1\

'(11s

= °thus

Then

-

«((2 - h~2)·

By the assumption

11 3 with (33 satisfying

a t'/\ (*).

y 2( = S

° thus

Then

Proceeding in this way we construct a formal power series in om

c;

=

wi th fl m satisfying

u -

_ m+l pk

a (~ i t 13 k) 1

1. 1

ilt'3 m

(*)

=P

and with the property that m+l

Y m+l

Using the remark made of the beginning, we can find f eo OO,8-1(U) such that () k'

T ~f'

t

Is =

e

k~

~

=0

ap

J.f

for

° s-l (U)

Set v = f then (l f f: -~ , has flat coefficients on S. ~ 0,8(U)/):'0,8(U).

k = 0, 1, 2,

. and we have that u This proves exactness at

(jV

This lemma tells us that the cohomology of vJ-l * (U)/'T * (U) is zero in any dimension and therefore the cohomology sequence of (2) gives a set of isomorphism HO,s(O*(S»

~ HO's(~*(S)

= HO,s(S).

-158-

A. Andreotti

Introducing this result in the sequence sequence

(1)

we obtain an exact

which is called the Mayer-Vietoris sequence for

U and

S.

b) The previous considerations can be repe ated replacing the space of C

-,. H°itl(U)....". H°itl(U+) • H°itl(U-)

--i'

H°itl(S) -

(Mayer-Vietoris sequence with compact supports).

9.3.

Bochner theorem. Let X be any (n-2)-complete connected manifold of complex dimension n ~ 2, for instance a Stein manifold. Let S be any connected closed cot' hypersurface i n X such that X-S f- U f+ is the union of two connected open sets f- and f+ t of which say f- is relatively compact.

=

Theorem f

2a s

(9.3.1).

the trace on in f-. ~.

Under t he above assumptions, any C ~ function the compatibility condition asf = is £f...1: C '" function on X- which is holomornhic

s atisfYin~ S

°

The Mayer-Vietoris sequence with comp act supports gives

- 1 59 -

A.... Andrec.tH

Now

HOkO(X)

= HOkO(X+) = °

as X and X+ are connected and and not compaet. HOkO(X-) as X- ~s · compact. By the duality Moreover HO'k O(X- ) theorem we get HOkl(X) ~ Hom cont (Hn-l(X, n),~)

=

=°

by the assumption th at Since

Hn-l(X, Jl. n )

the maps Hence

and

n

~

2

Hn(X, J\,n)

8 are topological

Hn _l (X,.n~) ~ Hom cont ~

and

X is

(n-2)-complete.

are finite-dimens io·nal, i,

homomorphis~.

(Hn-l(X, J1. n ) , a:)

nn Hom (rt..n-), (X,JL),

~).

And Here E* is the dual bundle of t he canonical bundle Hence the restriction map

E.

is surjective.

=

FOr X ~n this t heorem is due to Bochner, Fichera, Martinelli (cf. [16,1 [24,1 (41]). Note th :d no assumption is ma le on the shape or convexity of S.

9.4.

Riemann-Hilbert and Cauchy problem. These problems are ge ner ali zations of a pr obl em concerning holomorphic functions in several v ari ables f irst considered and solved by Hans Lewy (cf. [37~ [381). As s um e th at U is an open s et o f holomorphy in ~n. Then ·t he Mayer-Vi etoris seq uence (with closed su pnorts) sp lits in the s hort ex act se quences o ·~ HO,O(U) ~ HO,O(U+) i HO, O(U- ) ---'f HO, O(S ) ...".

°

•

1 .6 0~

A. An d r e otti

This shows th :3.t (a~

Every cohomology class on S can be wr i t t e n as a jump of a cohomology class on U+ and a cohomology class on U-

i.e. the so-called Riemann-Hilbert problem is always

solvable for an open set Moreover if

°

s >U

.0

f hoLomo r phy ,

the solution is unique while for

s = it is determined up to addition to the functions on U+ and U- of a global holomorphic function on U. (b)

Let us agree to say th at the Cauchy-problem is solvable from the side of U+ if HO,s(U+) ~ HO,s(S) is surjective.

Then we reali ze that if

U is an open set of

holomorphy the solvability of Cauchy-problem for is equivalent to the vanishing theorem HO,s(U-)

=

s >

°

°

and in this case the solution is uni que. If

s =

°

the Cauchy-problem is solvable if and only if HO,O(U) ~ HO,O(U-)

is surjective i.e., loosely speaking, contained in the envelope of holomorphy solution will be unique if for instance, connected.

is The

Example. Let us consider the situation of the example given at the end of section 9.1.' Here U = 1 x Xl2 + x 22 ~ is an elementary con4 vex set while U is the closure of the complement in ~2. The boundary S of U+ is strongly Levi-convex. Let a E S and let J1 be any neighborhood of a which is also a domain of

+"

holomorphy. Writing for fI.,.ft+,.n. - and SJ1, = nOS Vietoris sequence, we get the exact sequence

the Mayer-

-161-

A_Andreotti

We have i) HO,l(Jl)

= 0,

HO,2(.Q)

= 0,

n

as

is a domain of

holomorphy. 11)

Ho,l(Jl+) = 0. This fact is a consequence of the regularity theorem of Kohn and Nirenberg (cf. next section). It would be desirable to obtain a direct proof in this special case.

iii)

Since U- has a pseudoconcave boundary one realizes that for every point a €o S we can find a fundamental sequence of neighborhoods Jt. of a, which are domains of holo. morphy and such that HO ,0 (-J\)

~

HO,

°

(Jl-)

is surjective (and an isomorphism). Making use of this information the sequence of Mayer-Vietoris gives us the following isomorphism

HO,l(JC) ~ Ho,l(s.rJ The first tells us that given any

ell> functions on

°

s..Jl,

exists a satisfying the compatibility condition asu ~ ~t~h~e~r~e-=~~~=­ C 1» function u !.!! Jt+ which is holomorphic in Jl+ and such

!.!l!l

11 Is

(use is made of the assumption

=

u.

iii)).

-162-

A . A n d reotti

In connection with the second isomorphism (which is valid even if the assumption iii) is not satisfied) we remark th at iv)

01

1\ and in fact dim H01 ' (A ) = '-, provided.J~ lC is sufficiently small. In fact by a loc al change of holomorphic coordinates at a we may assume th at s..n. is strongly elementary convex (1.4, exerci se 2). Then the statement is a st~aightforward consequence of lemma (7. 2.2).

I.

H' (If)

0

An immediate consequence of this fact and the isomorphism established above is the following theorem first proved by H. Lewy [37]. Given on Lu for

any

!!

lR

3 the equation

~ oZl

point

iz

=

..L1!

1 aX3

a

~

lR

f

3 we can find a fundamental sequence of

neighborhcjods wiJ such th at for infinitely many f equation does not admi t any solution u E. C"'(w y ) .

Eo C"'( w >1)

9.5.

Cauchy-problem as a vanishing theorem for coho mology. Let us now consider the Levi form restricted to th e analytic tangent plane of the hypersurface S. Using the methods of proof of the vanishing theorem (7.1.1) and the regularization theorem of Kohn and Nirenberg, (s ee (3~, [321 and 5J) we obtain the following result

r

Theorem (9.5.1). For any point Zo ~ S at which the Levi form ~ p positive and q negative eigenvalues on the analytic tangent plane to S at zo' we can find a fundamental sequence of neighborhoods such that

l

tl zo'

U,)'v~N

all domains of holomorphY,

> n-q-l

8

or [

o

< s < p

•

-16 3-

A. Andreotti

Analogously one can find a similar fundamental seguence of neighborhoods ~ such that

tU " veW

s

)

n-p-l

2!

°

[

< s < q Moreover, if P > 0, we can select the seguence ~ U } v Way that the restriction

in such a

HO,O(U-J) ~ HO'O(U~)

is surjective, Le. U'l! is in the "envelope of holomorphY" of + U)I' AnalogouslY, if q). 0, we can select the seguence i. U'" 5 !B. such a w!Y that the restriction HO,O(U

is surjective, Le.

lI)

~ HO'O(U~)

is in the "envelope of holomorphY" of U~,

Uy

According to the remarks made in the previous sections this theorem tells us when locally the Cauchy problem for cohomology classes is solvable. Two special cases will serve as an illustration Case 1.

Assume th at the Levi-form is non de ge ner a t e with

°

<,

p < q

=

n-l-p

Marking only the cohomology groups which are ( possible) different from zero, the situation is illustr ated by the follo win g picture

u+

HO,o(U+)

HO,P(U+)

s

HO,o(S)

HO'P(S)

6151

pI

-

U

ls

HO,o(U-)

!S

HO,q(S)

~~51->

Ho,q(U-)

Moreover one can sho w by applic ation of Lemm a (7.2.2) as we did in th e generaliz ation 0 f Levi-problem to coho mology ~l2<Sses

-16~-

A. Andreotti

(point l:,) in the proof) that the cohomology groups we have marked are all infinite-dimensional. Case 2. and

Assume that the Levi-form is non-degenerate,

°

<:

=

P

n

is odd

= Tn-l

q

then with the same conventions the situation is illustrated by the follewing picture

u+

HO,O(U+)

HO,P(U+)

P

J,

HO,P(S) ~ HO,P(U+)

HO'O(S)

S

° Again the groups marked in the picture are all infinite-dimensional In this case in dimension p the Cauchy problem is not solvable from either side, only the Riemann-Hilbert problem is solvable in that dimension. Remark. In both cases in dimensions p and q we are in the presence of systems of first order partial differential equations Lu = f on S which for infinitely many C rP functions f satisfying the integrability conditions have no solution u of class C-:

9.6.

Non-validity of Poincare lemma for the complex f R~ (S),

On S

we can consider for any

the or es he a f

as .

the sheaf

QO,s

•

defined by

JL _ QO's(Jl.)

We thus get a complex of she aves operator

s

JS

QO'O IN"

os

~

QO,l 1.'V'

s,* = 2J -4

11 rl's QO,2

""'"

with differential

~

...

It is natural to ask of t his se quence of she aves is exact. Indeed in th at case
is

f = 0.

-165-

A

Andnaott,'

The answer to this question is in general negative as does show the following Theorem (9.6.1). ~ S be a locally closed hypersurface in a: n and let z o e S be a point at which t he Levi form on the . analytic tangent space of S ~ Zo . ~ p positive and q negative eigenvalues and is non-degenerate (so that p+q = n-l). Then in the complex

the Poincar~ lemma is not valid in dimensions holds in any other dimension.

p

From theorem (9.5.1) and the Mayer-Vietoris sequence we deduce that there exists a fundamental sequence of neighborhoods W v of Zo in S, v = 1, 2, 3, ••• , such that Ho,s(4J y ) = if s f. p,q. Thus the Poincare lemma for is valid in dimensions different from p and q. We have to show that this is no longer true in dimensions p and q. Let w be any neighborhood of Zo in S. First we remark that

~.

°

as

QO,p(w) ~ C"'(w)a:(p)

where of

= (n;l).

Indeed we can select a baSis for the space n (0,1) forms in a neighborhood Jl, of Zo in a: , Jl. n S = w, (p)

of the form YJ 1 t

•

0

•

,'I n-l'

ap'

Then a

£.

As such

QO,p(u)

C"'(w)

°loooop

5•

has a natural structure of a Frechet space.

- 166-

A . An dr-eotti

Given w we can select a fundamental sequence of neighborhoods ()J C co,v = 1,2,3, ••• , of Zo in S such that aP(",-,J) # as we have remarked in section 9.5. Set for LV = w or Wv'

°

Since d is a differential operator, it is continuous for the S topology of Q*(w). Therefore ZP(w) is a closed subspace of QO'p(w) and therefore a Frechet space. On its turn BP(w) QO,p-1(~) / Zp-l(w) inherits a quotient structure of a Frechet space. Consider for every v the following set of continuous maps

=

Zp(to.!)

r ~ZP(Wy)

ti

'\!

BP( w)

where Set

i v is the natural injection and E"\J

= 1: (0, /3)

E Zp(u;» <. BP(w)

I

r

v

the restriction map.

r)cd = i )M

1.

Then E vasa closed subspace 0 t ZpC.. : ) '" BP (w 11 ) has the structure of a Frechet space and we c an co mpl ete t !1e :'1r evi ous maps with the following commut at Lve diagr am ZPCu.)

r..;

_ ..- T ZPC w y.)

1'i

I'j

E~

- 7 BPC"",v) (j

1 With j an d d cont inuous. We h ave j C E ~ ) = r: ivC BP ( ~v» ' Thus t his s pac e is a con tinuous i motGe b y .-J. l ine ar map of a Frec het sp ace. By the BCUlach open mappin g theorem we mnst h ave on e o f th ese t wo ~ r o pe r t ie s

-167-

A_ A n d r eotti

1)

r-} ivCBP(UJ~»

either

= Zp(iA)

r~l iv(BP(w v) is of first category in zP( w). Now as we have remarked in section 9.5. we can construct for each v an e'Lemerrt ~\I E> Zp(w) such that r~ (4)v) ¢. i /BPC,l 1y»' i1)

or else

This rules out possibility

i)

for ev 7ry v.

Hence

is of first category. Therefore there exists an element every v i.e.

g

€

ZP(w)

such that for

such that the equation

3S }JV

=

g

cannot be solved in w)) although the integrability condition ~S g = 0 is satisfied in the whole of fA) • This shows that p nincar~ lemma cannot hold in dimension dimension q the argument is the same.

p.

For

Elcample. In the particular case 0 f the Lewy operator Lu = f in lR 3 it follo ws that given a €:iR 3 there exists a neighborhood lV of a and f
=

9.7.

Global results. We illustrate the type of global results one can obtain by the following example. For the proofs we refer to [5J. Let X be a compact connected manifold of complex dimension n , Let p: X ~ lR be a C 0' function and assume that

s

= fx ..

X

is a smooth hypersurface

I p (x) = Os

(dp # 0 on

S) dividing

X into the

two regions X-

= ~x

E

X

I

p(x) ~

o I,

X+

=[x

E

X

I fl (x)

:: 0

S•

-168-

A. Andreotti

We assume that the Levi form of f restricted to the analytic tangent plane to S is nowhere degener~te and has p positive and q negative eigenvalues (p+q = n-l). Then one can prove that dim dim

t

t

HO,s(X-) < ~ HO,q(X-)

=

if

s ~ q

<:p

and similarly

HO,P(X+) = Of! • t Let us agree to say that the global Riemann-Hilbert nroblem is almost alWayS solvable in dimension r i f f dim

has finite-dimensional kernel and cokernel. Similarly we agree to say th at the Cauchy-problem is almost alWayS solvable in dimension r from the side X+ iff

has finite-dimensional kernel and cokernel.

A straightforward appl i c at i on of the May er - Vie t or i s sequence in connection with the description of t he groups Ho , r (X! ) given above le ads to the follo wing conclusion:

-169-

A . Andreotti

!! p # q the Cauchy problem is of interest(l)

only in dimen~ p from the side X+ and in dimension q from the side n-l (n !!!.!!ll X- and it 18 almost alWayS solvable. If p = q = 2 be odd) then the Cauchy problem is of interest in dimension p bu' not almost alWayS solvable on either side, while the Riemann-Hilbert problem is of interest in dimension p ~ almost alWayS solvable. It is worth noticing th at these considerations can be extended to more general complexes of partial differential operators~ The operator ~s was first introduced in 'a different form in [35]. In this ' exposition we have followed [41 and [71.

(1)

In the sense that it leads to maps between infinitedimensional spaces.

-170 -

A . Andreotti

BIBLIOGRAPHY. (1)

Andreotti, A.:

[2]

Andreotti, A.

Theoremes de dependance algebrique sur les espaces complexes pseudo-concaves. Bull. Soc. Math. France 91 (1963), 1-38. . and Grauert, H.: Theoremes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. France

90

(1962),

193-259.

[3J

Andreotti, A.

and Grauert, H.: Algebraische Korper von automorphen Funktionen. Nachr. Ak. Wiss. Gottingen (1961), 39-48.

[41

Andreotti, A.

and Hill, C.D.: E.E.Levi convexity and the Hans Lewy problem. Part I: Reduction to vanishing theorems. Ann. Sc. Norm. Sup. Pisa 26 (1972), 325-363.

[5] . Alilireott:t,

-.A.

and Hill, C.D.: E.E. Levi convexity and the Hans Lewy problem. Part II: Vanishing theorems. Ann. Sc. Norm. Sup. Pisa 26

(1972), [61 Andreotti, A.

747-806.

and Huckleberry, A.: Pseudoconc ave Lie groups. Compositio Mathematica 25 (1972) ,

109-115. [7 J Andreotti, A.

and Kas, A. : Duality theorems for complex spaces. Ann. Sc. Norm. Sup. Pisa, to appear.

[8]

Andreotti, A.

and Norgu et, F.: Probleme de Levi et convexite holomorphe pour les classes de cohomologie. Ann. Sc. · Norm. Sup. Pis a, s. 3, 20 (1966), 197-241.

[91

Andreotti, A.

and Norguet, F.: La convexite holomorphe dans l'espace analytique des cycles d'une variete algebrique. Ann . Sc. Norm. Sup. Pisa, s.

3, 21 (1967),

31-82.

-17 1-

A. Andreotti

[10]

Andreotti, A.

and Yum-Tong Siu: Projective embedding ~ pseudoconcave spaces. Ann. Sc. Nor m. Sup. Pisa, s. 3, 24 (1970), 231-278.

[1~

Andreotti, A.

and stoll, W.: Analytic and algebraic dependence of meromorphic functions. ~ecture notes in Mathematics 234. Springel Berlin, 1971.

[12J

Andreotti, A.

and Tomassini, G.: Some remarks on pseudoconcave manifolds. Essays in topology and related topics. M~moires dedies G. de Rham, 1970, 8.5-104.

a

[13J

Andreotti, A.

and Vesentini, E.: Sopra un teorema di Kodaira. Ann. Sc. Norm. Sup. Pisa, s. 3, ~.5 (1961), 283-309.

[14]

Andreotti, A.

and Vesentini, E.: Car1eman estim ates for the Laplace-Beltrami equation on complex manifolds. Publications Mathematiques I.H.E.S. 2.5 (196.5), 81- 130 .

[1.5]

Barth, W.:

Der Abstand von einer algebraischen Mannigfaltigkeit im komp1ex-projectiven Raum. Math. Ann. 187 (1970), 1.50-162.

[161

Bochner, S.:

Analytic and meromorphic continuation by means of Green's formula. Ann. of Math. 44

(1943), 6.52-673.

[17]

Borel, A.:

Pseudo-concaVite et groupes arithm~tiques. Essays in topology and related topics. Memoires dedies a G. de Rham, 1970, 70-84.

118]

Bredon, G.:

Sheaf theory. Me. Graw-Hill series i n higher mathematics, 1967.

I

-172-

A. Andreotti

[19J

Cartan, Ii.:

Quotient d t un espace analytique par un groupe d1automorphismes. Algebraic geometry and topology. A symposium in honor of S. Lefschetz. Princeton U.P. 1957, 90-102;

[20J

Cartan, H.:

Quotients of complex analytic spaces, Contributions to function theory. Tata Institute of Fundamental Research, BombaY, 1960, 1-15.

(21)

ChOW, W.L.

and Kodaira, K.: On analytic surfaces with two independent meromorphic runc t.Lona,' Proc. Nat. Acad. Sci. U.S.A.

[22J

Coen, S·:

38 (1952), 319-325.

Sul rango dei fasci coerenti, Boll. U.M.I. 22

(1967),

373-382.

[23]

Docquier, F. '

and Grauert, H.: Levisches Problem und Rungescher Satz fur Teilgebiete Steinscher Mannigfaltigkeiten. Math. Ann. 140 (1960), 94-123.

[24]

Fichera, G.:

Caratterizzazione della traccia, su1la frontiera di un campo, di una funzione ana1itica di pi~ variabi1i comp1esse. At t i Accad. Naz. Lincei. Rend. C1. Sci. Fia. Mat. Nat. 22 (1957), 706-715.

[25J

Godement, R.:

Topo1ogi e algebrique et theorie des faisceaux. Hermann, Paris, 1958.

[26)

Grauert, H.:

On Levi's problem and t he imbedding of realanalytic manifolds. Ann. of Math 68 (1858) 460-472.

[27J

Grauert, H. l

Uber Modifikationen und exzeptionel1e analytische Mengen. Math. Ann. 14 6 (1962) , 331- 368 .

- 1 7 3-

A. An d r e otti

[28]

Gunning, R. :

and Rossi, H.: Analytic Functions of Several Complex Variables. Prentice-Hall Inc. Englewood Cl if f s , N. J . , 1965.

f29l

Herve, M.:

Several complex variables. Local theory. Tata Institute of Fund amental Research, Bombay, Oxford U,.P., 1963.

[3OJ

Hirzebruch, F.:

Topological metho ds in al gebraic geometry. Springer, Berlin, 1966.

[31J

Hormander, L.:

(32)

Hormander, L.:

An introduction to complex analysis in several variabl es. Van Nostrand, Princeton, N.J., 1966. L2 estimates and existence theorems for the

a

operator .

Ac t a Mat h . 113 (1965),

89-152.

[33J

Kodaira, K. :

On Kahler varieties of restricted type (an intrinsic characterization of algebraic varieties). Ann. of Math. 60 (1954), 28-48.

[34J

Kohn, J.J. :

and Nirenberg, L.: Non-coercive boundary value problems. Comm. Pure Appl. Math. 18 (1965),

443- 492 .

[35J

Kohn, J.J.

and Rossi, H.: On the extension of ho1omorphic functions from the boundary of a complex manifold. Ann. of Math. 81 (1965), 451-472.

[361

Levi, E.E.:

Studii sui punti singolari essenziali delle funzioni analitiche di due 0 piu variabi1i. Opere, Cremonese, Roma, 1958,

187-213.

-174-

A. Andreotti

1"37)

Lewy, H.:

An example 0 f a smooth linear partial differential equation without solution. Ann. of Math. 66(1957), 155-158.

[381

Lewy, H.:

On hulls of holomorphy. Co mm. Pure Appl. l1ath. 13 (1960) ', 587-591.

[39)

Malgrange , B.:

[40]

Ku1hmann, N.:

Uber holomorphe Abbildungen komplexer Raume. Archiv Math. 15 (1964), 81-90.

[411

Martinelli, E.:

Sopra un teorema di F. Severi nella teoria delle funzioni analitiche di piu vari abili complesse. Rend. Mat. e Appl. 20 (1961), 81-96.

[42J

Narasimhan, R.:

Introduction to the theory of analytic spaces. Lecture notes in Mathematics 25. Springer, Berlin, 19 66.

[431

Narasimhan, R.:

Several complex variabl es. Press, 1971.

[441

Ramis, J.P.

and Ruget, G.: Complexe dualisant et theoremes de dUalite en geometrie analytique complexe. Publications Mathematiques I.H.E.S. 38 (1970), 77-91.

t45J

Remmert, R.:

Projektionen analytischer Mengen. Ann. 130 (1956), 410-441.

[46]

Serre, J.P.:

Fonctions automorphes, quelques majorations dans le cas ou X/G est compact. Sam. H. cartan, 1953-54. Benjamin 1957.

[47]

Serre, J.P.:

Un theoreme de dualite. Comm. Math. Helvetici 29 (1955), 9-26.

Systemes differe ntiels a coefficients constants. Sam. Bourbaki 1962, 265.

U. of Chicago

Math.

-1 75 _

A. An d r eotti

[481 Siegel, C.L.:

Analytic functions of several complex variabl es. Lectures delivered at t he Insti tute ' f or advanced study, 19Lf8-49. Notes by P. Bateman.

[49)

Siegel, C.L.:

Einfuhrung in die Theorie der Modul fo r men n l ten Grqdes. Math. Ann . 116 (1939), 617-657.

[50]

Siegel, C.L.:

Meromorphe Funktionen auf kompakten analytischen Mannigfal tigkeiten. Nachr , Ak. Wiss. Gottingen, 19 55, 71-77.

[51J

Weil, A.:

Variates Kahleriennes.

Hermann, Paris,

1958.

r521

Zariski , 0.:

Some results in the arithmetic theory of algebraic varieties. Am. J. Hath. 61 (1939), 249-294.

[53J

Zariski , 0.:

Sur la normalite analytique des variates normales. Ann. Inst. Fourier 2 (1950), 161-164.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C . 1. M . E. )

pROPAGATION OF SINGULARI TIES FOR THE CAUCHY-RIEMA NN E QU AT ION S

J.J .

Cor so

t e n ut o

a

KOHN

Bressanone dal

3 al

12

gi u gno

19 7 3

- 17 9 -

J . J . Kohn

Introduction. These .lectures are intended as an i ntro duction to the study of several complex variables fro m the po i nt of view of partial differential equations.

More specifically he r e

we take the approach of the calculus of variations known as the a- Neumann problem.

Mos t of the material covered here

is contained in Folland and Kohn, [ 4 1, Hor ma nde r ( 11 ] and in the more recent work of the author (see [ 161 , 117] and

L201).

we consistently use the Laplace operator as in

Kohn ['14l , since we believe that this method is particularly sUitable for the study of

re~ularity

the induced Cauchy-Riemann equations. in finding regular solutions of the Riemann equations.

and for t he study of Our main emphasis is

inhomo~enious

Cauchy-

We wish to call attention to the ex-

tensive research on this problem by different methods from the ones mentioned above (see Ramirez [ 29] , Gr a u e r t and Lieb [8 J, Kerzman [ 13], 0vrelid [ 28], Henkin [ 9], Folland and Stein [ 5] ).

It would take us too far afield to present

these matters here.

Another closely related SUbject which

we cannot take up here is the theory of approximations by holomorphic functions (see R. Nirenberg and O. Wells [ 27] , R. Nirenberg [26]. Hormander and Wermer ["12 ] , e tc , },

- 180-

J . J . Kohn

The

Lecture 1.

a- problem and

Hartog's theorem

The purpose of these lectures is to serve as an introduction to the use of the methods of partial differential equations in the theory of several complex variables.

Let

zl"",zn be the coordinate functions in (n and let x J = Re (z J)'

Yj

= Im(z j)'

for a function u we define the derivatives u z and Uz by: . j j (1.1) :

and

Uz

j

= !(ux

J

+iU y)' j

The Cauchy-Riemann equations are then the equations Uz

(1. 2)

j

=0

for

j

= l, ••• ,n.

A function satisfying the system (2) is called holomorphic and the theory of several complex variables consists of the study of these functions.

A classical theorem states

that a function is holomorphic if and only if it can be represented locally as a power series in the coordinates zl, ... zn

(see, for example, Hormander

[111

theorem 2.2.6).

Here we will prove the existence of globally defined holomorphic functions, these functions will not be constructed by piecing together local solutions of (2) but rather by

..

studying the inhomogeneous Cauchy-Riemann equations. precisely, given functions

o(l""o(n

More

we wish to solve the

eq~ations

(1..) )

j

=

l, ... ,n.

Furthermore, we will want to investigate the dependence of

-1 81-

.T. .J. Ka h n

FUrthermore, we wlll want to lnvestlgate the dependence of the solutlon u on the OC j • For example, ln the dlscusslon of Hartog's theorem glven below, the crucial step ls to find a compactly supported u when the supported.

are compactly

~j

Slnce the operators ln (1) have constant co-

efficlents we have

UZjz k

= uzkz j

and hence the followlng

compatlbl1ity condltlons are necessary for the exlstence of a solutlon u of ()): (1. 4)

()(k-

Zj'

for all

k,

j

= 1,2, •• • ;n.

The equatlons ()) and (4) are best expressed ln terms of dlfferentlal forms c(

settln~

=lcl.dz , J

j

and we have (1.)')

dU =

0(.

(1.4' )

The problem of flndlng u

satlsfyln~

that satlsfles (4') ls called the

a-

()') wlth glven problem.

The followlng classlcal theorem due to Hartog lndlcates the profound dlfference between the theory of several complex varlables and one complex varlable. example of how the

a- problem

The proof ls an

can be used to prove exlstence

theorems of holomorphlc functlons.

- 182-

J . J. K ohn

and if

U

n

~

1);> 0

If

Theorem (Hartog).

and

2, then every ho1omorphic function defined on

has a unique ho1omorphic extension to the ball

That is, if

h

exists a ho1omorphic function ""h 'V

h

=

h

on

In case

U

is a ho1omorphic function on

then there

B such that

on

U

n

=

the function

1

in the above theorem, the restriction Ehrenpreis in [ 3]

= Z -1

h(z)

n

shows that,

2

~

is necessary

showed that the required extension

property can be obtainedfram the existence of a compactly supported solution

u

whenever 0/. has compact support.

'r he argument is as follows: which equals if

1

when

L I zjl 2::::::

Liz j \ 2

1-&/2

f

let

~

Coe

be a 1- 0/4

function

and equals

0

we set for

(1. 5)

1-

a <..

Ll

Z \

j

2

<:

elsewhere Clearly

satisfies

ci..

If there eXists

u

and has compact support.

(3)

with compact support satisfying

(3)

then, first of all by

'2I

Z

j

u

\2 >

(3)

is holomorphic in the domain

1_0/2,

and hence by analytic continuation

u

is zero in this domain.

-18 3-

J . J . Kohn

In

B we deflne

,-,J

h

by

h = ph -

(1.6 )

..., h

Then

ls the requlred solutlon slnce, by (3) and (5) "V

lt ls holomorphlc and slnce

u

subset of

u,

h

=h

ln an non-empty open

u.

and hence throughout

It remalns to be shown that when exlst

u E (:([:n)

whenever the

n

~

then there

2

CoOQU[.n).

o(j E

Followlng Hormander (see [11) 2.3), we deduce thls from the followlng classlcal solutlon of Namely, if

J1. C ([ 1

smooth and lf

b.Q,

u E: ("'" ( (l)

then

=~

n(z)

(1.7)

and

~ ~'l:~z

d't'

bn

(3)

ln one varlable.

the boundary of

Jl

rio)~ ~~-z

d'l:'

+

ls

A

~

d
Thls formula ls easlly derlved by uslng Stoke's theorem on the domaln obtalned by removlng a dlsc of radlus center

Z

from

follows that if (1.8 )

u(z)

Jl eX

fer it)

= -..L 211'1

= _1_

and then lettlng E. ~ O.

fS .n.

and 1f we deflne

~ 't" - Z

t

and

It then u

by

d'l A df

2nl 1

=m dlfferentlatlng and changlng varlables agaln, we obtaln

-184 -

J . J . Kahn (1.9 ),

=

uz(z)

=

1

2m.

=0

Since c<..

1

"21iT on

rsc

o(:r

('! + z) d'r"dr 'T

f5 ..n.

~d-r 'l-Z

"

dT we obtain

b.1't, applying (1.1 to 0<.

(1. 10 ).

It is clear that if a solution of support then the integral of n z,

0(

compact support whenever the satisfy

(4).

The desired

u

j

In the case of u

of

()

with

have compact support and

is then defined by:

IS

C

0<.1 ('I: 'Z2'··· ':?:n) 't' - Zl

d 'L" d:e

oI.1('t'"+

=

TOree that

u

we have

satisfies

()

we first note that by

further, using

obtain (1 .12) .

0(

~

(1.11)

(10)

is zero.

, however, we obtain a solution

2

has compact

(10)

=

= =

1

~

(4)

and

we

-185-

J . J. Kahn

In particular we see that

K

=

U supp

u

is holomorphic outside of

From (11) we see that

(o(j).

is 0

u

if I z2\

is sufficiently large and hence, by analytic continuation, u.

=0

K;

in the unbounded component of the compliment of

therefore

u

has compact support.

The above argument can be used to prove the following generalization of Hartog's theorem, due to Bochner (see Theorem.

n > 2

If

and if

Jl C \.. is a bounded con-

nected open set then every holomorphic function on a unique holomorphic extension to

JlL

union of of

':2 ': ).

.n.. ; where, r--

,--

.0..

11

has

is the

and the bounded components of the compliment

SL. b.a, the boundary of Il , is smooth

If we suppose that and that a function

h

is defined on

b

JL

it is natural r-...

to ask, when does there exist a smooth function on

it.

in

.0..

such that

h

=h

on

bll and

h

h

is holomorphic

The obvious necessary condition on

h

is that it

satisfies the so-called "tangential Cauchy-Riemann i.e. those linear combinations of the gential to b

Q.

More precisely, if

d

JYj r

and if

(1.13)

and i f Uf ( 1.1.'

a

La

j

1,

••• ,an

r-

::;

b _Q such that

are functions .s uc h. that,;on , -8L1

is any smooth extension of

L a j f zj

which are tan-

0,

Zj

°

equations~

is a real-valued

smooth function defined in a neighborhood of dr,. '::

defined

on

bn.

h,

then,

-18 6-

J . J . Kahn An equlvalent way of wrltlng thls ls:

(1. 14.'.).-

.fLc.

Theorem.

If

boundary

bJl,

lf

..Q

a:n,

n

>-.

2,

is a domain with a smooth

if the complement of Jl

is bounded.

Then a smooth function

can be extended to a smooth functlon h

ls holomorphlc ln

f9:rall

is connected and

J1,

h

h

n.

on

if and only if

h

on

b

Sl.

such that

satisfies (14)

that satisfy (13).

al""'~

PoHowing Hormander

en]

(theorem 2.3.2') we can

prove the above by first constructing a smooth extension

H

such that ( 1.1.5')

j

Slnce the supported where

p

u

=1

a -problem for

0(

j

E

=

It •.. t n •

can be solved with a compactly

Col (([ n).

we set d.

in a small neighborhood of

outside a sllghtly larger neighborhood. with a compactly supported sion

..v

h = pH - u

u

= a (j'H)

bSl and vanishes Uslng o(=-:au

we obtaln the desired exten-

as in the previous theorem.

The function

H can be constructed by starting wlth any smooth extension

r

of

h

and noting that (14) implles that:

(1.1E! )

rewritlng this we have

(1.17)

near

bit

-1 87-

J . J . K ahn

which implies that

and hence (1. 18}

setting 2

H ::

(1.19'

f - fo r - f 1 r /2

we obtain (1. 20)

as required. It should be mentioned that the tangential CauchyRiemann equations (14) have been studied extensively (see Boc hne r Kohn

( 2)

[15]

,Lewy [2 21

[ n1

• Kohn and Rossi

, Andr eot t i and ~ il ~ [-1] • . e.tc. ).

In fact, the famous example of Lewy of an equation without solutions is one of these.

We will return to this

equation later. Our main concern will be to study (3) on a domain without any restriction on the support of

cc ,

Suppose, for

example, that the c( j E L 2 cD.. ) then ..we wish to find L u~L2(rt) satisfying (3), by this we mean that we want to find a sequence of smooth in

uy

defined on..fl

such that

L2 eQ ) we have u:: 11m u y and o(j :: 11m u yzj • Suppose that there exists a point P E. b{l and a holo-

morphic function such that

f

f (P) :: 0

defined in a nei~hborhood and

f ~ 0

in

U(\JL

11 of

- {p J.

P

Then

-1 88-

.T . J.

Kohn

we claim that if a solution of ()) eXists in there exists a holomorphic function that

h

is an open set containing

V

ists no holomorphic function

=h

p = 1

h

P.

cannot be continued analytically over the point

That is, if

g

L then 2(fL) defined on Jl, suet:

on

V (\

n. =

F

and we chose

au=cX

0(

f

N

Jf

Now we define

::

F by:

~- U F~L2(5'L). e{

J

We set

=0

u E L2 (.~\) then we define the desired h by: h

such that

Vf"IiL

(in fact,

L 2 (rt)

e C;( u )

So if there exists

P) •

(1. 22.)

Now

P.

N so large that

and note that borhood of

J pf-· in lOin

V such that

defined on

To see this let

in a neighborhood of

(1. 21)

g

then there ex-

P

0<,,-:=

3F

in a neighsuch that

F - u

h $ L (0-) but for any neighborhood W of P 2 - W) and thus h cannot be continued past P.

hE L2 LCl The problem of finding a holomorphic function which cannot be continued past a point the Levi problem for

fl

at

defined in a ne1;;ghborhood of

h

on

P€:. bfl is called

P.

A holomorphic function

f

P E. b

rt. which has the proper-

ties hypothesized above is called a local holomorphic separating function at Theorem.

If the

P.

We have thus proven the follOWing:

a - problem

has a solution

UE:: L

2

CQ )

cf.. J E:: L2 (Q ) and satisfies (4) and i f there eXists a holomorphic separating function at P E- bSl then whenever the '

the Levi oroblem has a solution at

P.

-1 89 -

J . J . Kah n

It ls easy to see that a holomorphlc separatlng functP

borhood of

(In that case there ls a llnear holomorphlc

P

lf the demaln

Sl

10n exlsts at

ls convex ln a neigh-

Let (Lt' be the domaln glven by the

separatlng functlon). lnequalltles (1. 2J)

and.fl.2 Let

ls the ball of radlus ~ and center a,O, •.. ,O).

iL = n

lV

..0 2

then the orlgln is ttl b..a and _Q

ls convex ln a nelghborhood of the orlgln. toe equatlon (J) cannot

be solved ln

Thus we see that

n)

L2( for lf lt were posslble to solve lt we would obtaln also a solutlon of

Levl's problem at the orlgln and this would contradlct Hartog's theorem.

A modlficatlon of the above argument

JI..

shows that for the above , u

z2 ,. . . ,

uz

natural topologles (l.e.

n

the range of the operator

) ls not closed ln any of the

Lp' dlstrlbutlon) and thus the ~ - problem cannot be solved ln any satlsfactory sense.

a

-1 90-

J . J . Ko h n

Lecture

Pseudo-convexity

2.

nc

We will now study the properties of

jL

tr ~

n

which

insure that there is a local holomorphic separating function at

PE bCl.

As was pointed out in the previous lecture it

is enough that.fl

be convex in a neighbor!;!ood of

P.

However, the notion "of holomorphic separating function is invariant under holomorphic transformations whereas the "notion of convexity is not.

We will introduce the property

of "pseudo-convexity" by, roughly

isolating that

speakin~,

feature of convexity which is invariant under holomorphic transformations. We assume that

blL is smooth; that is, that there

eXists a real-valued function of

bil such that

fix the sign of r

(:3.1)0

r

>

The domain

dr '" 0

n

r

and

so that: 0

outside of

defined in a neighborhood r

=0

n

on and

b...C1. r

<

is convex if the Hessian of

We will

o

in D.. •

r

(1.e.

the matrix of second oartial derivatives) is non-negative acting as a quadratic form on the tangent vectors to if it is positive - definite then In

IR2

fL

blL;

is strictly convex.

this reduces to the familiar fact that the graph

of a function is convex if the second derivative is non positive, the general case may also be deduced from this by taking intersections with planes. In terms of the complex n this condition is expressed by coordlnate in r'L.

-191-

J . J . Kahn

wherever

(2.3) In other words we are restricting the

i

( rz z i

j

j

2n quad-

~

.::: ) (l:)

ratic form

rz z

2n

(P) (P)

n

To those vectors which are real (i.e.

tangential to

bJl at

P.

b

j

= a) j

and

It is clear that this condition

is not invariant under holomorphic transformations.

However,

observe that the condition implies that

>

(2.4)

0,

whenever

°

for

P E b.a.

Note that the above is invariant under holomorphic transformations and, in fact, we will express this last condition in invariant form.

~l,O(biL) be the subspace

Let

of the complex tangent vectors at

P

€

b ft which are of type

(1,0); that is, they annihilate the anti-holomorphic functions.

In terms of coordinates

Tp 1,0(bfL)

if and

only if it can be expressed as ( 2 • 6P

.~

L

=

'0

L a j az j

,

where

lr

zi

(P)a i

=

0.

-192-

J . J. Kahn

For each

P ~ bit we have the herml tlan form

x :r p

(2.7)

r

p

1.0 (b.S1.)

by

(2.8).

;\

<:

where

,>

p

p '

«oar) •

=

(L. L')

T , 1.0 (bi\.)

X

~

c

L;\L'> •

'p

denotes the contraction between contravarl-

ant and covarlant tensors.

The form

"\ A

I

p

ls called the

Levl form and the condltlon glven by (4) and (5) ls equlvalent to

A.

,p

(L, L)

~

o

for all

L

~

0 1. (bD.,). 1>

Thls condltlon ls clearly lnvarlant and we say that ls pseudo-convex 1 f

(9) ls satlsfied for all

1l

}>l E. b.Q and

that lt is strongly pseudo-convex lf the Levl form ls posltlve deflnlte at all

P E bfl.

Conversely. l f i l p~

b{l

then there exlsts a holomorphlc coordlnate system

on a nelghborhood convex.

ls strongly pseudo-convex and lf

U

of

P

such that

JL {\ u

ls strlctly

To prove thls we flrst note that strong pseudo-

convexlty ls lndependent of the choice of course. that

r

satlsfy (1).

coordlnate system

r; provlded, of

Then we note that for any

z

r

can be chosen so that (r z (p) ) ls 1 j a posltlve deflnlte hermltlan form on all n - t .uples . (a 1 ••••• a n). 1.e. wlthout the restrlctlon (5). this we set (2.10)

r

=

e

't.H

- 1.

To achleve

-19 3-

J . J . Kohn

where

R

is any fixed function satisfying

a sufficiently large number.

( 1)

and

is

T

Then we have

(2.11~ _

we , e compos e an arbitrary

n-tuples

as follows:

(2.12)

where

so that (al •••• ,a n) satisfies (5). = t:

(2.13) .

Thus we have

L ~ z i zj (P)

+ 0(" 'fl

9

1a j +

2

'['2

(2. la i , 2)~.

The error term is bounded by: (large const. ~fi\ 2 + small const.

(2.14) -

Choosing

L:

lai l 2).

T sufficiently large this is smaller than

the first two terms on the ri ght of (13) thesis, the first term is larger than

- since,

''C const.

by

hy po-

L. I ail

2.

Therefore. we have (2.15)-

In z i zj (P)b;'b j >

const."[ I b

i\

2

as desired. We are now ready to prove the following classical result. Theorem.

If i1.

is a strongly pseudo-convex domain and

-194-

J . J . Kohn

P!C: bil.

then there exists a neighborhood

holomorphic coordinate system whose such that

U

f\ft

U

do~ain

of

P

And a

contains

is strictly convex with respect to the

linear structure given by these coordinates. Proof:

It suffices to find a coordinate system such that

the form (2) is positive-definite even without restriction

(3).

Let

u1, ••• ,u

P

with origin at with

(~u

i

U (0»

be any holomorphic coordinate system n and let T be a function satisfying (1)

positive definite.

j

Expanding

~

in a

Taylor series, we have

(2.16)

2 qe ([ r u (O)u i +

r =

r

i

lr

+

ui

u (O)UiU j ) J

((Ilu i \ ) 3).

U 11 (O)uiU j + i j

Setting

(2.17 )

Z

i

= ui

for i = 1, ••• ,n-1,

zn = 2 2 'ru ( P) u i

i

,.. 2l:r

and

( P)uiu j

uiu j

we have

(2.18) Since r

r = Be ( z ) + n u U i j

(0 )

L r zi zj (0 )

Zi

Z

,+

.J

is positive definite we also have ,r

positive definite and thu s b Y (1 8) the

~essian

_ (0) ZiZj

is also

positive definite. The

followin ~

classical

theore~

shows that in case the

Levi form is identically zero the d oma i n is al so locally convex. rhe proof of t hi s t.he or-en is less elementary than t he one a bove.

- 1 95 -

J. J. Kohn

Theorem.

If

r

If in

is real, analytic and B nei ~hbor hoo d

U

coordinates

zl''''' zn

on

P ~ bil

the Levi-

u n blL) then there ex-

form is identically zero (i.e. on ists a coordinate neighborhood

of

V of

P

with holo morphic

V , such that the set

IT/ (\ b fl

consists of the points for which The two theorems above make it seem plausible that whenever a domain is pseudo-convex then in a neighborhood of each boundary point these exist coordinates with respect to which it is convex.

However, this is not true as is

shown in the following example (see Kohn and Nirenberg [14]) Let

fL C( 2

so that near the origin the function

~

is

given by: (2.19)

r

2 8 1S 6 Re (w) + [z ] lWl 2 + /z{ +"7 IZ!2 qe (z ).

=:

Since complex dimension

2

dimensional the Levi form

the spacer-1,O(bjl) 1 K 1 matrix.

~

is one-

In the above ex-

ample it can be shown that the Levi form is larger than const. (/W1 2 + / z I 6 ) near the origin. Thus, by suitably extending

r

we have

Jl

pseudo-convex and strongly

pseudo-convex everywhere except at

(0,0).

Now if there

were holomorphic coordinates on a neighborhood

u /\ .n..

origin relative to which find a linear function the zeros of

ae

h

h

of the

is convex, then we could

such that

which are in

OJ

U

He h(O,O) remain

= ° and

outs~de

of

J1

That this is not possible in this example 1s shown by the following result.

-196 -

J .J. kahn

Theorem.

If

h

is a holomor?hic function which is defined

on a neighborhood

there exist points (z1,wl)'

= h(z2'w 2) = 0,

h (Zl,w1) where

~~

h(O,O)

of the origin and 1f

U

(z2'w 2) E

>

'T(z l , wl )

;J

=

°then

such that

°and

r (z2'w 2)

<

0;

is defined by (19).

The proof of this theorem depends on the following ;Lemma. Lemma.

For each

(2.20)

a E. ([' the function

=

fA(Z)

\z\8 +

~5

f,

iz\2 '1 e (z6 ) +

defined by l1e(az 8),

changes sign in every ne Lghbo rhood of the origin. Proof:

t ~ 0,

since for

fa (t z) = t Bfa (z)

it suffices to

that the function

changes sign. (2.22)

Then we obtain

.!.l rr

=

7

( 211

sign then

._

So·2rf Ie: (e)I de .

~

0

Since ~2"'?; (9)d~ o

2rr .J) 1'" S 2; (O)e . d£>

D

= 2 IT, we see that i f

I,,:. ( fd)! d

does not chan!1;e

= 2 11 which contradicts the above

inequa 11 tv , Proof of the theorem:

Suppose

defined in a neighborhood of h(O,w)

!!!

qu i r-ed ,

°

then

If

r, (O,w)

h(Z ,0)

!!

can be

\e

°then

sign by the above lemma. h

:

Deranretriz ed

h

is a holomorphic function

(0,0) (w)

with

h(O,O)

= 0.

If

which changes sign as re-

h(Z,O) = fo
which changes

Otherwise some of the zeroes of RS

f011ows:

-197-

J . J .Kohn

q

z =

(2.23) · '

with

w =

I'l

0.

P

That is, for every '!

we have

w are

Evaluating

defln~d

by (2).

h(z,w) = r

°

when

z

and

on these zeroes,

we obtain (2.24)r =

<

P

Now, if

8q

controls the sign of:T . sign.

If

p

= 8q, r

sign and so

then the first term on the right for I'SI small and hence f'

changes

then the first three terms control the

changes sign by the lemma) wi th

a = ap •

If

p> 8q, then the second and third terms control the sign and the lemma can be applied with

= 0.

a

At this point we wish to discuss invariants of pseudoconvex domains which are not strongly pseudo-convex.

These

invariants play an important role in the study of boundary regUlarity of the be a

C

OO

PE-b.Q.

~ - problem (see Kohn

[16]

).

vector field defined in a neighborhood i,O(bR),

for each

1.e.

~

(2.25):.:

and

Denote by

=

L

of

L is of degree (1,0) and that

We suppose that

L(r)

Let L

°

the conjuge of

on

L

(defined by

-198-

J . J . Kohn

= ' t ,U) ,

IL(u)

(2.2q)

L

in terms of local coordinates

=

La J ~ aJ

For each non-negative integer

i

vector 'f i e l ds

tQ- ;(L)

(2.27.1

on

(rL..)

=

{fL

Le. the space of all ;f(L)

(2.2~)

'u + a;

we define a space of

as follows

1-J giT- with

f ~ +

= :t. k -1

k

(L) +

[1°

(L ),

oe

t , lI;E( ('{);

t: k -1 (L U,

that is the space of all vector fields of the form

where

IJ

r We say that ~

r

W' €- oJ... 'f"k-1(L) '

,

L

~

V'

and

v' r .

is of finite type at

if for some

P

we have

( 2 • 30 ) . .

; ( : (L )

where t'~ (U

T/,O(o"J)) + r O,·1(b1U, !"

denotes the vector space obtained by evalu-

ating alFthe vector fields in integer L

J:.. 0 (L)

at

that

.'

L

(A)

it

1 k (L)

and i f DO) is not ture for any is of infinite order at 2 If

L

'r he lowest

?

for which DO) is true is called the order of

.it

.

k

then we say

l'he following hold :

is pseudo-convex, then it is strongly

pseudo-convex if and only if for each non-zero

at

P c- b

defined in a neighborhood of

51

an d each L

is of

-199-

J . J . Kahn

order

1

at

(B)

If

P • U is open and all vector fields on

of infinite order at

E u

f'

n b n,

U

are

then the Levi form is

identically zero on The basis for the above properties is the following expression for the Levi form in terms of vector fields with values o n r 1,O(b..Q).


(2.)1)

<ar

, Lt.L>

,

[L

,E l

>

This formula follows from the fact that and the classical expression for the exterior derivative:

<~ ~'r, L 1\ r:)= -~,a -d<

=

<~ 'r, r > =

and we have (1, 0)

and

(2.)2) - .

Lemma:

d r

, e) =

< dr

ve
v G r 1, °(bn) +

r": 1 ( bn ) P

<(~ r

(2•) ) )

,I) + [«ar , L

) +

<()r,

[L,rJ)

(0,1) ,

<ar,c>=

p

>

since ar is of degree ° of dell;ree further

[

If

Proof:

r, u/\ L

)

0'

L (r)

=

0.

=°

Le.

VCr)

if

and on ly i f

on

bfl,

then

'J ) = 0

gy ()2) and by considerations of degree it is clear

that all

r O, l ( bfU p

satisfy (J).

On

the other hand (2.)41' .

dim dim q.

(

r 1 , O(.b{l ) + ~

Q.' [t bfU p

r ?, l (bfU t'

211 - 1

2n - 2

-200 -

J. J . Kahn

and since

(or)p

et: Tp ( b .0. )

is considered as a linear functional on

and is non-trivial (if it were trivial it would

be a multiple of dr ) we see that the subspace of (

V satisfying ())) is of dimension

of those

Tr(b~)

2n - 2

which

concludes the proof. The above properties are then

e~sy

consequences of ()1),

the lemma and the definition of the Levi form (8). We conclude the present discussion of order with the following lemma. Lemma.

If every non-zero local vector field on

Tp1,0(bft), is of finite order then there exists

values in

no non-trivial connected analytic vari Proof: P

e

V

Suppose

of

L

V.

such that

P

L

Q

for

E. TQ+O(yj

; [ :( L )

since

c cr TQ

~TQ. ('I{ )

L

C

~

(f ot

€

1,

~

Q

Then, clearly

Q 6 U() V. Q E U() V

°(V ) +

T 1,O(b .(L) +

b ~1.. and such

i.J ()

for all

(V )

r

crTQ(V)

therefore,

Then there is a non-zero

of degree (1,0) def1ned. :\1n a neighborhood

LQ,cri,O(bD.) that

: contained in bSl.

is such an analytic variety and that

V

is a regular point of

vector field U

b1L, with

T O,l(V .~

TQ. O,l(b.fU

is of inflni te order for all

Q

and we have and

e Un v.

To conclude this lecture we wish to mention some classical properties of pseudo-convexity.

tlQJ

section

2.6

We refer to Hormander

for .s ~v s t e ma t i c treatment of these.

-201-

J . J. Kohn

Def1nit1on.

A real-valued runc t ron

u E COO (Sl)

is called

plurisubharmon1c if the quadrat1c form (5)

2. u zi zj Siaj

1s pos1tive def1n1te. The follow1ng theorem (see Hormander

theorems

2.6.7, 2.6.11 and 2.612) shows how the notion of plurisubharmon1clty

leads to generalizing pseudo-convexity to domains

whose boundary is not smooth.

Jrl

Theorem.

If

boundary

bfl

n

(a)

is an open set in

~n

with a smooth

then the following are equivalent:

There exists a pluri-subharmonic function

JR. = { ZEn\U(Z)<

such that for every

f!

f

u

on

C E.

c}

is compact. (b)

Sl

is pseudo-convex, 1.e. (4), (5) are satisfied.

As is standard we extend the definition of pseudo-coovex to domains satisfying condition (Ja). says essentially that J( pseudo-convex domains.

This condition

can be exhausted by strongly

This need not be ture for pseudo-

convex dqmains in complex manifolds.

From condition (a) and

from the solution of the Levi problem for strongly pseudoconvex domains, it can be deduced that a domain

1l

in (0

is a domain of holomorphy (i.e. there eXists a holomorphic function on

1l.

which cannot be continued to a larger do-

ma.Ln ) i f. and only if

.0.

is pseudo-convex.

_202 -

J . J . K ahn

Finally we wish to c a l l att ention to the sult (see Ho r ma n :ie r r ll] if a domain in

t r '1'l'

\L

f o l lo w 1n ~

re-

t he o r-e m 2 . 6.'1 3) wht c.h shows t hat

w1th a s mooth boundarY is not ps eudo-

convex then it is not a do main of holomor ph y . Theorem.

c4 ,

nc

Suppose that

i.e. the function

some FE bfl.

r

and some

([ :n has a boundary of class

is of class

(al"" ,an)

c4 •

Su ppo s e t hat for

with

we have

iJ

then there exists a neighborhood if

u € C 4 ( U)

'Y+ = { 7'


(

y+

.

u-

b it..

then there exf s t s a function

v = u

.:< ) .;.

morphic function on extended to

V·n

on

such t hat

Ir

P s uc h t hat

a nd satisfies t he tan ge ntial Ca u c hy - Ri ema nn

equations {( l . ) v E.. C 1 ( Y )

U of

0

=

on

V () b .Il,

a nd

0 v

= 0

~

•

In particular, any holo-

[

~

uI

r (

~ )

<

o}

on

can be

- 2OJ-

J . J . K ahn

Lecture 3.

Formulatlon of the a-Neumann problem.

We return now to the study of the operator

a from

the

polnt of vIew of L2• Denotlng byap,q(fD the forms of type (p,q) whlch are C00 i n J2, that is restrictions of C
with

82

n,

we have

On these spaces we define ~ inner

.. 0.

products as

follows:

(3.2)

(u,v)

=

('P,\f')

:II

(9,0-) ..

SUVdV

.a..

LI

u,v~a:>,O({l)

for

'PiV;idV

for

2..n..~ei jifi jdV

'f,tEO..0, 1 (11)

for

e,
The corresponding norms are denoted by II II. We denote by Ll,q(Q) the completion ofap,q
a

in

(3.1) and T*, S* the adjoints of T, S then we have L2

O,O,('\)~

~L 1--- ~

T*

0,1(f'\\h 0,2«('\\ .1""~~ .l&.J.

S*

These operators are defined 001

their domains (not on

the whole space at the beginning of the arrow).

The domain

of T, denoted by Dom (T), is defined as follows:

0.4)

Dom (T)

= {UE-L20'°
with

~Uj}

Uj~~'O(Q), Cauchy in L20,1(Q)j.

- 204-

J. J . Kohn

By taklng \jIE' 1 (.rL) wl th oompaot support and wfCClClan we have (aw,lIJ)

(3.5)

= (w,

-

Therefore, lf uEDom (T) and

L'P1z

1

i.

1Uj} , {VjJ

are two sequencet

wlth l1mlt u and such that both {au ~ and {;~VJ\ are Cauchy; thus, settlng w

= uJ -

j

V j

ln (3.5), we obtaln

(3.6 )

so that for u"= Dam (T) we oan deflne Tu by

and lt ls lndependent of the sequence. Slmllarly, we deflne Dom (S) and Sf for all

~~Dom

(S).

The domaln of T*, denoted by Dam (T*), ls def1ned by: (3.8)

Dom (T*)

={lf~L20,1(Q)l3 c

> 0 w1th

\(Tu,cp)\ ~ cl\ull for all u~ Dom (T)} to deflne T* on Dom (T*) we see from (3.8) that the map u

~

(Tu,lp) ls a

bounded l1near functlonal on Dom (T) and

o

0

thus has a un1que extens10n to L2 ' (fl), slnoe Dam (T) is dense. This extenslon has a unique representatlve whloh ls by .defin1tion T* , thus we have (u, 'I'*I,f»

=

('ru, If)

,

- 2 05-

J . J . Kahn

If wl th I/J

1~Dom

=If

(T*) ls dlfferentlable then we can apply ().5)

and w wl th compac t support and we

0

btaln

(J.1O) Th1s can be thought of as a complex d1vergence correspond1ng to the complex gradlent and complex curl whlch are g1ven by

on func t10ns and on (0,1) -forms respec t1 ve ly,

~

S1nce

(J.ll)

ST

~

=0

2

= 0

and

we have T*S*

= O.

Th1s means that lf uE Dom (T) then Tu€ Dom (S) and STu and s1m1larly

I f ~ ~,

0

s* and T*.

The following ls a class1cal result 1n H1lbert space theory (see Hormander Theorem.

[111

theorem 1.1.1).

If A and B are HUbert spaces and T : A ~B ls a

closed densely def1ned operator, then the follow1ng are equ1valent: (a)

~(T), the range of T, i s closed

(b)

There 1s a constant C such that

(J.12)

liull

A

when (0 )

(d)

f c(lruji B , uEDom (T) n[<1«T*)],

L J


denotes closure.

closed

There 1s a constant C such that

- 206-

J . J . K ahn

Proof.

Since

(T*) is the ortho gonal complement of

11( T)

= { h f. Dom ( 'r )\ Th

= o}

we have

is aclosed one-to-one map hence by (a)

===>

(b) and similarly (since T

wish to prove (b)==}(d) .

\(If'

Tu)B\

=

I(T*'f'

iff Dam

u ) A\ ~

thecl o ~ed

= T**),

g r a ph theorem

(b)~(d).

We

From (b) we obtain: I\T*lfIlA llu tlAf. cIlT*tIlA11ru\lB '

(T*) and u f Dam (T)

n [61(T*)]

hence

which implies ().12') since

\\ 'fIls = i nf 1(tp , IP ) BI lI ~llB The proof is then complete since (b) '

:::;> (a) and (d) ~ (c)

are clear. Now let C be a third Hilbert space and we assume that we have closed operators S : B ~ C such that

densely defined T : A ~ Band

-2 0 7-

J . J . K oh n

(3.1)

ST

=

0

We define 2{C.B by

(3.14)

~

= 1( (T*) n'l1.. (S )

then we have: Theorem.

A necessary and sufficient condition thatR(T) and

(R(S) be both closed is that

(3.15 ) for all 'f'~ Dom (T*) () Dom (S) 1fi th If> .1
First we have the weak

ortho~onal

decomposition

formula (J. 16 )

which is an immediate consequence of ().1).

From ().1)

we also conclude that (iRJTU C ;}f(S) thus if gE Dom (T*)n [(R(TD

and ().15) reduces to ().12'). T*S*

then Sg

=

0

Similarly, since

=

0

we obtain ().12) as a consequence of ().15). trary

If' £ ~

wi th tp .l

Jt

For an arbi-

- 20 8-

J. J . Kahn

we use the orthogonal decomposition ().16) and obtain

f

+ ~2 •

= 1f

1

To prove ().15) it suffices to show . 2

1I

1 B

+

.2

nep 2II

B

~ cons t

. 2 (\\T*'f' . 1 A

·

II ·

,

. 2

+)15 '1f, ~

T2 .

),

which is the sum of ().12) and ().12'). The inequality ().15) is often obtained as a consequence of the following: Theorem.

Suppose that whenever

'f

k

Eo Dom (T*)

{If

k

n

1iS a sequence

Dom (S)

such that Ulp U is bounded. 1f k B

lim T*r.p = 0 in A and 1f k

lim :3 (IJ = 0 in C Tk

there exists a subsequence which converges in B. holds and

~:

1t is finite

Then

() .~5)

dimensional.

The hypothesis implies that the unit sphere ina( is

compact, hence

~ is finite dimensional.

hold then we could find a sequence .@ t=- Dom (T*) k

(\

Dom (5)

If () .15) did not

- 2 09 -

J. J . Kohn

with

ek 1. X

such that 2

liek 11 B

+ \\ s

2

t> k 1\C

).

Setting

we obtain

llT*lf II

2

k A

2

II S
+

1

<. K

3 which converges to Je andk \I If Ii = 1 we obtain a

hence thereexists a subsequence of { ~ an element If .1.

X

but since tp €

contradict1on. Corollary.

If the norm

on Dam (T*)

n

1l¥,11

B

+ '

1lT*!p1l

A

+

\Islj)11

B

defined

Dom (S) is compact then ().15) holds and

is finite dimensional. once we know that (3.15) is satisfied then we can remove the brackets in (3.16) and thus obtain (J.1?)

which is called the orthogonal decomposition. ~his

more explicit by introducing the laplacian

We shall make

-210-

J . J . Kahn

(3.18)

Lrp

with

= TT*.p

+ S*Stp for tpEDom (T*)n Dom (8)

T*t? E:Dom (T)

(3.19)

It = 'Yl(L) ,

since clearlyj(C?1(L)

and if

(3.20 )

(L'f .....)

.

Then

and Slf E Dom (8*).

LV'

=0

+

l\stp\\

B

then 2

= 0

C

so that Y'~X. The following theorem is essentially due to Gaffney

[ 6 ]. 'rh eor em.

The operator L, defined by (3.18) where Sand T

satisfy (3.13)i5 self-adjoint. Proof:

By a theorem of von Neumann, (I + TT*)

-1

are bounded and self-adjoint t1

= (I

+ 'r T* )

(I + S*8)

and

-1

-1

we set + (I + 8*S)

-1

- I

which is bounded and self-adjoint, we shall prove the theorem by showing that R= (L + I) -1 (I + TT*)

-1

=.

- I = (I - (I + TT*» -TT* (I + TT*)

which shows that

Firat,

-1

(I + TT*)

-1

-211-

J . J. Kohn

(j{ (I

(}((1 + 3*3)

s1m1larly

R so since ST

+ T'l'*)

=

=0

(I + S*S)

we have

-1 -1

-1

C

Dom (T'1'*).

- TT*(1 + TT*)

d(un c

Dam (S* S )

S*SR = S*5(1 + S*S) (}(.(R) C

Slm1larly

TT*R so that

(L + 1)3

= TT*(1

+ TT*)

c -1

+ TT*)

:i)

(J.21)

and

-1

and

-1

and

Dom (L)

+ S*S(1 + S*S)

Finally L + I 1s injective. s1nce -1 and therefore. R = (L + I) • We def1ne the spaces

-1

Dom (TT*)

= TT*(1

cR.(R)

and we have

C. Dom (S*8)

~ and

,1'by:

();"22)

r+ Jrr:

q : '- ><.

c ~ iL

by

(J.22' )

rhen (3.15) can be wr1tten as (J.?)

\\f\\

2

:So co ns t , q

(,p.t)

+ 3

. ((L + 1)/f.tp)':::;

= Dom ( T* ) () Dom (S)

and the herm1 tlan form

-1

for

~

1ff-5.

=

l\epli.

I 2

- 2.12 -

J . J . Kahn

~ 1s

Now observe that if dense in

B

then, setting

.~. we have 1'dense in .~.

=

{~~B \ fl Je) Since i f

which is close to;r and y

I

= ~"

y~~ +

-e

I

there ext s cs l' 'E l) where ~t-£ and

''I' l ¥ h e n c e 1'''e-1 and henceis close to .0' • lf

The following result is often called the Friedrich representation theorem.

r c ~ is dense and y: is a

Theorem.

If

Hibbert space with thE

norm q,

a Hilbert space with the norm

i\

\\

and i f (3.2)

is satisfied then there eXists a densely defined F Dom (F) C ~

such that

:~-7 ~

and

(3 .24 )

and all

'P(:1[

Furthermore, F is self-adjoint, onto and

Proof:

Given

~E .~

given by

~(I;#i)

so that T

=

then consider the function T""

(~,lji)

from (3.2)

we have

is a bounded functional and hence there exists a

unique K(c()E;1'which represents T«.,_ that is: (3.26-)

a:-.~~

-21 3-

J ..J . Kahn

Iff-9=:

for all

Us1ng (J.23) and (J.26) we have

2

\\KIi\\ S

canst. q (1<:0(, K'J"_) ~ const. \ k, Ko<.)\

f, const.

ilocl\ HKo(lI

hence

Furthermore, for

q(&X, K~) •

=

\\K-t(\ ~

const.

= q(Kp,

K~)

p<=-5we have

<:/. ,

(~, KP) I

Therefore, K : ~ --=r adjo1nt.

ll""- \l.

•

=

(~, K~) = (K~,p).

J'

1s one-one, bounded and selfSett1ng Dom (F)= ~(K) we define F = K- 1. a nd 1t

has the des1red properties. Defin1ng F by setting 1 (J. 28)

Dom (F ) 1

F

Dam (F)

F on Dom (F) and F

1

1

+Je, = 0 on

Je.

we obta1n that F

is a self-adjo1nt densely def1ned o?erator 1 We cla1m that Dom (F = Dom (L) and that F = L. It is l 1)

clear that Dom (L) C Dom (F l) and that F l = L on Dom (L) therefore 'Dom (L*)~ Dam (F l*) and s1nce both Land Fl are self-adjoint, we have Dam (F 1)

thato-t(L)

=q

=

Dam (L).

Thus we conclude

ani from (J.l?) we can conclude that

d?( Si')

", J( , .::,' -;) a nd :R.('r)

=

O« T'r*>.

-214-

J. J . K ahn

N

Further we define the operator

N

=

~.

0 on

L K on

B '-.7 B such that

J{.

5'

Denoting by H : 8 -7X the orthogonal projection into ~ we obtain the orthogonal decomposition of

D. 30)

cJ..

NoW i f Sa{

=0

=. TT*No( +

S*SNv( + Ho<.

we have from (3.30)

(3.31)

SS*SN~

= 0

and thus (SS*SNoe, SNoe.) = ~S*SN,( and hence

S*SNol. = 0

r/.. =

2

II

= 0

and

TT*No( + Hli

Thus we see that the necessary and sufficient condition to solve the equation d.. ermore, the solution gonal to

n(

= Tu is that So( = 0 and H4( = O. u = T*N~ is the unique solution

Furthortho-

T) •

These facts are summarized in the following theorem. Theo~..

Given that A, Band C are Hilbert spaces and

T : A -7 B, S : B ~C are densely defined closed operators such that ST = 0 and setting

;K='Il(1'* ) rJ '11(s)

£J =

and

Dom ('I'*) (\ Dom (S),

'j : {rr /l) \ tr 1 df.1

-215-

J . J . K ahn

we assume that

.for all

If W.

PJ

is dense in B and that

Then the space B spli ts into an orthogonal

decomposition

Furthermore, the operator L= TT* + S*S whos'e domain is

is self-adjoint and has a closed range and there eXists a unique bounded self-adjoint operator N : B -7 Dom (L) such that D.}2 )

where H

B4

I - Hand HN

= NH = O.

LN

~

Jf

is the orthogonal projection onto

-;.e.

It

then follows that each o:eR has the ortho;z:onal decomposition (}.}O) and that the necessary and sufficient condition for the existance of a solution u satisfying 'ru S

=0

and ~J.Je. then u

=

'l'*N"" .

=~

is that

It also follows that If

P : A -7 7'l(T) is the orthogonal projection onto neT) then

(3.33) Proof:

P

=

I - I'*,'H.

All that remains to be proven is (}.}3), that is,we

must show that the onerator P defined by (3.33) is the orthogonal projection of A onto n (T) . I'P = 0

a rid

J« ? /C"'

It suffices to show

I) 1.

71 (r) ,

- 2 16 -

J. J . K ahn

now

TP

=

T - TT*NT

since HT - O.

=

Flnally P - I

rR-(p:-

=

T - LNT =

=

T - T - HT

0

-T*NT and hence

I) C O{(T*)

.i. ll(T).

The above Hllbert space set-up ·ca n be applied whenever we complete the spaces~,q with a Hllbert norm, elther llke the ones

lntroduced ln 0.2) or with welghts.

up works for domalns in complex manifolds.

The same se t

In that case we

put a hermltlan metrlc on the manlfold thls lnduces an lnner product on the forms at each point, the lntegral of thls Inner product ls then the L 2- lnner product. lntroduce welg~ts here. Deflnltlon.

If [Lc

suppose that

~

iii ,

M

We shall also

a complex hermltlan manifold, then

ls a non-negatlve function, we form the lnner

product 0.)4)

where

~,t~a:;q(1J~> denotes

the lnner product deflned at

each polnt and dV denotes the volume element. L

p,q

p,q

(Q,A) the completlon ofC[

Denotlng by

under the norm assoclated

wlth ().)4) and by T and S the closure of the operators ,..,p,q-l

"3:(J:,

/y

p,q

~ LA-

and

P,q

a : {A.

(1 Q, q ~ LA.-

+ 1

'lie

flne the operator L as above and we say that the

then de-

~-NeumaIUl

problem for forms of type (p,q) on [L and weight A

ls to

-21 7-

J.J .Kohn prove the exlstence of an operator N as above. As shown ln the theorem, the exlstence of N then lmplles that the range of exlst ln general.

a

ls closed ln L2 so that N does not we shall solve the Neuma nn problem on

a-

certaln types of domalns by means of establlshlng estlmate

(3.16) and thls estlmate ln turn wlll follow from stronger estlmates.

Further, we wlll lnvestlgate varlous klnds of

regularlty propertles for N whlch wlll automatlcally yleld regularlty results for the

a - problem and

on holomorphlc functlons (3.33).

for the projectlon

-218_

J . J . Kohn

Lecture 4.

The basic a priori estimates.

We will proceed to solve the

~-Neumann

problem as set

up in the definition of Lecture) in the spaces (J.4).

We will assume that

n.

by

has a smooth boundary defined

by a function r as in Lecture 2.

First we wish to find the Denote by ~ the

smooth elements in.1)= Dom (T*) J' Dom (5). space of 'f'e.a which are C'" i-n

~iven

11 ,

Le.Z> = ~n&,q.

Then

1!fE:~ and 'II~ o..p,q we obtain, by integration by parts:

->.

f (e -~), YJ> S

(oyJ, i}-IJ)

(4.1 )

= (~).

ID , T

jl

=

=

S<,eY.(e

JL

(e~

j'(e->''f) ,'V)

A

~ 0,

e -;\ bSt

/'

r

(~-~ av ,

{~..

~ denotes the formal adjoint of

Before in the case

proceedin~

a

evaluated on dr.

has been normalized so that


(4.2)

dV

.(€(~,dr)'f" ~> d\/

(7'(S';dr) denotes the symbol of ~

Here we assume that

(4.)

- ' (>.)

where ~ ~ CCJII(.fi) , and

d 'v -

311J)

on

b11..

we will illustrate the formula (4.1)

p = 0, q = 1,11. C

(-2,jz

j

(: n

and

A = 0; 1 t reduces

,u)

U dS • +XJrf b1l z j j

S1nce the boundary term in (4.1) vanishes when ~ has comoact su npo r t 1n..o... we have

-219 -

J . J ; Kohn (4.4)

for all 'f'!'EQP,q-l

wlth compact support and slnce the set of

these ls dense, we conclude that

=e (\f,aI/J)

Then, slnce

j":(e

-).

'f)

= (T*\f1''jI)

(>-.,

ap,q-l

>-

for

(>.)

for 'fE;.~ and

we conclude that the boundary term ln (4.1) must vanish for al'l lfi
~€

cr (J',dr)cp = 0

(4.6 )

on brr

'fE:Dom (T*) whenever f((lP,q

Conversely 1t ls clear that and satlsfles (4.6). we conclude that

J!J

\f7f:t).

lf

Therefore, since Dam (5) ~ ctp,q, conslsts of all

f

"o~,q

for which

(4.6) holds. It wl11 be useful to express the ooerators

~

and

Jr

ln terms of a special basis of vector fields ln neighborhoods of boundary points. of

PEbSl. and on

1

Let w , ...

n ,~

U be a small neighborhood

U choose an o r-thcno rma L set of vector-

fields of type (1.0),

L (r) j

Let

o

L , ••• ,L 1 2 if

j

<

such that

nand

L (r) = 1 n

be the dual basls of forms of type

U.I'hen 1.f o/EQP,q

on

u(j Sl , If

(1,0)

ca n be expressed by

on

- 2 20-

J . J . Kahn

(4 . 8)

where I

= (i

J

= (J • • • ,J )

1

, ••• , i )

w1 th

p

1

q

1 <. 1

w1th

1

J

~

1

1

<

1 <. •• < 1 2

<

<.

1

j

J

q

p

< n

<: n

and IJ

co '!ben on

u()ll.

1

i

J

i

P

= (..,. 1\ • '11 6J 1\

WIt.."1\ os q .

we have au

. ~ en .

T

=~E

J

)c;;J

(u

JIJ

= c:~ Ej (toT

)w It.. W - + •••

= 2: ...

(If

1J

and (4.10)

where the

jfH -

IJ

J

H run over ordered

sent ordered ~

L

J.

IH

I<j H)

+ •••

(q-1)-tuples, <j H) re pre-

whose elements arecomb1nat1ons ot the

q-tuple

and the elements ofH.The dots represent l1near

From (4.10) 1t 1s 1mmed1ate that the cond1t1on (4 .6) on

un bfi is equivalent to

(4.11)

to T

IJ

=0

on

U bJ2.. whenever (\

nE-J.

- 221-

J. J . Ka h n

The fact that the operators

T*

and

are of the

3

form given above and that the boundary conditions are of the form (4.11) show that the argument of Lax a nd Phillips (see can be applied as in Hormander

f'31]

[l l) )

(proposition

1.2.4) to prove the following: Proposi tion.

:l>

is dense in

unde'r the norm

~

222 + If\! + \ If\\

II T*f\\ .»

\\3

o..} •

(A)

This proposition enables us to prove estimates for elements in ~

from which we can then deduce the crucial inequality

(3.15) of Lecture 3 for e l.emen t s in2\. n x n

Let

matrices defined by n

then

a

(4 .13)

ij

be the

k _

LL,r:J=2:a L +;> b L i j k-1 ij k ~1 ij k

(4.12)

Let

n

k

(8 k)

(c

ij

) c

k

ij

=

be the (n-1)

ij

a

n

ij

for

-b

~

1

k

ij

(n-1)

matrix defined by

c: r , j c: n-1 "

It then follows from the formula for the Levi-form ((2.31) of Lecture 2) that C

ij

is the Levi-form in terms of our

basis. The following formula is at the root of all the estimates which we will derive here.

If If, E j) and the

-222-

support of If lies in

J . J . Kahn

JlnU ' then

2 2 2

KT*lf"

(4.14)

(~)

S

+2-), .n,

+

-

+ US
(A)

= \\IfU

if

If

e

e

+

-A

4V

[jkJ I<jK> I
L5 e If' If. bn. jk !-<jk') I<:kK,>

_

(A)Z

-)..

dS

where

IIIfH

(4.15) and the (4.16 )

a

Qtt)

2

(?-)z

2

=

'2\\LID \\ j J IJ ('>I,)

2

+ \\ If\\

(/\)

are defined by:

A[jk:"\

'J

=

L

L(?\) + j k

~a

i

L (",). jk i

Observe that the norm defined by (4.15) is equivalent to (4.17)

We will derive formula (4.14) only in toe case q = 1 and A = 0,

p = 0.

to obtain th e gene r a l case one proceeds

in exactly the same manner, the calculations are then somewhat more complicated. From (4. 9) we obtai n

- 22 3-

J. J . Kohn

(4.18)

where

.

2

I\L If j
~alf'\\ = L ~\4'Il- = z

lllfn

l2 +

- 4.. If \ k j

0(\\4>1\

z

\\If\\ )'

_.

(O )z

Then (4.19)

II kj

~ III: If - I: 'f

j<.k

jk

2

211~

=

j,k

2

L

If' R -

j k

j,k

( ~ If ' j k

L

If) k j

and (I:~' ,I: tf j k k j

(4.20 )

)

To justify (4.20) we observe t hat if by

by part s

inte ~ration

(u , L v) = - ( L u , v) +

(4.21)

u,V

k

k

+

SL

bJl.k

.

o (\\u I! Il v )/ ) ,

j

<.

n

on then

k = n

since

'r he boundary does not appear in (4.20) since

Ife-:D implies t ha t

tfn = 0

then,

(r )uvdS

t he boundary term t hus appears only whe n Lk(r) =~kn.

cQO(lln U) o

bSL

(r

If ) . n

i s z e r o on biL; be ca use t hen

so the t e rm va nishes when

r lf = j n

Furthe r, we ha ve

j

0

on

b..D..

s in ce

L

j

j = n

a nd i f

is t.ange n tia l.

- 224-

J . J. Koh n

(4.22)

(L

Lip ,If)

= ([L

kjkj

(a

k

.LJ'f

.tp) +

.1kj

(L

L If'.Ip)

jkkj

Ltp.lp)-(Ltp.L-r)

kj n k

+ 0(

j

k k

j j

1l1f1l_lllf\\). z

where again no boundary term appears and we use the fact that (4.2)

(fL

If • If ) =

k k

j

0

q\\?\L \\tp\\) z

since the boundary term appears only when ~n

=0

on

bll.

Observe that 2

',\J
(4.2!j.)

k = nand

+ 0
).

and that (4.25)

(c

Summing on I

and

k

j

=Sf\ bJ(

j

c

l.P, ijidS +

kj k j

OQ\f\L~f\\)· z

and combining the above we obtain,

n cOl:lo (U,,5l)

.0,1

for tfe,Z)

(4.26)

If. '4' )

L

kj n k

2

1\~~11 + \~lf\\

2

2

= 1I1f1lz

+0 (

+

S

bll

c

II

If ~ dS

kj k j

- 22 5 -

J . J . Kahn

which is the desired formula (4.14) when

p

=

0,

A=

0

and q :: 1. The following theorems give estimates which lead to the solution of the Theorem.

If

(a)

for all

~-Neumann

PebCL

problem.

then the follow~ng are equivalent

~

..p,q

The Levi form at

P

has

positive eigenvalues or at least

n-1

q? 1

A :: O.

For

ei~her

q+l

at least

n-q

negative eigenvalues.

i t is strongly pseudo-convex, i.e. the Levi

In case form has

such that

with coefficients in

e;2).

(b)

for

U of P

There exists a neighborhood

positive eigenvalues, the inequality (4.27)

is an immediate consequence of (4.14) with

=n

q

being in~p,n means that

.

f

vanishes

on the boundary and hence again (4.27) is a consequenoe of (4.14).

In case

thus (4.14) with

we have ,2)p,q = oY,O

A=

is replaces by

2

0

and

J-cr= 0,

2

k\.alfJ\ :: lIlfll_ + 0 (lICf\lll'f>ll )

(4.28 )

for

q:: 0

z

nP'O

~€ VL.

•

z

Condition (b) for

q

=0

says that the

Levi form has at least one negative eigenvalue so that for some

i <. n

we have

c 11 (P) 4

0,

we can suppose that

-2 2 6-

J . J . K ahn

C

11

(P)

(4.29)

<

O.

uEC oo (lil"\ U ) 0 I I

For any function

~II: ull

2

=

, 1

+ O(\lu \\ lIu ll) Z

small enough so that

U

have

2 -«(L ,1]u,u) + \\L u\1 + O(\\u\Ln u\\) 1 1 1 z

. b{\. S c 11 lu \2 dS tak1ng

we

ell

<.

~Cll

(P)

in

U we

obta1n (4.30)

and from th1s (4.27) follows 1n case

= O.

q

A~

a final

1llustrat10n of how (4.27) 1s der1ved consider the Gase q = 1

assum1ng that the Lev1 form has at least two negat1ve

e1genvalues • .' Choose a basis form 1s d1agonal

~t

for

CU( P»

1-=. 1

m,

~

C11 (P) = 0

for

P

L 1 , ••• ,L n

and

assume that Ci 1 ( P )

0

for

m' z, 1 < n-1

small enough so that

\ ei i

1\I:(.j711 2 t III: i

j

1j

m < 1< m'

where

- Cii(P)\<

2

(4.31)

such that the Levi

lfl\2 1 j

m

> 2.

~ on

<

and Choo s e

U.

Then

u-sij )ln C 11f\ 2dS b 11 j

+ 0
1

~

t ,

j

<

m'

and set

U

0

- 227-

J . J . Kahn

£

11

=£

and

f

ij

=t

when

i ~ j

sUbstituting in (4.26)

we then obta1n (4.32)

+

+

+

\\J-crll

2

2

~ const.lllfll + z

5 f f..' (-t L.. c

bSL

L

j=l

c

IIIJI

b'j<m 11

jtm' jj ~Tj

I2

+ ,...

C

+€ c

) \lp \ 2

jj

If' M

j

L dS

17j 1j i.'·'I"jJ

By choosing E and l) small enough we get the coeffic1ents of I~JI

2

1n the boundary integral pos1t1ve and bounded away

from zero in

U.

On the other hand, the last term 1n the

boundary 1ntegral 1s bounded by cons t ,

L I4>J \2

where the

U

constant can be choosen as small as we please by making small.

Thus we obtain the inequality (4.27) in case

p

= 0,

q = 1, and the Levi form has at least two negat1ve eigenvalues.

To establish (4.27) for arbitrary

Lev1 form has at least

q + 1

ne~ative

q

when the

ei~envalues

we pro-

ceed in the same way. The proof that (4.27) implies (b) is given in H~rmander [io]

Proposition.

If condition (b) of the above theorem is

satisf1ed then there exists a constant neighborhood

W of

bll

C

>

0

and a

-228-

J . J . Kahn

(4.33)

for all 'l ~ 0, where

~

=e

and

sr

~p,q

If E- o

(\

COO (fl 'II U) ,

o

for some fixed large

s.

For the proof of this proposition see H~rmander[llJ • To establish

(4.33) in the case that SL

pseudo-convex, choose -) "ZiZJ

s

is strongly

sufficiently large so that

is positive- definite as in (2.10)' of section 2; it

(~

will then be positive definite in a neighborhood !I[jk]

Then from (4.16) we see that

W of bM.

is positive definite

and then (4.33) follows from (4.14) by use of a partition of unity. Finally the same argument also yields the following result. Proposition.

If ~E-c""(li)

in a neighborhood

W of

then (4.33) holds for all when

q

~

is strongly pluri-subharmonic bSl..

't?

and i f 0

and

n..

lff,'b,P,q

•

1.

In case there eXists a I\ 'E-C~(.(L)

is pseudo-convex

n c0

C>O (

fl (\ U)

which is strongly

pluri-subharmonic throu~hout tl, then the integral in (4.33) can be all taken overIL

80'-

thtt't we- have

- 229-

J. J. Kahn

(4.34) Hence by the results of the previous lecture we obtain the following result. Theorem.

If there exists a strongly plur1-subharmon1c func-

c"
and 1f .D- 1s pseudo-convex w1 th a smooth f\ , n-"\ ) pro bl em L p, q ( ~l .~~ has a b oun dary th en th e a-Neumann t i.on 1n

a

solut10n for

q

~

1

and

~

suff1c1ently large.

The condition that there ex1sts ).~ C""(J1) ~n,

pluri-subharmonic 1s satisf1ed in 2

\=\z\ ="Llzli

which 1s

b, taking

2

j

for example.

It is also sat1sfied for

fL

in a Stein

manifold, by taking

r.

N

2

=2:\h \ , 1

k

where

hl, ••• ,h n are holomorphic functions that separate points. Thus we obtain the operator N, for '[ C, and

->

for each 1r given by

(4.35)

T*NO(.

and L2 solution of the a-problem Now we have, if SO( = 0: 2

I\r*No
(T/,)

= (rr*N«, No£)

('tA)

(0(, Noc) (to£ )

(cont)nued

O~

next

Da~e)

~ 230-

J . J . Kahn

c:

C Ito(~,

-'t'-C

If (Az !.") 1 .1

2 (1:'>..).

ls the mlnlmum of the least elgenvalue of

IJ

we can ohoose

C = q

(J

-1.

In q:n

coord1nates so that the orl~ln "tlesln

~

=0

we can select the

Sl then settlng

d1ameter of..Q. (l.e. ~ = sup' \p-Q\, P, Q ~ SU

ohoos1ng .~_

(4.36)

= 1 z' 2, .~

_'1:'&2

<

Chooslng

er = q

Henoe 1n

en

+ ~ -2

we

have _ Izl 2

e'

C

~ 1

=q

and

for

zESt.

we obtaln

we obta1n the result of Hbrmander ( [10J

that there ex1sts an

L2

solutlon of the a-problem whlch

ls bounded as 1n (4.37). The above theorem, and also estlmate (4.37) can then be genera11zed to pseudo-oonvex doma1ns whose boundary ls not smooth by oonstructing an appropr1ate sequence correspond1ng to the exhaustnon of

Jl = Un.

c

(see Hermande r- [10]).

-231-

J .J.Kohn

Lecture 5.

Pseudo-differential operators.

In this lecture we will give a brief review of Sobolev spaces and pseudo-differential operators. and if

s

If

is a non-negative integer then for

nee lR n ue

c"
we define

!\ull

(5.1)

2

s

=

L

\\D""u \\ 2

\otl~S

and let H (it) denote the completion of c~(Ii) under s this norm. Hs (.0.. ) is called a Sobolev space and II u Us called the sobolev s-norm. These spaces have the following properties:

A.

H C£UCH LCl) s

further if

and i f

B.

If

if

s

>

t,

we have

s

t < t.;: s

then for any i)o 0

o

stant C(l)

for all

UEH

t

there eXists a con-

such that

ufH (fL). s u~H (fl). ~.

s

"11= 1.2 ••••

is

-2 32 -

J . J . K ahn

and 1f

II u)\ -<.

subsequence t

C.

C

tUv

J

independently of

1

then there exists a

-V

which conver~es in

H (il) t

whenever

< e, If

fl

has a smooth boundary and if

s

>k

+ tn

then

k _ C (0-)',

Hell> C s

this follows from the inequality

u~C~«(l).

for all u~C""(.rL)

A corollary of this property is that

1f and only if

u~nH

CO...>.

1\ \\

and

s

By dual1 ty we define tegers as follows. define

1\ ull

s

If

s

UECOO(D.. )

H s

and if

for negative ins

<0

then we

by:

(5.5)

\\u II s

= sup

where tbe sup is taken over all

(u,v)

II

v l\_s

vEC~(li).

From (5.5)

follows the generalized Schwarz inequality

(5.5' )

,\(u , v )\

S \Iull t\vlI s

-s

whioh is very useful. H (.Q.) s

is then again defined as the completion of

- 2 33 -

.T . J . Kohn

c" Cn.)

Itu II

under

and the properties

s

A,

B

still hold as

well as (5.2) and (5.). For

UEC

OO

o

en.)

the above norms can be expressed in

terms of the Fourier transform as follows:

(5.6)

II U\I I\s

where

co. CQ..) o ~

/ \

~

!\ u("S) =

I'

where

v

s

s

IIA U\t ,

!".J

n

c'' (~) (1 +

is defined by 2

log,)

i

A

u(~) ,

denotes the Fourier transform of

v

which is

given by

C _lx· 5

~ ('~) = ..) e

(5.8 )

v ( x ) dx •

Formula (5.6) can be used as a definition of s-norms for arbitrary s€~,of course it then applies only to compactl~ supported functions. Formulas (5.7) and (5.8) make sense for distribution With compact support. we

~ay

that s

We denote bV locally in

($u) (; L

2

q

u

is a distribution on

is locally in

u

f\

If

loc s

!ts (Sl ) .

U1.)

(S\..) rhe n

H

s


for all

UE ~!

s

then

if

:rE- Goa (fU. Q

the set of all

ir

II

Lo c (D-)

which are we have

-234-

J . J . Kahn

Su Eo

s

If U

c

P:C

n

n

(lR ) --7' Cr><>(IR)

00

o

R~ we say tha~

P

n

P

s

~

cl\ u]

1s of order

m

m

on

there ex1sts

o

lIS Pull

18 a l1near operator and 1f

1s of order

and every ~~ C 04 (Ii)

8tH

If

;r E- C oa CIt.). o

for all

H (.D...)

for every

C

1f for every

such that

u e-c00 (U) •

for all

s+m

U

0

m

then we say

P

1s of

order -00.

n

It then follows that 1s of order

m

Defln1 t1on.

A funct10n

symbol order m

and that

ao( E:' C""(/R )

I\ PE-C

m 60

1s of order

n

ua x Iii

n

1s called

)

1f for every compact

pa1r of mult1-1ndices ~ , ~

m.

KC'cR

n

and for every

there ex1sts a constant

C

(depend1ng on K,C< and ~) such that

SUp/Do(D0!p(x,~)1 -c C(l

(5.10)

x ~

K

The operator

n

n

P:Coo(lq )~C""(n

defined by

o

Pu(x)

(5.11)

where

Isl)m-l~J .

+

d~

d~, ••• ,df

n

and x '3

=I

x

j

f

j

1s called t he

-2 35-

J . J . Kahn

pseudo-differential operator with symbol ~ a~D~

In the above example

g:

nand

I\'TtI.

operator with symbol (1

1s the pseudo-differ-

\ot:.l6;m

~ a (x)

ential operator with symbol

-gp(.= ~«'1, •••

p.

I" 1= 'W\.

Sol..,

where

0(

is the pseudo-differential

~\sI2)m/2

•

The following are the properties of pseudo-differential operators that will be most useful to us. (A)

If

p 1s a symbol of order

(5.11) is an operator of order (B)

If

P

(0)

If

orders

p*

and

m

and

Q are pseudo-differential operators of

b

then

operators of

operator of order

P + Q

a + b

of order

is a pseudo-differential

Further the composition

order-~,

PQ,

is a pseudo-differential

and the commu tatoz u

[p,Ql i~

m.

is also a pseudo-differential operator

operator of order max(a,b). modulo

defined by

m. P

a

P

is a pseudo-differential operator of order

then its adjoint 9f order

m then

=

PQ - QP

a + b - 1.

The meaning of "PQ

modulo operators of order

-~"

shoul~

be interpreted as follows. If W is any proper neighborhood of the diagonal in \HnJ(IRn, (proper -means that if UC.IR n

.ass.

J . J. K ahn

•

is relatively compact then

U' = {(x,y)eW\XfUJ

un

and

= {(x,Y)fW1YfU

~ ,

are relat1vely compact)

Po of order

-00

, then there ex1sts an operator such that 1f Uemn 1s relatively compact,

then sett1ng

then for any

UEC

U =

Thus

P + Po

so that

00

o

0

(R n )

such that

tV

(P + P )(u)

on U

O·

=0

on

U.

has a un1que extens10n to an operator of

(P + PO)Q makes sense.

Further, 1f P'O

is an-

other such operator then

(P + P')Q - (P + P )Q

o

1s of order (D)

0

-OQ,

The 1nequal1ty (5.9) and the d1scuss10n above show

that a pseudo-d1fferent1al operator m .odulo operators of order

-00

P

can be extended, n) to UH s loc(R • A crucial

property of pseudo-different1al operators is that they are pseudo-local. We

. say that an operator

local if, whenever

u

P

then

P

is pseudo-

is distr1bution in the domain of

- 2 37-

J . J . Kahn

sing supp(Pu)

(5.12)

c:

sing supp(u) ,

where the complement of sing supp(v) is defined as those .po i nt s which have neighborhoods on which Definition.

v

is in

A pseudo-differential operator

P

ee

C •

of order

m

K there exists

is called elliptic if for every compact C such that

il p (x, ~ ) \ ~ cIs \m

(5.13) where

p

is the symbol of

Theorem. If of order of order PQ - I

P

large,

P.

is an elliptic pseudo-differential operator

m then there exists a unique (modulo operators pseudo-differential operator

-~)

QP - I

and

elliptic

I~ I

for

of ~order

are of order

-~.

Q such that both

Further, this

Q is

-me

In view of (5.12) we have the following: Corollary. such that

If Pu

particular

P

is elliptic and

=f If

n

P

UfH (1Ft •. ) and -N Further we have

is a distribution

then sing supp(u)x: sing supp(f), in

U€C~(Sn)

Proposition.

u

whenever

fEC~(an).

is elliptic of order Pu~H

s

n (IR )

then

m and if n

UE H (S). s+rn

-238-

J . J . Kahn

Hull

(5.14) for all

s+m

~ C(~Pu\l

s

+ \\u\\

-N

)

n u~H

s+m

(~).

n

Conversely, if (5.14) holds for all some fixed

N with

sand

-N<s<m

uec " (fi )

o

then

P

and

is elliptic

of order m, The estimate (5.14) follows immed1ately from the theorem and (5.9) since

= QPu

u

w1th

Q of order -m and

Proposit10n. order m > 0

If

2

Proof:

-~.

1s a pseudo-different1al operator of

< II u \i h P

K of order

and 1f

(5.15) then

P

+ Ku ,

+ II u II

C ( Pu , u)

2

is elliptic. It suff1ces to prove (5.14) with

s = 0

and

N = O.

We have

(5.16) /lull

m/2

2

=1\'/\ m ,

{[

ull

2 m/2

'S:

m/2J

~c \( ?, !\

1\t>uIII\ ull

m/2 m/2 2 C(p!\ u, (\ u) + \\u,1 m/2

u,

+ In

f\m/2

IIul)2

u)

m/Z

\

'1 J

+

c on t , next pa ze

- 2 39 -

J . J . Ka h n

< -

2 C{\\Ul\

+ La r ge c ons t , iiPu\\ -

m-~

+ small const. nuli

2 11

+ I\ui!

2

2

•

m/2

The desired estimate is then obtained by use of (5.). In case

P

= ::[

ao< 0"-

ellipti city is equivalent to

\oq~m

the condition

a.J \ol.l=m L,

ol

5 "

if

"0

0

All the above results can be generalized for determined sys tems.

If

P

differential operators and functions.

kxk

represents a

Defining 1\ Ull

t s. 18 )

2

/lull

s

u

matrix of pseudo-

represents a k-tuple of

by

s

k

= ~

j=l

\lu /I j

2 s

Then ellipticity can be defined by requiring (5.1 4) to hold for some -N < s < m.

For a matrix of differential operators

this is equivalent to requiring that the matrix of the principal parts be non-singular.

All the above theorems

the~

hold for the case of such operators. We conclude with the observation that all the above results can be "localized" in a natural manner 1.e. the notion of ellipticity can be defined on any open subset

of

Rn

In

and the natural analogues of the above results hold.

- 240 -

J. J . Kohn

particular we have: Theorem. If P is a differential operator defined on UClRn which is elliptic and of order m, 1.e.

L

P =

ao
with

a ol ~ C"'(U)

with

S = 1

,\ .q ~m

and (5.11) holds then for

J, S'E: C""(U) 0

in a neighborhood of the support of

1I5 u \l

s+m

for all

~ C(\\)"'Pu\l

s

we have

!

+ \I)'ul\

-N

)

C depends on -S, $' ,s and N.

UtC""(U), where

It

then follows that sing supp (u) C sing supp Pu. Proof.

Let )1 E:- - : (U)

hood of the support of the support of 3 1· tial operator on

be such taat $1

s

Then let

Rn

hood of the support of

-C'\}

and

Q

5'0.

taking

QP'

of order -me J'u

=1

=~P.ru

s + m norms we obtain

Let

=

in a neighbor-

in a nei ghborhood of

be an elliptic differen-

p'

which is eqU.al to

ential operator such that order

and 5'

=1

P

in a neighbor-

Q be a pseudo-differI + K where

rnen + K(.:s- u )

K is of

-241-

J. J. Kahn

1\:Su\l

(5.20)

s+m

~ 1\ p:sull

+ cons tvll uil s-N

~ \\SPuI\ +1\ \:),"s] u\\ + const.\l-Sull s s -N since the support of [p,,51

5 and since (5.21)

[p,~J

II\[p,;s]ull

is an operator of order m-1

= 1I1p,!]5

s

is contained in the support of

1

un

s

'S

const.II,~ uj( 1

we have

s+m-l

so we obtain (5.22)

j\:sull

s+m

~

II! ull 1

s+m-1

+ \lrPuI\

+ const.ll~ulf

s

-N.

Repeating the same argument with s and

s

by

and 'S1

.)1

"5 E: CDo(U) 2

0

by S2

2

= 1

in a neighborhood of the support of "S neighborhood of the support of "S. 2

U:S~II

s+m

k

=N+

and:5 = 1

1

After

s

+

Choosing

I

~const·(Il:rPuI\ +\l3

k

s -1

where

With ~

obtain

replaced by

II 5

ull

k k

ull

in a

k steps we

s+m-k - N.

s + m concludes the proof of (5.16).

-242-

J. J . Kohn

It rema1ns to show that 1f OIl

C,

where

the !

V

and :i'

Pul V

1s

C 6d

1s an open subset of V. 1n

Co

(V) •

To

and hence 1n

V

1s

do th1s we choose

'SuE-H

s+m

for every

C~.

Th1s same result holds for when system.

u}

Then us1lfll; a standard smooth1ng

operator 1n (5.19) 1t can be shown that s

then

P

1s an el11pt1c

-243-

.L.J... Kohn

Lecture 6. Inter10r regular1ty and ex1stence theorems. As 1n Lecture J we set

(6.1 )

(T*¥l,T*IjI)

for If Lemma.

'!bere ex1sts

51/1)

eX)

such that

C;> 0

2

2

IIfll1 ~C

for all ~~ e.t

+ (5."

From (4.14) we deduce the follow1ng

,'fEe~.

(6.2)

on

("A)

p,q

If

w1 th

ql\(f,'f)

+lIepll

on

b5l,

= 0

where

depends

C

/t.

Proof.

S1nce

(6.))

11111/ T

2 Z

~

=0 _

= L~'L1If

bil

on

IJ

1\

we , ha ve

2

also 2

0 \\ <. \\ 1 T Z -

const.l\f/l

2 (,,>-)z

and

(6.4)

2

"'flll ~ 1

const.(IICfIl

2 z

+

2

Il'fL ~ z

const.lltpll .

2

...

(),)z

S1nce the .bounda r y 1ntegral 1s zero we obta1n (6.1) from

(6.4) and (4.14). From th1s lemma we see that the operator

T*T +

ss*

.1s elliptic.,. ' of second or.d.e.r and hanaa we conclude that

-244 -

J . J . Kahn

whenever the a-Neumann problem has a solution

L~,q(.Q,r.) ~

is smooth.

given by

No(

then

is smooth on any open subset on which

Hence also the solution of the

T*No(, when o
and

SOl

a- 'problem

= O,is -smooth

wher.-e

Under these circumstances if "S and

<X.. is smooth.

'S' EC;Cn)

N on

we have, by (5.16) with' u = No(.

P = T*T + Ss*

and

(6.5)

'nleorem. hood

If

W of

x

function at least

Il.

is pseUdo-convex and if in a neighbor-

b.Q

there is a strongly pluri-subharmonic

and

n - q

q

~

1;

or if the Levi form has

positive eigenvalues or

q + 1

eigenvalues then for sufficiently large 1:' LP,q (il, 1:: A),

problem has a solution on xp,q

2

negative

a -Neumann

and the space

is finite dimensional and consists of elements which

.

are in COO on Q Furthermore, if (tp,q : I 0 for q~ 1. Proofs

the

ei~her

The case

W =Jl

W=

1L

follows from (4.)4).

then

To establish

the general case it suff1ces:to'show (by the results of Lecture that if

»

that if

l\lf"ll is bounded, 11m q

't'i\

then there is a subsequence

(v ,'fol

)

4'v..L dtp,q and = 0

such that

-245-

J. J. Kohn

in L~,q. that

in a neighbrohood

~1 = 1

Then, if r

Jl - W.

J 1e

To do this, choose a function V of

C~(1L)

bSL

such

and is zero in

is sufficiently large, we obtain the

following inequality as a consequence of (4.33).

(6.6 )

1I.:s"1'P1I

2

s const.q ~ canst.

• p,q for 'f'.~:b

('t',\)

(q

(-t-).)

where SEC ~ (.Q.)

. (Jr.-If', Jd)

(If,~)

and

!

+ II ~ If' II~

000

= 1

on

.n.. -

. V.The.

second inequality is obtained by noting that

(6.7)

<:: cons t ,

-

since

J0- -

.

ls,s) 1

C\IStp" + 1\50\iD\1 )

is a matrix whose components have supports in

V and hence are bounded by const.IS \.

A similar

o

calculation for

T*'1

establishes (6.6).

Further, from

(6.2) we obtain (6.8)

\t:r~l\

o

where S' fC""

o

0

2 1

~ const. q

(0.)

(lfl,If) +

lIS' 5&'11

2

0

Ln.) and 5' = 1 on the support of S •

Combining (6. 6)

0

and (6.8) yields

0

-240 -

J . J . Kahn

Apply1ng (6.8) to the sequence

It j 0 Lf'elll

that tfv

J

{
have to conclude

1s bounded and thus there exists a sUbsequence

.r0
such that

tha t for every such that S'lf'v

J

J

r

converges 1n

G Co" Ul.)

thus we see

L~,q (fl.)

there ex1sts a subsequence Lfv

converges in

p,q L (Jl).

r =5 •

setting

2

and applying (6.9) to the differences I.fv

-

i

LPlJ

J

J

o

we see that

a Cauchy sequence and hence converges in Therefore Cf V

=0

e E= L~,q(..o.

e

so it also converges to qt")- (G.e)

converges to

J

)

and

Since

and 8.Ct:,q

we conclude that

e=0

as

required. Th1s solution of the

~

-Neumann problem can be applied

to prove existence of holomorph1c funct10ns as 1nd1cated in the first lecture. Theorem.

It gives the follow1ng result.

Under the hypothesis of the prev10us theorem and

1f there eX1sts

P~b~

such that at

P

there exists a

local holomorphic separat1ng funct10n, then there ex1sts a holomorph1c runc t ton on

SL

any domain which contains ~:

which cannot be continued to P

in the 1nterior.

. As 1n the first lecture we construct a function

which is not 1n L cn.. ) but such that 2

01..

=
is in

F

-247-

J . J . Kohn

L (il ) , what is more the oon~truotion oan be so arranged 2 that if F~ is a small translation of F in the direotion b n.

normal to 80(=0 '9l..

then d..

= lim a F" •

and that d...l.dt°,l.

Then

= T(T*Nod

I:: L

and

T*No<

2

'!hus we see that

(nJ

thus

h

=F

-

T*N~

is the desired holomorphio funotion. Another applioation of the

a-Neomann

prove the Newlander-Nirenberg theorem.

problem is to

To do this. we first

define almost oomplex struoture. Definition.

If

Sl

is a differentiable manifold and

CT

denotes the oomplexif1ed tangent bundle then an almostoonplex struoture on

Jl is given by sub-bundle T1• O oftT

wlthth~ following property.

gate of

If

T1,0

~.1

denotes the conJu-

then 1.0

t:T

=T

(:T

T

Jrl

If

1.0

0.1

~T

+ r

i.e.

1.0 T (\

0.1 T

=0

0.1

is a conplex manifold then the underlying al-

most-complex structure is given by the vectors of type (1.0). It is clear that if there is a complex struoture associated with a given almost-complex structure then it is unique. Definition.

If

..fL

is almost-complex with t he r almost-complex structure giTen by T1• O is called integrable if for any two local vector fields

Land

L'

with values in

-248-

J . J . Kahn

1,0

T

'I

[L, L'j

the commutatox Clearly if

1,0

T

also has values in

1,0 T

is the almost-complex structure

associated to a complex structure then it is integrable. Conversely we have 1,0

[ 25J i.

If

T

an integrable almost-complex structure on

st

then

Theorem.

(New1ander, Nirenberg

a complex structure such that

T1,0

gives

Jl

has

gives the associated

almost-complex structure. First observe that this theorem is strictly local, that is, it suffices to show that given a point

P ~

n.

there

h 1, ••• ,h n defined in a neighborhood U of P, such that dh 1, ••• ,dh n are linearly independent and such that for every vector field L with values in T1,0 we

existr function

have L(h ) = L(n

(6.10)

Then

j

j

).~O

for

J = It ••• ,n.

hi, ••• ,h

are a complex coordinate system and it is n clear that a function u defined on an open subset of U satisfies the equations L(u) = 0 for all L with values in r 1 , 0 if and only if u satisfies the Cauchy-Riemann equations with respect to the coordinates

h 1, ••• ,h n•

On an almost-complex manifold the space of form (l a natural

bi~radation

has

-249-

J . J . Kohn

by

TT

p,q

:

0. ~~,q

a

Proposition.

and

d

= 2-

rr

= Lp,q

p,q+l

p,q p+hq

~

+ TO,l.

Denote

the correspond1ng projection mapp i nga ,

We define the operators

(6.11 )

= T1 , 0

~T

decomposItion

1nduced by the

by

IT

d

IT

ap,q

d

TT

p,q

For an almost complex structure the following

are equivalent. (a)

integrable

(b)

For every form

~

of type (0,1) we have

Erdlf = °

2,0 (c)

If


is a (O,l)-form then

(d)

If

u

is a function

(e)

Proof:

d

=~

+

a

By a standard

d

2

~

2

d

a If>

°

u::

= a = ° and p'3 ::: 2

form~la

+

=«)~

-a d •

for an exterior derivative we

have

(6.12)

(d1, L"L')

= L«~,

L'»

- L'

- {If' [L, L~/ choosing

Land

values in

T1,0

L'

«If,

L»

,

to be any two vector vields with

we have

-250-

J. J . Kahn

since If

is of degree (0,1).

if and only of that

(a)

alent to

is equivalent to (b)

T1 , 0

then

2,0

Cl If =

hence i f

u

(b).

since for forms

holds i f and only i f

ofT d

is of type

L, L'

have by (6 • 11 )

2 (lu=

'!hus we see that

11 dlf 1,1 . 2,0

L

Clearly

3l.f =

o.

and

= 0

which shows (c)

is equiv-

(0,1)

of degree

~

and

dlf =

(1,0) ;

1T 1.

2,0

TT dr

0,2

we

so that

(c)

By (6.11) we have are vector fields in

L'

(6.13)

since -au = . du 1,0

we obtain

(6.14 ) this is zero if and only if that (c)

(e) is equivalent to and

(e)

[L,L'] (a).

function .u

of type

So

and their conjugates extend to forms of all (f).

On an integrable almost-complex manifold a satisfies the equation

~u

= 0

du = 'Ou and this in turn 1s equ i va l.e n t to L

(1,0).

Finally t he conditions

types and hence are equivalent to Corollary.

is of type

(1,0).

if and only if ru =

a

for all

-2 51-

J . J . K ahn

If

JL

ls a domaln ln an lntegrable almost-complex

manlfold then we can put thls manlfold a hermitlan metric, 1.e. a

metrlc whose lnner product lnduces a hermltlan lnner product on (T such that T1,0 ls orthoAermlt14~ ;

gona.l to

TO, 1.

'!ben all our results ell the a-Neumann

problem generallze wlth exactly the same proofs. Let

2n UC ~

be an open subset on whlch ls deflned an

lntegrable almost-complex structure • . We choose real coordlnates

x 1 , ••• ,x 2n

such that the ball

B={xe:R2n ls con talned ln

2n

\

:2: Ix I 1

For each

U.

j

t €: LO,

by

for each

j

(1,0)

21"\.

= L.. a

b,)

tE(O,l]

«

iJ

Let

basls for the forms of type

(6.15 )

2

j

k=1 k

1

we define the map 1

fJJ

t •••

n

,6.1

be a

1.e.

(x)dx

k

we set CAl

j

t

= t

-1:f..*

j

~ ·w ~.

these define an lntegrable almost-complex structure on for each

t~ (0.1) we shall denote thls by

ooordinates we have

(6.16 )

t.J

j

t

2n

= '>

j

a (tx)dx

j(;1 k

k

B • t

B

In terms of

- 252 -

J . J . K ah n

We denote by

B.

Note that

t

d , t

B

°\

j Z

j

w

we have j

= dz

°

the operators associated with

has a complex structure, in fact setting

(6.17)

dz

.a·t

j

=d

z

j

= La

°

°

j

so that

° ° ° Now ~

.

=LIz

j

k

k

B

are independent.

and the function

(O)x

\

°

2

-

(} z

j

=°

and the

°°

is strongly pseudo-convex

is strongly plurisubharmonic

°

on B.

Hence for small

and ~

is strongly plurisubharmonic (in the sense that

.

0'

t,

B

t

is positive definite). which varies smoothly with a ~ <0 t ~ ~

a fixe.d

2

t

~lfll

('t.q

for all If

~

t

B. t

.1,0

1J 2

where

t

t

Hence, when

L ' 1 (B ,'r,x) 2 t ~ and~'

~ const.

(l\T*lfl\ 2 t

t

t

° when

we conclude that there exists

such that (4.)4) holds in

LO,l(B ,tA)

the space

°

Choosing a hermitian metric

with a constant independent of

(6.18)

on

rt

is strongly pseudo-convex

t

i.e.

. 1,

'J)

t

2 \.;

+ \\5 \f\\ ('t>') t t ('t:f.)

denotes the norm in

II

with respect to the hermitian metric the a-Neumann problem on

"jO,l has a solution and dU t

= 0.

have supports in the interior of

Hence if Band J'

=

1

~253-

J . J . Kohn

On support of j then (6.5) holds independently of liN 0<11 t s+2

(6.19)

<;. const.

HI'S'olll

and

0.

for all of.. G L 0,1 (B) 2 t

a u(x)

(6.20 )

t <.. k

= Lb

t

t,

i.e.

+ \\"t.\I) .

s

Further, we have

aU

1

(tx) _ _ W

axk

i

where (6.21)

Hence 1f

U,,=C(B)

k

j

k 1

k

~b (tx) a (tx)

=~

we have for any

0(

11m

(6.22)

t~O

uniformly.

z

and we have

-

d

interior of

=1

S' =

Z

j

t t

In (6.19)

5'

ClI_

t

u =

o"'d

0

1

u

Now we define the function

(6.23 )

orig1n,

o ()

j

j

t

= z

j

TiN ~ z

z

j

t

by

j

t t t 0

0

= O.

choose

3", 3 o )

and

=1

S'

with supports in thE

B

such that

1

1n a neighborhood of the support of S a n d

o

in a ne1ghborhood of the

1n a neighborhood of the support of

o

J;

then

- 254-

J . J . Ka hn

115

(6.24 )

o

(z

J

- z

t

~I\"SN "3 zJ U t t 0 s+l Now for

~

\O(,~

0

yu

J

~ I,\T*J':N "§ z ~ s tOttOs

l - t'OJ

<... const. Ij'~z\\

- J\ +HJz\.

s +

t 0

large we have

'"

(6.25) sup

V

s

J

\

1

D~(Z

j

t

j

J

j

- z )( ~ const.R! (z - z tOO t t

)U

henoe, since the right of (6.24) goes to zero as see that if gradients of 8

to

=

t z

1

to

t

o

1

t -70

is sufficiently small then on

, ••• ,z

n

to

we

V the

are linearly independent.

Thus

admits holomorphic coordinates in a neighborhood of the

origin and hence so does 8

s

are given by

z

j

to

(xt

8, -1 0

1

in fact the coordinates on

), which oroves the theorem.

-255-

Lecture

J . J . Kohn

Z.

Boundary regularity.

In this lecture we will assume.n.. complex manifold with a

C~

is a domain in a

boundary which is pseudo-convex

and such that there exists a strongly piurisubharmonic function in a neighborhood of

b 11 •

We wish to discuss smooth-

ness of solutions of the a-problem and of the 3-Neumann problem in the ca~sed domain Jl, boundary.

i.e. up to and including the

We will restrict our attention to the a_problem

for functions or equivalently the forms. =~v

~-Neumann

A nautral question to ask is, given a (Ot1)-form when does there exist a solution u sing supp (w)

(7.1)

property in .n.. ;

.n. .

=a such that

of~

sing supp ( i.

It is easy to see that every solution in

problem on (0,1)-

u

has this

however, we wish to interpret the above

The problem is more delicate there for i f

solution and

h

u

is a.holomorphic function then u + h

also a solution which in general will not be smooth on

is a is b(l.

The following example shows that it is not always possible to find a solution satisfying (7.1). that in a neighborhood

(7.2) Let

11

r e CS( U

such that,

(7.3)

I ) ,

r

c,

U=

wlth

U of (0,0)

(zl,z2)~U U'

we have

I Re ( 2: 2 ) .<- 0

a neighborhood of (0,0) and

= 1 on a neighborhood .{

Let Il c ~ 2 such

_..E.... U = J(z2)= z2

V

il'

of (0,0) and let

U

-256-

J .J .Kohn Then the support of ''"

is contained in

We will show that every solution of in .!L - W'.

where

U and

for some posit1ve numbers f(z ,z ) 1 2

is contained in

S" (z ,z ) (. 1 2

I

3u =

\z

I·~

n

and

A

U'

~

with

= A,

U'.

a<

h(z ,-g) 1

is bounded by

~O

0

Re(z ) 2

<.

0

~

h

=u

-

by

K on the circle zl

< He(z ) <. 0 02-

.J'

then

~2'

h

h

1s holo-

to the line

we have

= u(z

u

1

(z , -'f,)

+ ~1~__

,-~)

~

K in ~ -W'

= A,

h(O,-b)

Re(z2) 1

2. S

which is a contrad1ction, since IZl/

-~ ~

= 0, - ~

1m (z ) 2

Let

If

h(zl'-~) on

For

and the set

I I z 1I

0<

iii.

the set

0,

morphic; consider the restriction of

=-

1s unbounded

are so chosen that

~ 0

=

A, 1m(z ) 2

1

does not intersect

zi

~

WIt is a neighborhood of

simplicity we suppose that

(7.4)

W = (U' - V)

then

h(zl'-~)

is bounded

= -0;

on the other hand

-K

h(O,-$)

is an average of

= A.

The theory of interior regularity of elliptic operators extends to boundary regularity for the so-called coercive boundary value problems.

Suppose

Q

is an integer-differ-

ential form on k-tuples of functions on

.fL

expressed by:

-257-

J . J . Kohn

where

=

u

(u , • •• , u ), v = (v , ••• , v) 1 k 1 1 k

,0 u ,v a 1 j E C (jt.). j j'

Cl>4 funot1ons on

Suppose that

bfL

such that

and k

»: h

B

1s a

matr1x of

B

1s of constant rank.

'!'h en 1f

k

j=l

for all

U

~ k 2 ) ~ cons t , Q(u,u) + L.. \\u \\

.:2

Z \Iii II j

wi th

1

j=l

BU

=0

on

j

Q

b Ij, , we say that the

str1cted to the space of k-tuples

satlsfY1n~

Bu

=0

re1s

ooero1ve. Theorem.

Suppose

1s coerc1ve.

Q w1th the boundary cond1t1on

on

r--:

a3

c5

denote the complet1on of

the r1ght s1de of

(7.7).

Then 1f

UE-

under the norm

cr; and sat1sf1es

Q(u,v) = (f,v) Then

for ~ll

s i.ng supp(u)

C

s Lng supp(f)

and 1f

3, :S' E then

=0

Let

03 = {u \ Bu = 0 and let

Bu

C""(0.),

o

.s'u

5 u~H

L <..~U

2

ea)

s +"2.

and

and

S' t; H s (J2.)

~1ven by

- 258 -

J . J . Kahn

n1 ulls+2 'So

canst.(ll.:s"f11 + IIs'u s

U) .

The ~-Neumann problem 1s coerc1ve (i.e. (7.7) holds

Q

when

=q

note that

(7.10)

act1ng on

only when

1mp11es that

(~.1~)

q(lj>,lf)

b)

5 I 'PI. 2

2

$

q = n-1.

cons t , (~lfl\_ +

dS +

n b~l.

z

To see this

2

Ulfll )

hence 1f (7.7) holds then 2

1\.lf.1I S 1

cons t ,

S

2

(IItfll_ + z

bU,.

IfI

2

2

dS + Illfll )

and th1s is true if and only if ~ = 0

on . bit.

We do however, have the following global regularity result (see Kohn 117] ). Theorem.

b Jl.

Suppose

there ex1sts a function of ~k

bit.

problem 1n

N L

then

p,q 2

p.q

N (9-

q

2

A wh1ch 1s

Then for every integer

such that if

Corollary.

Jl

is Coo,

k

pseudo-convex and if C""

there ex1sts a number

g1ves the solut10n of the ~-Neumann (il,~~)

w1th

q ~ 1

k _

) C C (.Q)

e

if

I'r

."

>- Itk

Under the samle hypotheses on

1, and if

au

= t::J..

1n a neighborhood

1n

L2

n

then for each

ap.q

if

ocE

k

there ex-

- 259-

J . J . Kogn

k _ ists

u

(fU

EC

k

such that

~u

k

=~.

Here we give a brief outline of the proof of the above theorem.

First we set

(7.12)

p,q and each of €:- L (iU

Now for each

2

q.,p,q there exists a unique lfrE-.u

such that (fA p, q

for each tj; f-
A

s that in fact

such that

)\lRt: II s

(7.14 )

'0

'-r

~ H (n)

s

~ const.

To prove that l.f. E H (ll), 'l: s the assumption that

there exists

whenever -r :> A ls

and

Hc
we first prove ·(7.14) under

is smooth.


s

•

For

PEb.D.

choose a

"boundary coordinate system" in a neighborhood U of P that is, we choose functions x 1 •••• ,x 2n-l . such that 2n1,dr 1 dX •••• ,dX are ' . UlJ f; ~ r l y., independent t.hrougnou t U we say that and

x 2n = r

x 1, •••• x 2n- 1

is the normal coordinate.

lf~j) and the support of index r:I..

are the tangential coordinates

0:'f' f- V. where

ip

lies in

Note that if

U (I {l

then for any

- 260-

J. J . Kchn 01.

= (-L)

D b

101- I

-~----

and

IJ

tX ~ . 0( D4'= L. D"

b

IJ

W

b IJ

where

=

~

Ltp W IJ

IJ

starting with the inequal1 ty (4,;3')

-r IIlfll

(7.16 )

sUbstituting

01.

D

b

rtf., or

2 (t"~ )

~ C Qoc(lp,tP) ,

for

I.f in (7.16) we obtain

We wish to estimate the right-hand side of (7.17) by using (7.1;) with an appropriate

\f f:

i>.

Integration by parts

gives (7.18)

+

o{"L II D~!'f. 'i\l 1'~I~s b (TA) '!:

+ Cit'

L..\\ DTj ' cp.'[" If

('1 <.s

where the 4 "through m-tuples, :r'f:C~(U(\?L) with runs 0

.:5' =

1

-261 -

J . J . Kahn

on support of

S.

Further, we have by (7.12) and integra-

tion by parts

= (D.,(5tl, D" S .\ + 0 ( c.-l\o<: II b b 'Pc ~ s

The fact} b~

1\ Cf....\\

. ~ s-l

)

is non-characteristic (which is a ~on­

sequence of the ellipticity of the "laplacian") implies:

Assuming (7.14) is true for

s - 1

in place of

obtain from (7.18), (7.19) and (7.20) With

~

s,

we

sUfficiently

large

Ttle fact that system 1.e. in

tp-r;

E-c(
and

=

cY-.

are connected by an elliptic

to solve for the normal der1vates

0(

U as follows

(7.22)

~

=L a

KL IJ

r:J.

KL

-262-

J.J.

where

F

is an operator of first order.

IJ

of (7.22) and use of (7.21) the 2

(7.23)

1I"!4'~"

("t,q

Kahn

Differentiation

yield

~ cons t ,

2

\lcLll

The desired estimate (7.14) is

s

t~en

obtalned using a

partition of unity w1th (7.23) for nelghborhoods of boundary po1nts and (6. 5 ) for neighborhoods of interior points. ~ctualll l>!'Ove

that CP"

E H (St)

s

To

we use the method of

"ellipt1c regularization" (see Kohn and Nlrenberg lt9] ). Cons1der the form

where ~en,

r

>0,

!"f'L

the l.) ) are a parti tlon of unity and '1', v

clearly

Q

'"

is coericlve 1.e. (7.7) is satisfied.

Hence, 1f we denote By under

for all

Q~

""p,q

Z

the completion of ~ ..... p,q then the unlque
. p,q

Iff- .3

,

"""t>,p,q .

is in

C t>O (Ii)

and hence

:l5 p,q

Now,

the same derivat10n is that of (7.14) yields

where the constant is independent of

~ when ~

is small.

-2 6 3-

.J . J. Kohn

Definition. is

We say that the a-Neumann problem in

!-subelliptic

hood

U of

P

at

PfbJIL

L

p,q 2

(ii)

if there exists a neighbor~

such that

v: 'I'll

(7.29)

2

b

~ canst. Q(V,'f)

for all lfE-ilP,q (\ Co (Urn. ) where

and The estimate (7.29) 2n

j=l

in particular if holds). arly ' £'

t -1

L \\ f\

(7.)0)

£

b

is equivalent to

~"Sif II

a xj

=1

then

2 .

s canst.

Q

is coercive (i.e. (7.7)

If (-ell ipticity holds and if

s

Theorem.

Q(tr./f) ,

'£. <;:. 1

t hen necess-

Pfb11.

the

ti If ..Q

Neumann on

is such that at each

LP,q (fl

3"-

) is 'i.-subell1ptic then it is solnable "

and (7.27) holds. From the theory of Sobolev spaces it follows that the norm given on the left side of (7.30) is compact in LP,q(Un.n.)

from this it follows that the ~-Neumann problem p,q (fl). The pseudo-Iocalness depends has a solution in L 2

2

-264-

J. Jc.Kohn

It then follows that there ls a sequence that the arlthemetlc means of the sequence Cauchy sequence ln

Hs

0 v ...-" 0 such

(Cf; v J

and slnce they converge to

are a ~~

ln

thls lmplles that lfrr f Hs • 'ro pass from the smoothness of f/f1 to the smoothness of N't(l( reQ~lres a functlonal an-

L

2

alytlc argwu.nt gl ven ln Kohn ' [171 • We now return to the questlon when the a-Neumann problem has boundary regularlty.

More preclsely. we want to

show that. under certaln clrcumstances. the operator

N ls

pseudo-local; ln the sense that slng supp N ~ C slng supp 0<.. where the slngular support 1.s consldered ln

If

U ls a boundary coordlnate

.D..

nel~hborhood

x 1 ••••• x 2n - 1 • x 2n = r s .., 2n u~C""(U(l.R) we define !\ u E C (lR ).

wlth boundary coordlnates sell( and

IR_2n

o

1 ••••• x 2n ) = {(x

I

x

2n "S. 0

J

b

-

as above, then where

•

~ 2 ¥or-' > /\ u«( .r) = (1 + I ~ I ) "u(~.r" b

where

S=

(~ ••••• S2n_1)

and

~(~,r)

denotes the partlal

Fourler transform deflned by

(v(f.r) \.- = 2f

~n-

where

and

-lx·f

1e

v(x.r)dx

1 2n-1 dx ,- dx • •••• dx

-265-

J. H . Kahn

on the follow1ng: suppose ~~b

is the solution of

(7.31) Then if "S ,:S'fC;(U(\n.) of :5"

and 1f

! k e Hs rn )

with

then

:S'

u- 2

J\

b

=1

on the support

)'fE- Hs~2 (Jl)

and we

have H.-2

11/\b

(7.)2)

There

S
II s+2

s const.I\3"'cXll •

e~flmt~es ;arl!J:Obta1ned ,b y

Ar:;

cr epl ac ing

then proceed1ng by induction on

~ 1n

; (7 ;jO) ,"'With

m with the same

type of argument as in the previous theorem. Theorem. n-q

If the Levi-form at

pos1t1ve e1genvalues or

ptbSl

q+1

negat1ve e1genvalues

then the a~Neumann problem 1n L~,q(Jl) at

either has at least 1s t-subel11pt1c

P.

~:

Due to the results of the prev10us lecture 1t

suffices to show that (7.33) for all

J

t 2 (2 2 2) !t(\bu\l 6 const. l\ulI~ + b.f\. [u ] as + ~u\\ UE-C;(U (\

iL).

L1, ••• ,L be a basis of the vectorf1elds in T1,O n U such that LJ(r) = 0 and Ln(r) = 1 throughout U. Let

on

It 1s then eas11y verif1ed that 1f for each r, Pr 1s 2n 1 pseudo-differential operator on nl which var1es

- 26 6-

J . J . Kahn

smoothly w1 th

r

= const.

...:Lu {


as a

the operator

Pro

Now 1t suff1ces to

by the r1gh t-hand s 1de of (7.33) w1 th

axj

d

S1nce these

2n.

vectors

U C \R.2n

of the same order at

b

<

p+-(J\, '3) e c

1ts symbol

acts as a pseudo-d1fferent1al operator on

bound \\ /\ -t

j

(1. e.

r , x and ~ ) then on

rune mon [Ln,PrJ

'r

are l1near combinat1ons the

aXj

and

Ll' ••• ,Ln_ l ,L l , · · · ,Ln _ l

N

= Ln

- Ln

1t

therefore suff1ces to bound

1\ 1\ -f L u\{ ,\Ill -i L u~ 2 b

and for

b

V\ -'NU\\2.

and

j< n

j

j

The second of these 1s

1mmed1ate.

-I

(7. 34) \\ f\ L u 1\ j

where

po, QO

2

=

o (L U,P u)

~

cons t ,

j

=

0

0

(u,P Lu) + (uQ u) j '

(~u II ",ul\- +

z

I\ull

2

are operators whose restr1ct1ons to

const. are pseudo-d1fferent1al operators of order

r = O.

F1nally:

(7.35 )

-~

111\

2

Nul\ =

(L -L n

n

0

)u, P u)

in view 'of the above remark this is bounded by the right side of (7.33).

-267-

J .J.Kohn The followlng theorem ls proven ln Kohn Theorem.

1.16].

If J(L1S pseudo-convex and lf ln a nelghborhood

PfbSl

each non-zero vector fleld of degree (1,0) and values ln T1,0(blL) on bil ls of flnlte type at of

P

and lf the Levl form ls dlagonallzable ln a nelghborhood of P

(l.e. ln C)

the

c

lJ

=c S

then there eXlsts £

~

on

q ~ 0

LP,q({l)

with

0

11 lJ

throughout a nelghborhood)

such that the F-Neumann problem ls

€-subelllptlc at

P.

-26H-

.1. .1.Kohn

Lecture 8. The induced Cauchy-Riemann equations. In this lecture we will take up systematically the b U. which arise in the extension problem

euqations on

discussed in lecture 1.

We define " on

(8.1 )

Observe that if (8.2)

q

=0

then o/€

.,p,q

is equivalent to

~

If=~r;\~+re near

p,q-1 where ljJfa

and

p,q

ee&..

bn.,

From {8.2) it follows that p,q+1

(8.3)

Therefore we have the following diagram (8.4)

O-"~

o

~

i

p,q+1 ~ d

p,q+1

Cl

r

~

d3 p,q+1

.

f

~,q ~p,q -'7 --'"

., 0

~b

p,q

, ---" B

0

~

where rn.p,q

UJ

and

a"b ~

b

p,q

I~

is the mapping induced by

the space the

I1P,q =l./I..

0,0

S

f = 0

a.

In case

is naturally identified with is necessary for extending

f

p = q = 0 C~(bJl)

to a holo-

morphic function. We define the following

cohomology groups:

and

-269-

J . J . Kohn

(8.6)

then (8.4) 1nduces a sequence p,q

(8.7)

H

p,q

(~) ~ H

(Q.) ~ H

p,q

«(f, )- - - j " H

p,q+1

(~ )

0,0 0,1 1l1ustrate, we define the map H ($) -", H (~). 0,0 f~H (n), f 1s represented by a funct10n 1n -C~(bJl) To

If

wh1ch we w111 also denote by 1f

rv

f

f

and wh1ch has the property "" 0,1 the cohomology then afe ~

1s an extens10n of f '" 0,1 class" of 'iff 1n H (~) then 151ves the 1mage of f under 0,0 0,1 the map H (~) ~ H (,\:). It 1sclear that th1s 1mage is ,..... 1ndependent of the extension f and that f can be extended to a holomorph1c funct10n on JL if and only 1f 0,1 (~); that 1s there eX1sts is cohomologous to 0 1n H 0,0 such that = ;-g. The des1red extens10n then 1s gE If.

ar

f

- . g.

p,q

Proposit10n.

H

(~ ) '.:t H

p,q

o

(Q), where

cohomology of forms wh1ch van1sh on Proof:

If

If

~

r p,q -e

p,q H U{)

o

b~.

then from (8.2) we see that

1s the

- 2-'W-

J . J . Kahn

and that p,q

(8.11)

F(J{'

t;,

)C

From (8.9) it follows that

F

n-p,n-q is an isomorphism

;,,P q

(8.10) it follows that i f Iff:- J(. ,

Theorem.

If

is connected and if the Levi form has at

bJL

least one positive eigenvalue at each f€-C()
with

(ll. )

oo

hEC in

b

such that

Let

tv

and i f

then there exists a function on

h = f

0tJ

and

bSl.

olgous to zero 1n 0.1 n,n-1 (~) ~

at

(8.12)

be an extension of

f'-C (J\.)

seen the desired

so 86

=0

~ f

pE-bSl

h

is holomorphic

Jl .

Proof:

H

aF'f

= o.

Thus we p,q (Q) induces an isom?rphis~between H

have proved that F n-p,n-q and H (~).

then

and from

J{

h

exists if and only if

0,1

H

.

f. as we have

-'" af is cohomo-

By the previous theorem

(~).

Let I.f' ••••• f. 1

k

n.n-1 be a basis for df

1s cohomologous to zero if and only 1f

fri;..1/1j

= if'f

= 0

for

"'{ f EC"" (b.R )[ abf

=0}

jL

j

bR

j

= 1 ••••• k,

Thus we see that the space (8.13 )

tf =h has dimens10n

~

k.

coonE) hol.}

on b.a • h Thus

dependent cosets 1n ~

2

f, f , ••• ,f

so that 1f

k+l

have linearly

-2 71 -

showing that b It ,

J . J . Kohn is cohomologous to a form that vanishes on

~

Further,

~cy

= ~(rCa
which concludes the

e)

proof. Proposition. p,q H (~) and n-p, n-q

H

represents a cohomology class in

(It)

classes of Proofl

then

~

If

represents a cohomology class in

~

I.f>

and

depends only on the cohomology ~

= ae

with

" .o.fP

t7<>.."

~I1A~ =

J ~e

"(8.8 )

bn..

n,n-1 since e/\'t'f~ when ~ Theorem.

A

~

=

, q- 1

then we have

d((1,,'/I)

and hence

Ie '4J = a

bIt

A

the same calculation yields the result

= a-f. If at each point

PEb..o.

the Levi-form has either

n-q positive or q+1 negative eigenvalues then n-p,n-l H (~) is finite dimensional and is 1somorphic P, q

to

H (OJ • Proof:

By our previous results the condition on the Levi-

form implies that problem is ~-subelliptic and hence the ")pp,q space ~ is finite dimensional and all its elements are p;q nn-p,n-q in C on Jl.. We define a map F: 0.,. ---" LAby (.><

It is easy to verify that

(a .10)

-272-

J . J . Kahn

m

P(f) = ~ a f j=l j

(8.14)

j

w1 th

a

j

E. ([ ,

a

m

~ 0

,

has a holomorph1c extens10n F to fl 2 k+l Similarly, Ff. F f ••••• F f are linearly dependent and then

P(f)

hence there 1s a polygon1al holomorphic extens10n C;. nothing to prove.

Q so that

If

If' f

f

Q(F)f

has an

1s constant there 1s

1s not constant then it takes on

infinitely many values and thus

F

and

Q(F) taka,on ,1nfl-

nitely manY .values and ,bence c the set of zeroes of tb"1n•

Then ofro III ' P(f) = p(~)

(8.15) on

Q(F) is

.

.

bJl.- {zeroes of

Q(F)}

holds on ·"'~.- i ze roes of

we conclude that

Q(F)j.

F

= ..

Thus we know that

~

wr q/Q(F)

is locally bounded and hence can be extended to holomorph1c function on

II

which g1ves the desired extension of

f.

p,q From (8.7) and from the identification of H (00 n-p.n-q P.q with H (~) we see that H (~) is finite dimensionn-p.n-q-l P.q al whenever H (a) and H (~) are finite dimensiona1.

Thus the following condition implies the finite P.q dimensionality of H (£): The Levi-form has e1ther at

least ~r

max (n-q. q+l)

non-zero eigenvalues of the same s1gn

at least mtn (n-q. q+l)

signs.

non-zero eigenvalues of opposite

In fact this depends on the "internal structure" of

b.fl which can be defined abstractly as follows. Def1nition.

If

X is a real

C~ manifold. we say that

X

-275

J . J. Kohn has a partiallY almost-complex structure of codimension k

1,0

if there exists a sub-bundle T (X)CcfT(X) 0,1 - 1,6 setting T ( X) = T' ( X), we have 1,0

(a)

T

(b)

dim

0,1

(X) ()

T

T(X)

=2

such that;

(X) = 0

1,0 dim T (X) + k

(c) · If

Land L' are local vector fields with values 1,0 in T (X) then eL, L'j has also values 1n 1,0 T ; . (X)

Under these circumstances we define the space of exter1,0 0,1 ior forms Q on T (X) + T (X), on this space this sum: b induces a bigradation

(8.16 )

and we define the operators (8 • 17)

d

'. op,q ~ I1 P+ l,q and

b

b

-,

Iv(.

b

p,q

-0 b : (;i

1>-

-:r

abp,q+l

by setting

(8.18 )


= LU,

u

1,0

a function

L~T

(X)

and extending this to forms in the usual way. tors have the same formal properties as stricting ourselves to the

~~e

and

of co-dimension

def'ine the Levi-form at Pe X as follows. defined in a neighborhood of

a

P,

These operad.

1

Rewe can

Take a real l-form

which annihilates

-274-

J . J . Kahn

1,0

T

0,1 , (X ') + T Of.)

the Levi-form, then is given by

(8.19) for

1,0

L~T

~

,

(X).

Since at each point the space of such

is one-dimensional the numbers of non-zero eigenvalues and the numbers of eigenvalues of the

and opposite signs

s~me

are invariants. (T(X)

NOW we can put a hermitian metric on 0,1 1,0 (X) orthogonal to T which makes T (X) and we

,.J-

obtain the adjoint of

b

i"

of

b

.

By methods analogous

as for the a-Neumann problem we can prove the following

In]).

result (see Kohn Theorem.

Let

X be a compact partially complex manifold

of co-dimension

1

with dim

(8.19) has either at least

'X =

2n-l

and if the Levi-form

max (n-q, q+l) eigenvalues of

the same sign or at least min(n-q,q+l) opposite signs.

eigenvalues of

Then, setting

(8.20) where

{j

b

is the adjoint of

(8.21 )

~

and

ob =.va b b

+~J

b

b b

'

then there exists a completely continuous self-adjoint operator N

b

L

p,q 2

~

p,q v

(,1\\) -') L ' 2

(I'c)

such that

N

b

-27 5-

J . .J . Kohn

is pseudo-local and (8.22 )

C/N

b b

H

where

=I+H

p,q Furthermore, i f Ii E. L

o,q

p,q L

is the othogonal projection of

b

Xb .

b

on,

d

2

= 0

0(

b

on

and

0(.1

on . I? , q de

«=:a(,J"N.,t). b b b

then

The theory indicated in the above theorem is based on the estimate 2

IIlflll. <.:. const.(\\

(8.2))

l!"

for 'ff-

2

~ 'fll b

+ ,I\~~l\ b

2

2

+ (I'fll )

p,q

ab

Observe that

dim x

and q = 1

=)

the conditions in

the above theorem cannot be satisfied because then the Leviform is

1

x 1.

can be represented

In this case

locally by single first order operator.

In fact, i f X. eC

is given by (8.24 )

then to the equation

r = Im (z

1

au = f b

)

+

lzI

2

2

is the Lewy equation which

has no solutions for" ¥lost" functions

f.

The following natural oroblem was posed by H. Lewy.

2

- 216 -

J . J . Kohn

Suppose

L

ls a complex vector field ln L =

(8.25)

)

'iJ

1:-

J=l

a

J

tR? , 1.e.

~ ) ae.e (fR.) J

ax J

do there exlst non-trlvlal local solutlons of the equatlon Lu = O.

Recently, L. Nlrenberg (see ['3 2] )

has

found an

example of such a vector fleld for whlch the only lo cal solutlon of

Lu = 0

are

Nirenberg's example

u = const.

What is lUore,ln

the vector flelds

are llnearly independent.

L,

1. and LL, LJ

It ls stlll an open questlon

whether on X whlch satlsfles the condltlons on the Levl-form for

q = 1

au = 0 b

glven ln the above theorem the equatlon

has non-trlvlal local solutlons.

To conclude these lectures wewlsh to polnt out an appllcatlon of these results due to Kerzman (see [30]). Namely, lf

H : L (Il) ~ 2

If 0,0

denotes the orthogonal pro-

Jection map onto the space of holomorphlc functlons, and lf N ls pseudo-local then

H ls also pseudo-local.

can be expressed as (8.25)

Then the pseudo-locallty of K

€. [h'([i >
H lmplles that (b O ,.. bn) } .

Now

H

- 2 77 -

J . J . Ka hn

REFERENCES (1]

A. ANDREOTTI, and C.D. HI LL, several articles to appear in Ann. Scuola Nor m. Sup. Pisa.

[2J

BOCHNER, S.

"Analytic and meromorphic continuation by

means of Green's formula,"

Ann. Math. (2) 44,

652~673

(1943).

[31.

EHRENPREIS, L. theorem,"

(41

"A new proof and extens ion of Hartog's

BulL Amer. Math. Soc. 67, 5007-509 (1961).

FOLLAND, G.B. and KOHN, J. J. the Cauchy-Riemann complex,"

"The Neumann problem for Ann. of Math. Study Vol.

75, Princeton Univ. Press, 1972.

[ 5 J FOLLAND, G.B. and STEIN, E.M. mates for the

a

b

"Parametrices and esti-

on strongly pseudo-convex boundaries,"

Bull. Amer. Math. Soc. (to appear).

[61

GAFFNEY, f·r~P , , "H ll be r t space methods in the theory of harmonic integrals," trans. Amer. Math. Soc. 78(1955) 426 - 444.

[7]

GRA{JERT, H,0 "Bemerkenswerte pseudo.konvese Manni,dAl-t-

;tigkei ten," Math. Z. 81 (1963) ~ 377 - 391.

[al

______ and LIES I., "Das Ramirezsche Integral und die Gleichung af

=

im Bereich der beschrankten Formen,

Rice University St~dies, (to appear).

-278-

J . J . Kahn

{9]

HENKIN,G"Integral representations of holomorphic functions in strongly pseudocomvex domains and certain applications," Mat. Sbornik 78 (120): 4(1969), 611-632 (Russian), English translation in Math. of the U.S.S.R. April 1969, 7(4), 597-616.

(10J

HOl1l'1ANDER, L., "Estimates and existence theorems for the i operator,"Acta Math. 113 (1965),89 - 152. , "An introduction to complex analysis in sev-

[11]

eral variables, Van Nostrand, Princeton, 1966. , and 'wERI'lER, J., "Uniform approximation on

(12)

compact subsets in

(1:;]

KERZMAN, au

=f

n", l1ath. scand , 23 (1968), 5-21.

N., "Holder and LP estimates for solutions of

in strongly pseudoconvex domains, Comm. Pure

APpl. Math. 24(1971), 301 - 379.

[14]

KOHN, J.J.

"Harmonic integrals on strongly pseudo-

convex manifolds," I, Ann. of Math. 78(1963),112-148; ibid. 79 (1964), 450-472. , "Boundarles of complex manifolds," Proc. Conference on Complex Manlfolds (Minneapolls), SpringerVerlag, New York, 1965. , "Boundary behavior of

on weakly pseudo-

convex manlfolds of dimenslon two," J. Diff. Geom. Vol 6, 523 - 542 (1972).

-279-

J . J . Kohn

{17]

KOHN, J.J., "G
[18J

1\ '

and NIRENBERG, L. "An algebra of pseudo -

d1fferent1al operators," Comm. Pure Appl. Math. 18

(1965), 269 - )05.

[191

,

and NIRENBERG, L., "Non-coerc1ve boundary

value ' pr obl ems , Comm. Pure APPl. Math. 19 (1965)

443 - 492. [20)

.,_

, and NIRENBERG, L., "A',pseudo-convex doma1n not

adm1tt1ng a ho10morph1c support funct10n," Math. Ann.

201, 265 - 268 (197)).

[2ti

,

and ROSSI, H., "On the extens10n of holomor-

ph1c funct10ns from the boundary of a complex man1fold , Ann. of Math. 81 (1965), 451 - 472.

f221

LEWY, H., "On the local charact~r of the solut10ns of an atyp1cal l1near d1fferent1al equat10n 1n three var1ables and a related theorem for regular funct10ns of two complex var1ables," Ann. of Math. 64 (1956),

514 - 522.

[231

,

"An example of a smooth 11near part1al d1ffer.

ent1al equat10n w1thout solut1on, Ann. of Math. 66

(1957), 155 - 158. (24 J LIEB, 1., "E1n Approx1ma tlonssa tz auf streng pseudokonvexen Geb1eten, Math. Annalen 184 (1969), 55-60.

-280-

J . J. Kahn

[25]

NEWLANDER, A. and NIRENBERG, L., "Complex analytic coordinates in almost-complex manifolds," Ann. of Math.

65 (1957), 391 - 404. [26J

NIRENBERG, R., "On the H. Lewy extension phenomenon," Trans. Amer. Math. Soc., (to appear). , and WELLS, R.O., "APproximation theorems on differentiable sUbmanifolds of a complex manifold," Trans. Amer. Math. Soc. 142 (1969), 15 -36.

[281 0VRELID, N., "Integral representation formulas and LP estimates for the equation -u

= f,"

Math. Scand.

29 (1971), 137 - 160. [29J

RAMIREZ, E., "Ein Divisionsproblem in der komplexen Analysis mit einer Anwendung auf Randintegraldarstellung," Math. Annalen 184 (1970), 172 - 187.

[3 0]

KERZMAN, N. ,"The Bergmann kernel function: differentiability at the boundary,"

Math. Ann. 195 (1972)

149-158.

[311

LAX, P.D. and PHILLIPS, R.S., "Local boundary conditions for dissipative symmetric operators."

Comm.

Pure APpl. Math. 13 (1960), 427 - 455.

[32]

NIRE~BERG, L., "Lectures on linear partial differ-

ential equations," Mimeographed notes. Courant Institute (1973).

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C . I. M. E .)

THE MIXED CASE OF THE DIRECT IMAGE THEOREM AND ITS APPLICATIONS

Corso

tenuto

a

YUM-TONG

SID

Bressanone

dal

3 al

12

giugno

1973

THE MIIED CASE OF THE DIRECT IMAGE THEOREM AND ITS APPLICATIONS Yum-Tong Siu 1

J o. Introduction In these lectures we will discuss the so-called mixed case of the direct image theorem and its applications.

The

starting point of the direct image theorem is the following finiteness theorem. (0.1)

Theorem (Cartan-Serre)· If ~

sheaf on a compact complex space finite-dimensional over

~

for

is a coherent analytic

X, then ~

>

•

H~(X,7)

is

0 •

The proof is obtained by Schwartz's finiteness

theore~

on the perturbation of a surjective operator between Frechet spaces by a compact operator (see [8, VIII.A.19] and [6]). In 1960 Grauert proved the following parametrized version of the above theorem which is known as the proper Case of the direct image theorem [6]. (0.2)

Theorem (Grauert).

If

n: X --->

S is a proper

1 PartiallY supported by a National Science Fbundation Grant and a Sloan Fellowship

-284-

Y-T .Si u

holomorphic map of complex spaces and lytic sheaf on '] under

n

X, then the

is coherent on

v th

'1

is a coherent ana-

direct image

S for

v>=

R n* 7 of V

0 •

Grauert's proof uses the power series method.

The

idea is to expand a v -dimensional cohomology class in a power series in the variables of eral case to the case where space).

S

(after reducing the gen-

S is an open subset of a number

The coefficients in the power series expansion may

not be cocycles, but, by using descending induction on

])

Grauert showed that they can be approximated by cocycles. Then he used induction on

dim S and applied the induction

hypothesis to the approximating cocycles to get the coherence

~th

of the

direct image.

About ten years later Knorr [13] and Narasimhan [18] gave simplified presentations of Grauert's original proof. Recently Kiehl [10] used nuclear and homotopy operators and a form of Schwartz's finiteness theorem to obtain a new proof for an important special case of Grauert's theorem. Then Forster-Knorr [3] and Kiehl-Verdier [12] succeeded in obtaining new proofs of Grauert's theorem along such lines. Their proofs make use of descending induction on JJ , but does not

u~~

induction on

dim S.

This opens the way to

generalizing Grauert's theorem to relative-analytic spaces and such generalizations were carried out by Kiehl [11], Fbrster-Knorr [4] and Houzel [9].

Y-T . Si u

In 1962 Andreotti-Grauert (1] generalized the theorem

df Cartan-Serre in another direction.

They introduced the

concepts of strongly pseudo convexity and pseudo concavity and proved finiteness theorems for spaces which are strongly pseudo convex or pseudoconcave.

X is said to be strongly

space x

~

U

X there exists an embedding

of x

,N

A function

p-pseudoconvex if for every

onto a complex subspace of an open subset

\ j

yo -r

f ..

such tha t

i

of an open neighborhood

T

cf

and there exists a real-valued

( dZa21J1 dZ

on a complex

~

and the

has at least

)

every point of

function

G of

~ on

G

N x N hermitian matrix

N _ P + I . positive eigenvalues at

G (where

zl' ••• ' zN are the coordinates

of (N).

(0.3 )

Theorem (Andreotti-Grauert).

space and ~

f:

-> (a*,b* ) C IR is a proper

X

a* < a# < b# < b*

convex on

If>

I ~ < a#}

and

Suppose

t

codh? S;' r

b#}

such that

map·

is strongly

Sup-

p-pseudo-

q-pseudoconvex on

I 'f' ~ b} '" I ~< b}- for b# < b < b* -

is a coherent analytic sheaf on

IT< a#}- . Then, for p

on

H (X, "]) ---.-) .• ; I a

a* .~ a < a #

l'

and is strongly

is finite-dimensional over (: )I

X is a complex

Suppose

.)

and

< 'f <

~ )J

X

such that

< r-q ,

)!

H (X,"])

and "1'

b ] , __ }

is an isomorphism for

b # < b ~ b* -

(For the definition of

codh"] , see (A.I) of the Appendix- )

- 286-

Y - T. Siu

A holomorphic map n: X - > S of complex spaces is called stronglY

if

exists a

(p,q)-pseudoconvex-pseudoconcave if there

map f: X - > (a*, b*) C 1R. and there exists

a* < a# < b# < b* i)

nl (a

~ l' ~

such that bJ

is proper for

{f ~ b 1 = {SO < b J-

ii)

fj' is strongly

iii)

a* < a < b < b* •

b# < b < b* •

for

p-pseudoconvex on

ep is strongly q-pseudoconvex on

iv)

We introduce the following notations.

Fbr

a* ~ a < b ~ b* ' b

[a < 1 < b J

Xa

For a coherent analytic sheaf "] R)I(n:

v

b

"I

X , R (n a )*T denotes

on

)*(1I x: ) •

The so-called mixed case of the direct image theorem is the following parametrized version of the theorem of Andreotti-Grauert. (0.4)

Con jecture.

Suppose

n ; X - > S is a strongly

(p,q)-pseudoconvex-pseudoconcave holomorphic map

l'

and

a* < a# < b# < b*)

sheaf on

X such that

If < a#J. S

and

,)

Then, for

and

"l

dim S ~ n p

vb

Rn*"J-> R (n a

~

is a coherent analytic and

v < r -q-n, u-r

)* 7

(giv~n 'with

codh

7

R n* "l ,)

~ r

.

1S

£E

coherent on

is an isomorphism for

- 2 87-

Y- T . Si u

This conjecture so far has not been completely proved. The special case vex case.

{'f < ~ 1 = Rf is called the pure pseudo con-

The special case

pseudo concave case.

{'f

> b# 1 = f5 is called the pure

Partial results for these two pure cases

were obtained by Knorr [14] and Siu (24,25].

Recently the

pure pseudo convex case was completely proved by Siegfried

[21] by using the methods of the new proofs of Grauert's theorem and the pure pseudo concave case was completely proved by Ramis-Ruget [19] by using the methods of the new proofs of Grauert's theorem together with duality.

Unfortunately these

methods cannot be applied to the mixed case, because any induction on the dimension of the direct image is impossible. A partial result on the mixed case was obtained by Siu [26].

m these lectures we will prove the following improved partial result of the mixed case which is good enough for the known applications. (0.5)

Main Theorem.

n: X --> S is a strongly

Suppose

(p,q)-pseudoconvex-pseudoconcave holomorphic map (given with

If

and

sheaf on {'f<~l

a* < a

< a# < b# < b*)

~

,

X such that

.

Suppose

< a < a# and

and

dim S

~

7 n

is a coherent analytic and

codh(£) '1? n . S,n(x) x

codh

1

for

x<;:X.

~ r

.Q.!1

Let

Then the following

conclusions hold. i)

(Rv(n~)*7)s is finitely generated over

(OS,s

for

-28 8-

Y .-T . Siu

s ~ ii)

iii)

s

P:i- v < r-q-n •

and

V

~

)*7

b

-> R (n a p < ,; < r-q-n • ,J

R n*o1

b'

->

RP(n , ) * '1 a

p < r-q-2n

iv) for

P:i-))

is an isomorphism for

RP (n b)

a

'~

*'1

R'"

is an isomorphism.

-

(n~ )*1 is coherent on

s

< r-q -n-l •

Fbr the applications, only conclusions i) and iii) of the Main Theorem are needed. by the power series method.

The Main Theorem will be proved If we couple the power series

method with the methods of new proofs of Grauert's theorem and duality, we can improve conclusions iii) and

iV) to the

following, but we will not discuss it in these lectures. (0.6)

Theorem-

Under the assumptions of

p < r-q-2n , then, for S and a*

~

R,) n* 7

a < a#

-->

and

p ~]1 < r-q-n , R

11)

iii) iv) v)

'1

if

is coherent on

R'" (n~ )/1 is an isomorphism .f or

b# < b

~

b* •

The Main Theorem will be applied i)

(0.5)~

lIn*

to

the following:

extending coherent analytic sheaves blowing down strongly

I-pseudo convex maps

blowing down relative exceptional sets obtaining a criterion for the projectivity of a map extending families of complex spaces.

(For applications ii), iii) and iv), the pure pseudo convex case of the direct image theorem suffices.)

-289-

Y-T .~u

Fbr coherent sheaf extension, we will not obtain the best known result of extension from Hartogs' figures ~e

[23]~

will only obtain the result of extension from ring domains

(22] (which implies the extension across subvarieties

[29,5]).

The proof of coherent sheaf extension by means of

the direct image theorem is not the simplest approach.

A

very simple proof of the extension from Hartogs' figures was given in [27] which does not use the power series method of Grauert and does not use the method of privileged sets of Douady. The smooth case of the local result on blowing down strongly

I-pseudoconvex maps was obtained by Markoe-Rossi

[17] and the general case of the complete result was obtained in (25J.

The results on relative exceptional sets and pro-

jectivity criterion were due to Knorr-Schneider [15].

The

special case of the result on extending families of complex spaces where the parameter space is a single point was obtained by Rossi (20) and the general case was due to Ling [16l Now we give here a brief sketch of the main ideas of the proof of the Main Theorem.

In the actual proof, for

technical reasons, we use sheaf systems to construct complexes of Banach bundles to calculate the direct image sheaves, but, here in this sketch, for simplicity, we compromise the accuracy by calculating the direct image sheaves by the usual ~ech coch.in complex.

In this sketch there are

also other compromises of accuracy in some minor points for

-2M-

Y - T . Siu

the sake of simplicity.

The proof of the Main Theorem has

three key steps. The first key step is the existence of privileged sets for a coherent sheaf, i .e. if

s.

nt9P --> l)q

homomorphism on an open neighborhood

G of

0

is a sheafin ([n

which

is part of a finite resolution of the, given sheaf (wh~re nO is the structure sheaf of C n), then there exists an open polydisc neighborhood lowing.

If

P

of

in

0

G satisfying the fol-

s ~ r(p, noq) . i s bounded (in a suitable

sense) and the germ of s at 0 belongs to the image of s

and

is the image of a bounded section v

can be so chosen that

C-linear.

s

~>

e , then

over

v

of

v

is continuous

P

The existence of privileged sets can be proved in

three ways (some of which are valid only for certain senses of boundedness).

The first proof by Cartan uses the Weier-

strass preparation and division theorems and it is usually used in the proof of the Closure-of-M:>dules Theorem [$, II. n]. The second proof by Grauert uses power series expansion and it is used in Grauert's original proof of the proper case of the direct image theorem.

The third proof by Douady uses

holomorphic Banach bundles and it is used in his solution of the module problem [2].

In these lectures, we present Dou-

ady's proof, because it works for all senses of boundedness needed for our purpose and because the idea of Grauert's proof occurs in the second key step of the proof of the Main 'lheorem and we will see it there 8.I\YWay. leged sets gives rise

to

'!he existence of priTi-

'lheorem B with bounds and Leray's

-21]1-

Y-T . Siu

theorem with bounds, which are used, together with the bumping techniques of Andreotti-Grauert [1], to construct complexes of Banach bundles to calculate the direct image sheaves. The second key step is the analog of the Hauptlemma . of Grauert's original proof of the proper Case. is an open neighborhood of open polydisc in

«n

0

in 4: n.

centered at

0

SUppose

'l.t '" {U i} , l{ '" {Vj

tions of Stein open subsets of relatively compact in some

A (f)

f •

can be described as }

be suitable collec-

X such that each

Ui •

S

be the

with polyradius

Roughly the analog of the Hauptlemma follows.

Let

Suppose

Vj

is

Let

rUin n- l ( .6 (p») } {V

j

n

n- l (A(p»)} •

Then there exist

for some

fO

satisfying the following.

small (in a suitable sense), every l,

~

a

i'"l i

l,(i) +

when restricted to ~(f) , where

p sufficiently

~ l(?Jl(f} ' "1)

written as

t, '"

Fbr

67

can be

-292-

Y-T . Siu

ai

G:

r(A(p},

?

G: c"-l(\(f}, 7) •

nf!))

1tt>reover the bounds (in a sui table sense) of

ai

dominated by a constant times the bound of

•

~

and

we

7

are

will look

upon this as a generalization of the existence of privileged sets.

Instead of lifting bounded sections in the map

induced by 8 , in this case we consider the lifting of bounded sections in the map

defined by

->

~ a. l,(i} + i=l ~

6?

(restricted to l(P}) • It turns out that the ideas of Grauert's proof of the existence of privileged sets can be generalized to this case when we apply Leray's theorem with bounds.

For this we have to

use induction on dim S, but we do not need - any descending induction on -V.

The reason why we Can avoid this descending

v

which is so essential in Grauert' s original

induction 'on

proof of the proper case is that we look upon the analog of the Hauptlemma

as the generalization of tpe existence of

privileged sets and we use both the surjectivity and the in-

- 29 3-

Y- T . Siu

jectivity statements of Leray's theorem with bounds, whereas Grauert did not use the technique for proving the existence of privileged sets in his proof of the Hauptlemma

and he

used only the surjectivity statement of Leray1s theorem with bounds.

Our approach is simpler and gives the best possible

result in the finite generation of the stalks of the direct image sheaves. After the above two key steps there is still one obstacle to proving the coherence of the direct image sheaves. Suppose S is an open neighborhood of 0 in C n• To get the coherence of the direct image sheaves by induction on we need the following statement on global isomorphism. exists an open polydisc neighborhood

U of

0

in

n,

There

S such

that

is an isomorphism for all sheaves where

-§

of the form "l/(t n-c)m7, are the coordinates of (( n. The third

t

••• , t n l, key step is to obtain this isomorphism statement.

U:a lUi} set

Dof

is a Stein open covering of S let (,t(D)

define sheaves

)/

13

lUi

:a

v

n

X.

n-I(D)}.

D

1-->

D

1--> eV ( U(D) ,-9)

(§) ,

oJ

B

~ (9)

Suppose

Fbr any open subThe presheaves

(1t ( D) , ~ )

on

s .

We derive the isomor-

phism statement by constructing a sheaf-homomorphism

- 294-

Y . T . Siu

-e (7 ) V

- > ~v-l(7)

on

U which is

a

s

right inverse of

»

This right inverse gives rise to a right inverse 13"(9) _>fb l1- l C§ ) of () for sheaves .g of the form 7j'ctn-c)mj. f>v(§)

The existence of a right inverse

_>~"-l(§) of

~(U, ~)IC~)) v+l

R

.P.

n*~,

()

implies the vanishing of

which, together with t.he coherence of

R"Vn*

-g ,

_U-l .P. ••• , K n*~, yields the isomorphism

The construction of a right inverse

S :11("]) - > ~v-l (7)

of

()

is based on the generalization of the following observation. For an open polydisc G in a: n , a continuous lC-lin-

ear map

on

is induced by a sheaf-homomorphism and only if ~[tl'

~

••• , tnJ·

G if

is linear over the polynomial ring We show that, under the additional assump-

tion of the vanishing of

)1+1 )I+n_l "1 (R n* '1 )0' ••• , (R n*)O'

the lifting

in the analog of Grauert's Hauptlemma

can be done in such a

way that it is linear over the polynomial ring ([t l, ••• , tnJ.

When we have the finite generation of v l1+n CO (R n* "])0' ••• , (R n* trr~)O over no' we can apply the

above argument to a complex of the form

'& h ('])

Ph @ n([)

in-

-29 5 -

Y. T . Siu

stead of to

for

));;; p. :i-

~A('1)

v+

and obtain the isomorphism

n.

It is the third key step that makes the

additional assumption of

p < r - q - 2n

necessary, because

we need some room to get a right inverse of

6.

It is also

the arguments of this step that necessitate the introduction of complexes of Banach bundles for the calculation of the direct image sheaves, although such an introduction streamlines the presentation elsewhere as well.

In these lectures some tedious details, which are obvious and can easily be filled in, are left out, especially in

§ 3, § 4, and § 5 • Deta Us of this nature can be found in

[13, 18, 24].

There is an appendix at the end which deals

with homological codimension, flatness, and gap-sheaves. Consult the appendix when these concepts are mentioned or their properties are used. Now we list the notations we will use in these lectures. ~

the set of all positive integers

~o

N U IO}

~*

~ V ICll}

IR+

the set of all positive numbers

nfD

the structure sheaf of ([n •

-296-

Y- T .Siu

The components of a c;: q:m are denoted by a ••• , am. l, m a, b ~ IR ,by a < b we mean a i < b i for 1 ~ i ~ m

For

and by

a n

~

b

we mean

to

fRn ,p r: '=. +

.D.(tO ,

(tpt

p) O

~ ~

i

~

m •

~

a: [tl,

pa

v

r ..

b

0

n

v1

].

a: n

with

e

Pn

PI

+ ••• + )) n

II

1

1

()t

)1

n

.t d to, f)

~ IR~ , ~N(b)

with center

The

IlJn • in 1'rI 0

(1, ---,1) ,we denote

general, for

t

••• J

center to and po1yradius o lJ o lJ (tl-tl ) 1 ••• (tn-tn)n

When to .. 0 , we denote

IR.N

1

the open polydisc in

dt

(N

for

are always denoted by

,and

11> I

When

bi

denotes the polynomial ring

r: II"'n

'=. \L.

~

occupies a special position in these lectures.

coordinates of ([ n ([ t]

ai

n

simply by .6(P)

Ll (p) •

.6..

simply by

denotes the open polydisc in

and po1yradius

b.

Fbr

< b in

0 ~ a

,

GN(a,b)

..

{z

~

,6N(b) Ilzil

In

>

ai

for some

1

~

i

~

N} •

- 29 7-

Y - T . Siu

The closure of a set denotes the sup norm on

«N

r

,

L

2.(G,

JD)

functions on

If

G •

/loU

G

is an open subset of

G

2 L

denotes the set of all

holomorphic

G.

The stalk of a sheaf ~

7s

G-

is denoted by

G

at a point

'

rr

U is an open neighborhood of

then

fs

denotes the germ of

f

at

s

is denoted by

sand

f ~ nU, "]) ,

s.

A complex space may have nonzero nilpotent elements in its structure sheaf.

The structure sheaf of a complex space

Ox •

X is denoted by

ideal of the local ring

For

oX,x

, ...... X,x means the maximum and sometimes (when no confu-

x~X

sion can arise) it also means the ideal sheaf for the subvariety

[x]

of

X.

n: X --> S is a holomorphic

Suppose

map of complex spaces and

Then MN S,s means also X generated by the inverse image of

the ideal sheaf on

s

~

S.

S,s when no confusion can arise • For a coherent analytic V sheaf "] on X, R t denotes the yth direct image of

n.

..,.,."

under then

R"

Y is an open subset of

Ii.'

n.

cr. (71 Y)

is simply denoted by

Suppose U· lUi J and )( = (Vj

X and

CT'

= nlY

,

Il

R (1""* "].

J are collections of

open subsets of a complex space X and? is a coherent analytic sheaf on

Vl«)( means that each

X.

tively compact in some we can define

•* l, that

T

*l,

~

is also denoted

t;

=

7

on ?il

VT(i)'

ev nt, "J) by C, I U.

if

Ui

is rela-

For every t, ~ eliot,

'1)

by means of the index map For

T*l," T*?

V

l" 7 ~ e (1(', "l) , we IViI denotes Vi Ui.

"t •

say For

-298-

a complex space

Y,

Y x 7Jl. deno tes

{Y x lIt} •

When we have a sheaf-homomorphism analytic sheaves on a complex space same symbol

e

(X, "] )

->

. 7/J7-> jr

induced by

is an ideal sheaf on

e

--> -g

of

X, we sometimes use the

to deno te also the maps

r

(where

e : '1

r

(X,

.q )

-g/J§ X)

and other similar maps

when no confusion can arise.

- 2.99-

Y .-T . Siu

Table of Contents

§o

Introduction

Part I

Construction of Complexes of Banach Bundles

18

h

Privileged Polydiscs

18

§2

Semi-norms on Unreduced Spaces

31

§3

Theorem B with Bounds

40

§4

Leray t s Theorem wi th Bounds

48

§5

Extension of Cohomology Classes

55

§6

Sheaf Systems

65

Part II

The Power Series Method

75

§7

Finite Generation with fuunds

75

§a

Right Inverses of Coboundary Maps

107

§9

Global Isomorphism

111

§ 10

Proof of Coherence

119

Part III

Applications

135

§ll

Coherent Sheaf Extension

135

912

Blow-downs

141

§13

Relative Exceptional Sets

155

§14

Projectivity Criterion

157

§15

Extension of Complex Spaces

160

1

Appendix

164

References

178

- 30 0 -

Y-T . Siu

<XlNSTRUCTION OF <XlMPLEXES OF BANACH BUNDLES

PART I §l

Privileged Polydiscs

(1.1)

O D· A(t ,P)

Suppose

Define i)

B(D,EQ)

EO-vaJ,ued uniformly bounded hoIonor--

phic functions on the set of all

D,

EO-valued uniformly continuous holo-

morphic functions on iii)

the set of all

on

Eo is a Banach space.

as one of the following:

the set of all

ii)

and

D,

EO-valued holomorphic functions

D with

1I

where

0ll

Eb

is the norm of

B(D,E ) o is simply denoted by B(D).

EO

In any of these three cases

is a Banach space.

B(D,~)

If

neighborhood of with fiber pose

'1

EO'

n-

and

U is an open

E is the trivial bundle on

we denote

B(D,E o)

also by

B(D,E).

U Sup-

is a coherent analytic sheaf on an open neighborhood There exists an exact sequence

- 30 1-

Y - T . Siu

m~ m~ 0-> n1:1 -> ... -> n1:1 -> nVYO ->7-> 0

on an open neighborhood of Definition.

(a)

DI.

7 -privileged

D is an

neighborhood i f

the induced sequence

o _> B(D)

p

m

P

_>

B(D) 0

is split exact, (b) When

Coker CL (a)

-->

"lois injective. t

is satisfied, one defines

B(D,'])

as CokerCL.

This privilegedness is said to be in the sense of Cartan, Douady, or Grauert according as

B(D)

has the mean-

ing of i), ii), or iii). The definition of privilegedness and

B(D, 7)

independent of the choice of the resolution of

7,

is because,

by using Theorem B of Cartan-Oka, we can easily prove that any two finite free resolution of borhood of

n-

7

on a Stein open neigh-

become isomorphic finite free resolutions

after we apply to each of them a finite number of modifications

[8, Def. VI.F.I], i.e. after we apply to each of

them a finite number of times the process of replacing it by its direct sum with some finite free resolution of the zero sheaf which has only two nonzero terms

( of. [8 , p.202,

- 30 2-

Yo-To Si u

(1.2)

For Banach spaces

EO' FO we denote by L(EO' FO} the Banach space of all continuous linear maps from EO to F • O

Suppose

S is an open subset of ([ nand

holomorphic Banach bundle on we denote by set

U of

S with fiber

EO.

E is a For

s

~

S

Es the fiber of E at s . For any open subS we denote by EI U the restrictio n of E to

U• Suppose

F

is a holomorphic Banach bundle on

S with

FO• A map '(: E --> F is called a bundle-homomorphism if for every open subset U of S for which there are

fiber

trivializations a: El U

~ > U

there exists a holomorphic map

x

A(' )

EO

from

U to

L( EO,FO}

such t hat

for

s~U

denote by

and

tlu

x ~ EO •

U of

the bundle-homomorphism El u

induced by '(

For any open subset

.

For

s

~

- > Flu S we denote by

'Is

the map

S we

- 30 3-

Y-T . Siu

induced by 't • Suppose

e

o _> E(m) _> E(m-l) _' _> •.• _>

E(O)

is a complex of bundle-h9momorphisms of holomorphic Banach bundles on

S.

If for some

So

~

S the sequence

-> . .. _> is split exact, then there exists an open neighborhood

So

in

S

U of

such that the sequence

is split exact. To prove this, it suffices to prove the case where m· land

E(l), E(O)

the closed subspace of

are both trivial bundles.

Let

H

which complements

Let tr : E(l) ~ (S X H)

->

be the bundle-homomorphism induced by

E(O)

e

and the inclusion

map S

(j

So

x H '-->

is an isomorphism.

S x (H EB Im

es) o

=

E(

0)

•

Since the invertible elements of

be

-304-

Y -T . Si u

L(E~l)(f)H, E~O))

o

f'orm an open subset, there exists an open

0

neighborhood

such that a: I U

So

U of'

phism

(i.e. (~IU)-l

(1.3)

Suppose

is a bundle-homomorphism).

Q ) is an open subset of'

S (respectively

7

{.n (respectively ([, N) and S x .Q.

is a bundle-isomor-

is a coherent analytic sheaf" on

For

we denote by

7(s)

the sheaf'

••• , t are the coordinates of' {n. l, n be regarded in a natural way as a sheaf' on

where

t

7(s)

can

n.

For

p ~ 1 , we denote by

bundle on ([ n Let pose

s~ S

whose f'ib'er is

n: S dl and

'1

-> is

S

B(S2. 'n+P)

the trivial

B(.Q)P.

be the natural projection.

n-f'lat at

{s}x.Q

and '1

Sup-

admits a

f'inite f'ree resolution

on

S x.Q.

centered at z.

Suppose z

z ~.Q and

G C C.Q is an open polydisc

which is an '1(s)-privileged neighborhood of'

Then there exists an open neighborhood

such that, f'or any open polydisc

DC U

U of'

centered at

s

in s,

S

- 30 5 -

Y - T . Siu

D x G is an 1-privileged neighborhood of

(s , z) •

To prove this, consider the following sequence of bundle-homomorphisms induced by (*):

Since

G is 1(s)-priyileged and since by the

"1

at

{s} x Q

o ->

n+N'"

I

n-flatness of

the sequence

(rlm(s) -> ... ->

((ll

n+NI:J

(s)

-> ":1 .1(s) ->

0

induced by (*) is exact, we conclude that the sequence (#), when restricted to the singleton

{s}, is split exact.

(1.2), on some open neighborhood of exact and Coker a

B(D)(

G'n+Jl i )

(0

in

S, (#) is split

is a holomorphic Banach bundle.

that, for any bounded open polydisc B(D, B(G'n+NvPi))

s

By

D centered at

Observe s,

is naturally topologically isomorphic to

~

i :;:. m).

Hence, when

in a sufficiently small neighborhood

U of

D is contained s

in

S,

... is split exact and

to

B(D, Coker a).

B(D XG, "J)

is topologically isomorphic

To show the injectivity of

-- 30 6-

Y -T.Siu

->

B(DlCG,'1)

'1(S,Z) ,

it suffices to show the injectivity of

(where

is the sheaf of germs of holomorphic sec-

V(Coker~)

tions of the bundle

Coker~)

suffices to do the case

n f

Suppose

f

is nonzero.

integer

k

such that

~

~

and, by induction on 1. Ker

n, it

Take ~

•

There exists a maximum nonnegative

with

By

the

n-flatness of

"J, g

Since

G is an

lows that (1.4) subset {N+l ,..,

'1 on z

g(s)

Suppose

~

~

• z, it fol-

0 , contradicting the maximality of

7

and extend '

and if

Ker

F(s)-privileged neighborhood of

k •

is a coherent analytic sheaf on an open

S2. of a:;N. ~ x .Q.

~

7

If

G is an

IdentHy(N

with the subset

0 x (N

of

trivially to a coherent analytic sheaf GCC.Q is an open polydisc centered at

'1 -privileged

neighborhood of

z ,

~,

- 307 -

Y:-T . Siu

for any bounded open disc

DC ([ centered at

""

.2.!! "I-privileged neighborhood of

0 , D x G is

(0, z) •

Tb prove this, we can assume without loss of general-

ity that there exists an exact sequence

on

Q.

O _>

Define (0Pm (9

~l

"" Q.

(QPm-l m>

~l

-

by N

Q.

l

Q.. J

where

==

(Q.l,Wj

C

(_ljJ-'w ) (1

< j

~

m)

Q.j-l

w is the coordinate of

c

represented as a column vector, and

an element of Q.. J

N+lJP

is

is regarded as a

matrix of holomorphic functions which are considered as functions on (( x Q.

([ x.Q independent of w •

The sequence is exact on

Let

-> B(G,

lJ

p.

J)

(1 ~ j ~ m)

be a continuous linear map, which, when composed with the map

- 308-

Y-T .Siu

P 'l

B (a, N(D J- )

a. , gives rise

induced by

a projection

to

J

A corresponding

(where

where

p

a

,

can be given by

= 0)

-1

is independent of

t

and one denotes also by

~j

the map

it induces.

Hence B( D )(

"" a, "1)

B( a,

~

., )

and the result follows. (1.5)

Suppose

tered at

to

D is a bounded

0

pen polydisc in ([n

cen-

and suppose

0 ·-> I '

17"7

":r"

->~->-

->

0

is an exact sequence of coherent analytic sheaves on ' an open neighborhood of

D-.

If

privileged neighborhood of

D is an

7

t o ,then

,

-privileged and D is an

7 "-

'1 -privileged

Y - T. Siu

neighborhood of

to.

Tb prove this, we take finite free resolutions on P

,

a

,

m

0-> nI[)m_> 0->

" a "m Pm

->

(!) n

,

->

PI' a I

-> n " a" /API I -> nI;J -> (9

Po'

f!)

n

" //,\PO

I;J

n

-)

n-:

'1 - ) 0 I

"

->1 ->

°.

We can construct the following commutative diagram

000

o

J, 0->

,,,Pm

nI;J

~ ->

~lf\P

0-> nv

~

n

,,,Pm _

O -> nI;J

... v

->

->

~

o

where

i)

,

t

a

tf\PI

nI;J

n(!) I

1

->

1" P

J,

t

~

1

-> n~lo -> 7' -> 0

-> nV

a

m m ->

,

PI

->

,,,PO

nI;J

l

nf!)

n

P

->

tI7

iT

-> 0

I

~

0 -> "1" _> 0

10 0 1 01

II

l!'I v",Pj = ,I\Pj v w mPj ~ n n n

and, except in the last column,the

vertical maps are the natural injections and projec11)

tions. a j is of the form

(a~ oJ

being a sheaf-homomorphism).

- 310-

Y-T . Siu

Let t

p.

B(D, nlD J) be a continuous linear map which, when composed with the map

,

,.."

a. .: J

induced by

t

a..

J

,

(

B D,

p.

nl[) J)

-> B(D, n(!lj-l)

gives rise to a projection

Let

~~:

t

B(D,

be a similar map.

•

n(~lj-l) ->

Then a corresponding

can be defined by

Since clearly

o ->

, ) ->

B(D, '7

B(D,7)

->

" ->

B(D,'1 )

0

is exact, the result follows.

Before we state the principal theorem on privilegad polydiscs, we have to introduce a terminology. is a statement depending on for f

f (; IR~.

Suppose

We say that

sufficiently strictly small, if there exist

Sf

Sf holds

- 3 1 1-

Y - T. Siu

(.,)1 (; R+

1R +i-l

and positive-valued functions (.0i (fl' ••• , Pi-l)

(1 < ]. .

~

n)

Sr

sue h t h at

holds for f

<

(1 (1.6)

Theorem.

Suppose

an open neighborhood

U of

ficiently strictly small, borhood of Proof.

'7

on

satisfying

i ~ n) •

is a coherent analytic sheaf on 0

«. n.

in

Then, for

f

suf-

is an 7-privileged neigh-

~(p)

O.

Use induction on

n.

By shrinking

U , we can

assume that there exists a nonnegative integer t n "i s not a zero-divisor of

t~

70 t

for

d

to (; U

such that Then

~: .. t~7iS n-flat at U n {tn .. OJ , where nO: ([n -> tC is the projection onto the last coordinate.

By induction

hypothesis, when (Pl, ••• ,Pn-l) is sUffici~ntly strictly small, the polydisc GCa:: n- 1 with polyradius (f , ••• , fn-l) and l

centered at

0

U n {t n .. OJ and for the coherent analytic

is relatively compact in

is a privileged neighborhood of

0

sheaf

U n {t n .. OJ. By (1.3), when an open disc DC 4: centered at 0 is sufficientiy small, D x Gee U and D x G is

on a

~

-privileged neighborhood of

O.

By using (1.4) and

- 3.12 -

Y. T . Siu

applying (1.5) to the exact sequences

0-> 0->

we conclude that

(1

->

~ j

< d)

7-

D x G. is an

t

-privileged neighborhood of

-313-

Y-T. Siu

§2

Semi-norms on Unreduced Spaces

(2.1)

Lemma.

complex space

Suppose ~

is a coherent analytic sheaf on a

X and Dee x

x G: U , then

f = 0 •

Proof.

:x;..G: X. -u

Take

Then there

such that, i f

fG: r(U,"])

p

exists a nonnegative integer for some open subset

is an open set.

U of

D and

f

Let

0 = i Qi

n

7x .

mension of the radical of Qi·

Qi

open neighborhood of x O• dimx

Supp

o

x = Xo

Let

x

be the di-

is the stalk at of ~

o/~

~

i

Xo

defined on some

•

ki • (!2i)X

(see (A.7) of the Appendix).

conditions hold for

ki

Then

((~i) [k i -1] '7)x for

for

be the primary decompo-

si tion of the zero submodule of

of a coherent analytic subsheaf ~i

xG:......",P+17x X,X

Hence these two

in some open neighborhood of

Since we need only prove the lemma for each

xo

•

7'/~.' we can ~

assume without loss of generality that k

and

We can also assume the following: i)

X is a subspace of ~.

X has pure dimension

-314-

Y-T.Siu ii)

~: ~

The projection

coordinates makes iii)

--> D

onto the first

k

X an analytic cover over D. X is defined by the ideal sheaf

The reduction of

generated by holomorphic functions

J1

gk+l' ••• , gn

on A. iv)

The unreduced structure of

X is defined by

We are going to prove that it suffices to set n

= ~Ix.

p

J' .t .

= L.

Let

Let S be the sUbvariety of X outside which

t l, ••• , t k, gk+l' ••• , gn form a local coordinates system. Let T be the set where ROn* 7 is not locally free. Since

it suffices to prove that f

x

f = 0

on

U -

b.1Wn~+1 7x -Jl.,X

means that 1;

0.

t

la.l=.t 1

n-l(TUn(S)).

...

. 1 •••

Since

a.

g n n

= ~k = 0,

=

0

one has

There

-- 3 15-

Y. T. Siu

fX

in

'"

'7x .

Let

such that element of and equals

r

W be an open neighborhood of

n-1(W)

nu

n-1(W)

is closed in

.

n(x) Let

in

s

,

D

be the

r(n-l(W},"l) which equa~s f on n-l(w)nu 0 on n- 1 (W) - U • The element f* of

(W, ROn* 7-)

corresponding to

t

,

satisfies (y ~ W) •

f* '" 0

It follows that (2.2) (n

SUppose

and

p

on

w-

Q. E. D.

T •

V is a subvariety of an open subset

is a nonnegative integer.

Define Jrv(p)

sheaf of germs of holomorphic functions on derivatives of order JV(p)

~

p

We prove by induction on the theorem of Cartan-Oka.

p.

as the

G whose partial

vanish identically on

is a coherent ideal sheaf on

G of

V.

Then

G. The case

p '" 0

is

We can assume without loss of

generality that

f l, ••• , f k on G. Let i n-tuples of nonnegative integers a with

for some holomorphic functions be the number of

lal '" p.

Define

-316 -

Y-T .Siu

by the matrix

and define

by the row vector

Let n

be the quotient map.

Fbr, every

and, for

Then

g ~ J'V(P-l)x g

(OJ.

Can be written as

'"

p,

1~1

If g It follows that to V,

~

~ (D f.) ~

(Iff.) ~

i"'l a

g ~J1v(P)x

k E a.

i"'l

k

E a.

'"

x

x

•

if and only if, when restricted '"

0

(I~I

=

p) •

- 317-

Y-T .Siu

(2.3) ~

n

Suppose

,p

V is a subvariety of an open subset

is a nonnegative integer, and

r Suppose P

II f

G of

ilL

(V ,

n

C9

L is an open subset of

For x

as follows.

for some open neighborhood

rlno v .

/.1 (p))

•

'I" V

U.

We define a semi-norm

U , there exists

~

n of x

such that

IV

f

induces

Let

(leLl

~ p) •

Then {feL (x) }

/eLl

is independent of the choice of p

[r II

The semi-norms

N

f.

pll' ilL 2!!

space structure when

r'(v,

L} •

define a Frechet

V •

rtv.

that every point of

V

r(U, nl9/JV{p)).

x~

L runs through all relatively compact

Cauchy sequence in {f~IU}

p,

l)/lv{ p ) )

We prove this by induction on

such that

Define

= sUP{/feL(X)llleL/ ~

L

open subsets of

~ p

nl9/fy(p))

.

p •

Let

be a

{f]I }

It suffices to show

admits an open neighborhood

U in

converges to some element of We can assume that

G is Stein and

V

Y - T. Si u

Jy(p-l) for some holomorphic functions

sl'

000,

sk

on

G

0

Con-

sider the following commutative diagram of quotient maps

Y\

/7

n'!J!Jy(P-l)

By induction hypothesis, f'

of

reV,

nV/<1y(P-l))

converges to some element

7(f~) 0

By applying the open mapping

theorem to the map

induced by

y,

we can find

such that

f

Since it suffices to show that

,

fv -

~(gv)

converges in

r(y, n~;I5V(p)) ,we can assume without loss of generality that ?(fv) = 0

0

For some holomorphic functions

aVi

on

G

- 31 9-

Y -T . Siu

From the definition of

plH L ' i t follows that, for

is a Cauchy sequence in ]) • [ 8 , II •D. 3 ]

T heo rem

on

G

such that, for

By

lal

= p,

the Closure-of-Modules

there exist holomorphic functions

/(11

=

ai

p ,

converges to

])

--> (]).

(2.4 )

Suppc ~e

as

order of

Then

k

converges to ep(Ea ~s.) i=l ~ ~

X is a complex space.

Define the reduction

X as the smallest nonnegative integer

that, if

for some open subset

U of

X and

in

PO

such

-320-

Y-T . Siu

for all

x

~

U , then

f · O.

A complex space is reduced if

By (2.1) a rela-

and only if its reduction order is zero.

tively compact open subset of a complex space has finite reduction order.

X is

SUppose the reduction order of be

~

P<

00.

Let U

'"U

a relatively compact open subset of a Stein open subset

Fbr an element f of r (U, taX) define IIfllU as follows. Imbed ""U as a subspace Y of an open subset G of

X.

of

(N

by a holomorphic map ~ •

r(~(U), lOy)

Ilfl~

corresponding to

.. inflpl/r*II!(u) /f*

Two different choices of equivalent semi-norms. ferent

p's

f ~ r'(u,

f.

~ r(~(U),

Let

(9X)

SUppose

t ~

,

be the element of

nV/5y(p))

,

induces f } •

U or its embedding give rise to (However, semi-norms defined by dif'When we write

U and a fixed

rb(U, lOX)

with

f

Define

may not be equfva Lentv )

we assume that a fixed are chosen.

Let

IIfllu < 00

~

II·"u'

(and a fixed

p)

denote the set of all •

r( ~(tO 'f)

)( u, CD n

be its power series expansion in

«

t.

xX

).

Let

The Grauert norm

0 of f is defined as sup 11r11 11 u • Suppose Ilfll U,t ,p 11 ~ .. lUi} is a collection of open subsets of X such that

Ui

is relatively compact in some Stein open subset of

X.

- 3 21 -

Y . T . Siu

For

i} ~ eft

l,={t,.

~O··· k

(A(tO,olX 7Jl, I{)

' 4 : n )( X

)

define

°

11E.11•• = VI, t, P When

n =

°

J

II E, 1/

i

O

sup

' ••• ,i

'tk, t

°'f

k

/lSi

i

0· •• k

°

II U. (\ ••• nU. ,t 'f ~O

~k

is simply denoted by

~ (ll, (OX l denote the set of all I!E,II < CXl •

E, ~ eft (a, 0x l

II~lIVl

.

with

VL

For a holomorphic function power series expansion

define

g

on

A(tO, pl

with

Let

~

u.

322 -

Y - T . Siu

Theorem B wi th Bounds

C3.1)

Lemma.

of lC N and

subset Q wi th

Suppose

Stein.

~

X is a complex subspace of an open Gl C C G2 C C.Q are open subsets Hi a Xn Gi Ci .. 1,2). Then there

Let

exists a continuous map

1J' :

r (ACtO, f)

x

linear over ([ t]

i)

e"f is

~, ID.

([

n

)( X

)

r (ACtO, f) x Gl,

->

n+ NID )

such that

the restriction map from

is the quotient map. 11)

1l'1C~) 1IGl' to 'f' :; c 11E,11~ ,to 'f'

C is a constant

f.

independent of ~, to, and Proof.

where

By considering the power series expansion, we observe

that, it suffices to prove the special case where Take open subsets Stein.

Let

t,G;: r C~,

where

,

H .. X(\G


,

,

n" 0 •

Gl C C Gee G C Gz with If

C is independent of

If

H .. XnG

and

There exists

open mapping theorem to

If

~

•

G"

Take

?G;: fCG' , (0x)

such that

and comes from applying the

- 32 3-

Y -T. Siu

,

Let?

be the projection of ?

ment of the kernel of

r

L

Define 'Y'(~) = ? ' . of

7

,

2 (G , N(f)

->

Then 311 (s)

onto the orthogonal comple-

,

r(H , (9X) •

is independent of the choice

and satisfies the required conditions. Lemma-

(3·2 )

Suppose

X

is a Stein complex space and

c

Ul C C U2 C x are open subsets with U2 Stein. Then the restriction map r : r b (U2, rb(U l, (0x) factors through a Hilbert space. Proof. Stein

U1

0

U2 can be identified with a complex subspace of a pen subset Q of some ce N. Take open subsets

Gl C C ~ C C ~ CeQ with ~ Stein. Let ~= U 2 ~. From (3.1) we obtain a continuous linear map C

n

such that the composite map of

is

r.

Hence

r

factors through the Hilbert space

- 32 4-

Y -T . Siu

(3·3)

~.

Suppose

Gl C C G:2 C C X

Vi,,· {uJaI) j with

X is a Stein complex space and

are open subsets with

G:2

Stein.

be a finite Stein open covering of

UI1) C C UJ2).

Then for

k

~ 1

Let

GlJ ('lJ" 1,2)

there exists a conti-

nuous map

linear over

ct t]

0 map and II T(E,) 11 , ~ 7-1.,t ,f stant independent of Proof.

bf

such that

c; ,

CIlE,1!

.

U2 ,t

to, ~

the restriction ~ag:g;:,.,r.:;.ee=..;s;;.,.,;.w;.;:i;..::t~h~:.:..=..~;...::;.:===

~

0

'f f .

C is a con-

By considering the power series expansion in

t , we

observe that it suffices to prove the special case where n • 0 • t " Gl C C Gee Gee

Take Stein open subsets

,

t

Take a finite Stein open covering 1t .. lUi]

, VLI • t {Ui])" of G

(respectively

UJ1) CC U~ CC U~ CC Take

~ (; z~tt2'

" G)

G:2 •

(respectively

such that

uI2) • There exists


~ (;

rf - 1 (Vl,'

(0X)

such that

°7 • s; Ill' 1/7/1 vl where

C

.< c I E. Ilvt "

is independent of t:

and comes from applying the

-325-

Y - T Siu

open mapping theorem to k n Z (tt , (OX) •

-> By (3.2) the restriction map

factors

~hrough

7

jection of

a Hilbert space

H.

Let

7r

be the pro-

onto the orthogonal complement of the kernel

of the composite map

7

choice of (3.4)

7r Ittl

CC

'l-t))

=

~

•

Then

f(~)

is independent of the

and satisfies the requirements.

Proposition.

Gl

with

=

~(~)

Define

Suppose

X is a Stein complex space and

C C! are open subsets with

{uJ)))}

U~l) C C u~2). 1.

1.

a: ([ll

Let

AX!

-

Coker a

Stein.

Let

A.

o in A

satisfying the following: A(t

-> f()~

AxX

be a sheaf-

is flat with respect to

AX X ->

for

~

be a finite Stein open covering of - GJJ ()J = 1,2)

homomorphism such that

(a)

Q. E. D.

Then there exists an open neighborhood .Q. of

O,

fl C .Q,

there exists

'Y':r(~(tO'f) x ~,Im a) -> r(A(tO,P) x Gl' linear over ([,[ t]

such that

restriction map and 1/ 'J!(7)

II

d.Y;

Gl ' t

agrees with the

°,f ~ c 11711

~'t

0;

,p

- 326 -

Y -T . Siu

(b)

for


zk

A(tO, f) C .Q and k ~ 1 there exists (A(tO,1') x 'lilz ' 1m a) -> c;k-l(.A(tO, f) x Vll ' 1m a)

linear over

a: l t]

II f(E,) II

restriction map and where

C

Proof. disc.

[) 'f agrees with the

such that

C Ill-I! ° 1\ ,to:i,p . Vl. 2 ,t 'f

is a constant, independent of l;, to,

and

Consider first the special case where

f .

X is a poly-

We can assume without loss of generality that there

exists an exact sequence

-> Let

(a)m

(respectively

(b»

for the case L ~ m •

PI

(9

a

.AxX

(b)m) denote

-> (a)

Po

(f)

AxX

(respectively

By using (1.6) and (1.3) to ob-

tain local solutions of (a) and by piecing together these local solutions by Cech cohomology, we conclude that (a)l holds and that Cb)m_l clude that Calm and

implies (a)m.

(b)m~l

imply 1b)m.

case follows by induction on

From (3.3), we conHence the special

m.

Fbr the general case, we prove (a) first.

We can

assume without loss of generality that i) ii)

X is a complex subspace of an open polydisc

p., and

there exists a commutative diagram of sheaf-homomorphisms

- 32 7-

Y T . Siu

a.

->

f!JPO A~P

jquot. ~

a.

-> such that

N

Coker a.

Po

({)AXX

is isomorphic to

Coker a.

under

the quotient mapThen (a) follows from (3.1) and the special case. let

For (b)

m be a positive integer such that no more than

bers of

vs.

can intersect.

m mem-

We can assume without loss of

generality that we have an exact sequence p (0 m

AxX

->

rhen (b) follows from (a) and (3.3). (3.5)

When the flatness condition on

Coker a.

is dropped in

(3.4), the conclusions of (3·4) remain valid with the following modifications. i)

'Y'

and 'f

are linear over
over
Fbr a fixed

to,

f has to be sufficiently strictly

small. iii)

C may depend on

However, i f

to

and

p.

is not a zero-divisor of any stalk of

- 328 -

Y -T. Siu

Coker a ,then

C can be chosen to be independent of

(3.6)

X is a Stein complex space,

Suppose

analytic sheaf on Ax X, and epimorphism.

X.

fine the Grauert norm U,t

0

'f

AXX

A(t O'P) C C A

Let

compact open subset of

111:II

'f : oP

For

7

-> "J

fn •

is a coherent is a sheaf-

and

U be a relatively f 6: r(A(t O, f) x u, "}) de-

as the infimum of

where

A different choice of

rp

would give an equivalent semi-norm.

When we use such a Grauert norm. we assume that a fixed

'f

is chosen. Suppose Vl "" .[ Ui } is a collection of relatively compact open subsets of I . Fbr

define

When

(J.7J

to = 0 ,II~ II

'lil, t

Proposition

0

'f

is simply denoted by

(Theorem B with Bounds).

11~IIVl 'f Suppose

• I, Gv

- 329 -

Y -T. Siu

U"

are as in (3.4).

(JI .. 1,2)

analytic sheaf on A and suppose

a.:

lC

(OP .AliI

I

'1 is a coherent

Suppose

flat with respect to

-> 7

-> .c.

n:.a ll I

is a sheaf-epimorphism.

there exists an open neighborhood.Q

of

0

in

.6.

Then satis-

fying the following: for A(t O,pl C

(a)

n

linear over ([ [ t] tion map and (b)

there exists

a.1j! equals the restric-

such that

/rr( 7)/I Gl't0 ,r

~

c /I? /I

0

G.2 ,t ,p

A(t O ,pl C.Q there exists

for

linear over tion map and

a: [t]

6 rp equals the restric-

such that

Ilr(~)11

l\ ,t

0

'f

~ Cll~"

0

1712 , t ,f

;

C is a constant independent of Z, , t o , and

(1h!m '1

'f

is not

n-flat, for a fixed

to, ([ -linear

f . 'f

and

p

sufficiently strictly small but may not be , 0 linear over <[ Le l and C may depend on t and f. If exist for

t n - to n

is not a zero-divisor of any stalk of .

be chosen to be independent of Proof.

fn

•j

Follows from (1.6), (1.3), and (3.4).

'1 ,

C .£1!!!

- 33 0 -

Y -T . Si u

§4. (4.1)

Leray's Theorem with Ibunds

First we examine the diagram-chasing proof of the

usual Leray's theorem without bounds. herent analytic sheaf

o~

Suppose

a complex space {U(l}11

1

is a co-

X and

c; A

{Vi} ·' r: ~

are Stein open coverings of with an index map

: I

~

--->

'"'- I

X such that A.

1( refines Vi

Leray's theorem states that

the restriction map H

is an isomorphism.

L

(tt, 7) ->

Define

1.

H (1(,1)

cf"v (tt,

l()

as the set of all

such that

is skew-symmetric in (la, ••• , i O' ••• , i)l.

Define

I1

p

and skew-symmetric in

- 33 1 -

Y ~T .

ef'v (Vt, 1.() 6 1:

-> ef+1,v (1.t, l( )

62 : ef'» (U, l()

-> ef,Y+l (11, l( )

e1 : c

~

or ,F}

8 : efClJl ,F) 2

->

CO ,v (l7l, l()

-> ef,O.(7Ji,

I()

as follows:

. Iuo

.. c,.~O· •• ~)l

a.

('IV .

~O

Consider the following commutative diagram:

n···()v.~).1

Siu

- 3.32 -

Y-T. Siu

a

1

rex, "1 } ->

0->

0->

J

CO (1(,

a

1)

1"1 }

CO (Lt, 7

9

1 '1) -> co,

e2 'If1

°(lJl,

1(')

-2-> 6 ..J:...>

cl(Vl,

CO , 1

61 cl,0 (1t, J.( ) ->

62/ 1 6 (a, )() -> cl (a,

6/

1

'If

,1

J

l( )

61 ->

62

62/

'V

_6_> ...

e2'iI1

62

6J

el 0-:>- cl (1(', 'l) ->

a

'If

A sequence

is called a zigzag seguence if f*d f

JJ

(;

,f -))-1

zJ(l(,

(;

, i

* (; z

f I. ,

"l)

cv,l-v-l(lt,l()

(lot,

"l)

The proof of Leray's theorem consists in showing that the

- 333 -

Y-T. Siu

correspondence (cohomo.l rvy class of f*,t) (where

f*,£

Hi <7,t,

f t ,* )

are the end-terms of a zigzag sedefines an isomorphism between H1(')(, '1) and

quence)

"1 )

and

<;--> (cohomology class of

f£,*

which agree s with the map Hi ('1..t, "1)

defined by restriction.

->

Wi (l(, '7 )

This is shown in the following

three steps. a)

Fbr every

f*,L

~ zl{l(,"})

one can construct by the

Theorem B of Cartan-Oka a zigzag sequence with first term.

Likewise, for every

a zigzag sequence with bl

If

f l

f*,£ ' f )1.1-)1-1

sequence, then

,* (0

.

f l,* ~ Zl (Vl, "] )

there is

~)J

<.£-1), f£,*

is a zigzag

if and only i f

For the "if" part, let

with

Construct inductively (0 ~

such that

as the

as the last term.

f*,£ ~ Bl-l('l(, "l)

f,£,* ~ B1-l(lt, "1 )

f*,t

)J

~.£ -2 )

- 334 -

Y - T . Siu

e1g * , £- 1

+ f

&1 g )./-1,£ -li-l

0,£-1 + f

li,l-1I-1

The construction is possible, because of the Theorem B of Cartan-Oka and the following equations:

&1 f v-l,£-)l-2

+ 6 f 2 1I,}->l-1

0

Finally, since

it follows that

f 1,* '" &(-gt-l,*)

• The proof of the "only

if" part is analogous. c)

The correspondence agrees with the restriction map,

because, if

-

-

~~5

Y- T . Siu

f *, .e' f

the sequence

,

(f* ;.).

lJ,!-v-l

(O~V<£},f£,* defined by

.

~O···~

is a zigzag sequence. (4.2)

Proposition

(Leray's Theorem with Bounds).

X is a complex space and DxX ,

a,

7

flat with respect to

1('

Suppose

is a coherent analytic sheaf on

n ; .6x X

->

Suppose Vi., l( ,

A.

are finite collections of Stein open subsets of

X,

each of which is relatively compact in some (but in general not the same) Stein open subset of (Surjectivity).

(a)

lUI

C C

° in !

~

1

11(1.

t::.

Suppose

,

1{

X.

,

< < 7Jl,)( < < )(, and

Then there exists an open neighborhood .Q

satisfying the following.

For

A(tO,PlCQ and

there exists

(Cf,1j'l: zl(D(tO'fjXI(, 1)

->

Z£ (.6(tO ,f) x Vl,1) ffiC f - l (A(tO 'f) x)(',

linear over ([ l t]

such that

of

"1)

- 336 -

Y-T . Siu

Eo

Max(IIPU,) Il,~ ~

Vl.,

where

t

0

'f

'

0

Ilr(~l II", 't

,

A ( t o , pl

n

° )\~

lUI C C /l(l.

° in A

Suppose?il < ~

[(2,'7l

,

7..t, 1« < tt; f2£

satisfying the following.

c;:

!" to, and f.

,

and

Then there exists an open neighborhood Q

there exists a map

1

x 1('

c 11l,11)(,to , f

t ,f

C is a constant independent of (Injectivity).

[b ]

L~

+ 6 y(l, )

9' (l,)

=

Zi (A(tO'fll<

e

of

A(tO,fl C.Q and

from

vr.' ,7)ec£-1

(A(tO,Plx\(,1)

~= 67 on A( t~ pl xl(]

!Q.

cl - 1

(A ( t O' f ) x 17l, 7 ) linear over ([,[t]

6e(!,,?)

/le(!,,?)

J/,t""t,p °~

.. !,

on

L::!.(tO ,fl x 7J1.

° ' 11711If,t°,'f' ),

c Max (II!'I/, n,t,e

C is a constant independent of (When

e

'1

is not

exist for

p

n-flat, for a fixed

and

7,

°

f

t , and

to, ([, -linear

C maY depend on

to

is not a zero-divisor of any stalk of

can be chosen to be independent of Proof.

!"

T' 1f,

sufficiently strictly small but may not be

linear over 4: [t] t n - t~

such that

fn

and

7 ,

f.

If

then

.)

Fbllows the Same line as in (4.1) except that (3.7)(b)

(Theorem B with bounds) is used instead of the Theorem B of Cartan-Oka·

Q. E. D.

- 331 -

Y-T . Siu

~.

Extension of Cohomology Classes

(5.l)

Andreotti-Grauert [1] proved that a

k-dimensional co-

homology class with coefficients in a coherent analytic sheaf

7

can be extended across a strongly

k-pseudo convex boundary

and across a strongly (r-k-l}-pseudoconcave boundar y if codh? ~ r .

We need the corresponding result with bounds.

So we are going to examine one key point

~f

Andreotti-

Grauertts result in such a way as can be carried over to the situation with bounds.

The precise statement of the situa-

tion with bounds is given in (5.2). Suppose

X is a complex space and

analytic sheaf on ~)( X. are open subsets.

Suppose . 1. ~ 0

and

is a coherent Xl' DeC X

Assume the following.

H£ (A x (Xl" D) , '1) .. 0 Wt(AX D,t)

7

in case

t ~ 1

-> Hl(A x(Xl" D), 7) is surjective in case /, .. 0

if+ l(AxD,7) Let

-> if+l(AX(Xln D),"l) is injective

~ .. Xl U D.

(A x Xl) U (A x D)

is surjective for

'1

From the Mayer-Vietoris sequence of

on

it follows that the restriction map

)I

..

1. and is injective for ]) = 1

1

+

Such an argument cannot be carried over to the case with bounds.

So we look at the conclusion in Rnother way.

Intro-

- 338 -

Y - T. Siu

duce the following additional assumption

With this additional assumption, it is easy to see that one (respectively tt 2 )

can choose a Stein open covering ttl Xl

D) such that tt

(respecively

12

:

.a

ttl' U2

of

covers

Xl f"I D and (* ) Let ~.a ViI fl1J[2.

We are going to show, in a way that Can be

carried over to the case with bounds, that ~

H (~xU,7)

is surjective for

v ... 1.

->

~

H (Ax~1,7)

Let us consider surjectivity first. is clear. Assume Since H1(A x 'Zk

12

1. ~ 1.

,"1) ..

v

and is injective for

0 , there exists

7 <;:

1. + 1 •

The case 1, ... 0

; <;: Z.l(A

Suppose

»

x

U l , 7) •

Cl-l(,A)C

Vt 12 ,"1)

such that

l,1.6 x V!12 Extend

7

trivi ally to

67·

? <;: Cl-l (.6. U l , 7) Since ~ - 6"7 (*), ~ - 6;; can be trivially extended x

on A x 'U12 ' by to some element of Zl(Alltt,7). is

0

Now let us consider injectivity.

l,<;: ZL+l(6)CU,7)

such that

,

67

Suppose

Y.-T. Siu

for some 7" ~ C1. (A

?'

~

CI(06 x

Then there exists

such that

z,t2/1)

II

vt1 , 7).

67"

~ ~

on

A

tt12

x

•

Then

o on L ~

In Case

1 , there exists

Extend

7~

l;

1 C (4 x

,..

l;

trivially to

u ,1)

, 7"

l, = 6? on

A x

!,

= 67

on

(A x

A x 1.t

(Ax 'lJl , 2

on

~12' 1)

12

such that

•

"1).

Define

Ax~

1.t.

In case

1 A x 1.t • 2

R.

= 0 ,

e <; Z£ (A x 1..t 2 , '1).

Can be extended to

Then

I. 1

~ C -

,.. + 6 t; on

7= 7

? ~ Ct (A x tt , '1)

on

•

12

by

C· Then

t; ~ Cl - l

7, -?"

6(, =

Ax tt

(7' - 7") IA x Vl12

Define

by

A x

Vl •

Now we consider the strongly

p-pseudoconvex Case.

Assume that there exist i)

a biholomorphic map

r

embedding an

0

pen neighborhood

- 340-

Y- T . Si u

Ii c C

X of

disc

P

D onto a complex subspace of an open poly-

a: N

of

cf functions 11 ~ 12 on

real-valued

11)

Xl .. {erl

b)

d)

such that

< OJ D .. D(\ {Cf2 < O}

a)

c)

X

SUpp ('PI - f2) C C D (J)i-",-l . t h e restriction of a ) 0 ~s P

(i" 1,2)

whose restriction

cf function to

on

x <;: lt P- l

is strongly plurisubharmonic for every e)

1i

pn({x}xa: N- p+ l )

there exists an exact sequence

(I) .

o ->

,..Pm n+NI:.r

.

-> ... ->

....PO n+NU

->

0 .., R (.('1/AX D)

->

0

on L:::. x P , where ::; : A x D -> A x G is defined by the identity map of .A and '( • Then, by an intermediate result of Andreotti-Grauert [1, p.217, Prop. 11],

It follows that ering

the

(t),t

is satisfied for

l6>

p.

In consid-

situation with bounds, one has to apply the open

mapping theorem to

( 26> for a Stein open covering )( of analogous to

to

P

n Uf'"l < 0 J.

the proof of (3.3) and one has

the sheaf-homomorphisms of (#).

p)

to

in a way apply (3·4 )(a)

-341-

Y-T .Siu

For the strongly

t

that there exist case such that

p-pseudoconcave case, we can assume

,and

~l' f2

Xl'" Xl U D

'"

as in the

{
>

{'fl

> oI

p-pseudoconvex

o}

and conditions c), d), e) of the p-pseudoconvex case are satisfied, where

m ~ n + N - codh'"

on

Ax D •

and,

moreover, the Hartogs' figure

• '"

p(H f 2 > O}, where P '" P I X P II with PIlC
is contained in P'C
Again, by the proof of an intermediate result of

Andreotti-Grauert

[1, p.222, Prop. 12],

is surjective for t < N - p. tions of

(t)t

Hence the first two condi-

are satisfied for i

«

n+N-m-p. To obtain the

third condition of (t)£ for 1. '" n + N - m- p - 1, we argue as follows.

For every given

....

e > 0, we can assume that t, 'fi ,

HI ,H" are so chosen that, H1'

C H'2 C C. P ' • Q2:

contains

P n{ ~l

t h e Hartogs '"

> E.}

I

for some open polydisc f igure

(pi X (p" _

(cr ,

H"))

U (H2 x p ll )

[1, pp , 219-220]) •

show that the restriction map

It suffices to

·342-

Y - T. Siu

is injective.

II

Suppose

P

=

N-p" TT P.

j=l

and

J

j-l N-p I " U .. P x TT P x (P - HjP TT PJJ j j /pI It ~"j+l

"

"

.. ..

lt1 1%2

"

{U~l) , U 1' {U~2) , U ' 1

An element C,i <;: HN-P ('l~.' NlD)

N-p

TT

j=l

0

fi

j

with

... , UN_pI

... , UN_pI =

1,2)

is represented by a

U~ i) ()

on

II

H

(1 ~ j ~ N-p)

U n ••• n UN_ p, 1 if and only if the holomorphic function

holomorphic function ~i ..

(i

"

H

and

2:

f(i) za p+1 a + • • .a p 1 N p+l ap+l'··· ,aN"< - 1 on

U~i) (I U1 (\ ••• n UN_ p g.

function

1

on

U 1

(I •••

can be extended to a holomorphic

n UN-p

.

where

GO

f.

1

=

2:

apfol' ••• ,aN"

_CD

is the Laurent series expansion in the last nates

zpfol' ••• ' zN

of

a:: • N

Hence

1:

N- P

coordi-

is injective.

When

- 34 3-

Y -T . Siu

0 , the function

~i =

ZN-p(Ui, N~)

fi

when regarded as an element of

is the coboundary of

(i)

{h .

.

JO·· ·IN_p_l

} c;:

CN-p-l(Lt., NO) ~

where h

h

(i)

1\

01 •• • v ••• (N-p)

(i) 1 •.. (N-p)

=

2:

f

gi (i)

? 0 a p+ l ap+L l ? 0 a p-+: l

• • .a N

z

a p+ l

p+l

a p+)I :i- -1

It follows that, if

fl

is the restriction of

f2

and

fl

is the coboundary of some

and if sl ••• (N_p_l) then

and

f2

is the coboundary of

the sup norms of

Sj

j

0··· N-p-l

(2 )

[h .

j

JO··· N-p-l

and

f2

}.

are all

Hence, if

:i- e , then

- 344-

Y- T . Siu

the sup norms of

h

(2 )

. j 0··· IN_p_l

are

~

P"

and

H"

constant depending only on

Ce, where

C

is a

Now, to get the

situation with bounds, one need only apply the open mapping theorem to

(t < for a Stein open covering ~ of

N-p)

,.., P (l {Cf2 > O}

in a. way

analogous to the proof of (3.3) and apply (3.4)(a) to the /

sheaf-homomorphisms of (#).

(5.2)

Proposition

Ebunds).

1:

Suppose

(Extension of Cohomology Classes with

X is a complex space,

X -:> (a*,b*) C (-lXl,m)

cf function,

is a

a* < a#< b#< b* ' p,q" 1 , r" 0 , and analytic sheaf on 6 x X i) il)

strongly iii)

iv)

b# < b < b*

p-pseudoconvex on

"]

ii

flat

a*

{'f > b#}

and

{" < a#}

the natural projection (Supp1)

jection

For

for

q -pseudo convex on

is proper for

is a coherent

such that

{ 9' < b ]" = {1 ~ b ] Cf is strongly

"1

n ~x{a

~ 'f~ b})--->

< a < b < b*

with

respect

to

the

pro-

n: 6 x X -> 1:::.

a* < a < b < b*

let

x~

=

{a < " < b}.

Suppose

- 345 -

Y - T . Siu

n. c c c:

is an

pen neighborhood of

0

0

Vi, 1.(, Vi.' , )('

and

X , each of

are finite collections of Stein open subsets of

which is relatively compact in some (but in general not the same) Stein open subset of

X.

Then, for

< a l < a#

~

and

b# < b l < b* ' there exist a* < a2 < al and b l < b 2
2 ~ b2

Suppose

a2 ~ '" a < a ~ al '

I1(' ICC

, ' < < '1/' ' 1.(' " <.<" (.11., 1( ~,

'"

b

Xb

a

,., b (n x I ) n Supp'i C C .Q x \)e\' Then there

IVlj C C IN , and

a a exists an open neighborhood .Q . of lowing property.

For

~(tO,P)

cn

° in.Q'" and

with the fol-

pSi.


there exists a map

(rr ,1J') : Z t ( A (to, f ) x l( ,"1) -> linear over

Zl(A(tO'fpc?Jl.,'1)(BC£-l(A(tO,f)lCl(" ,7)

a:. [t] E.,=

Max

where

such that '1'([,) +

6'Y'(~)

on

A(tO,P)X

(1Irr([,)II U, t °, r ,IIJV(E,)/L, CIIE,1I)( ,t°, f " ,t°'F )~ .

C is a constant independent of

,

1"

(b) (Injectivity). Suppose 'Vi < < ~ , 1« jttj CC I~ , (SixX b ) n Supp"lCC QX\\fI, and

..

a

A(tO,f)

C

°

t , and

,

f •

< 'lk ,

a

(.Qx X~) () Supp"} CC

a borhood Q.

i

!2!.

n

Qxlu'l. Then there exists an open neigh.!.!:!..Q with the following property. For

° -

and

p <

£. ~ r - q - n there exists a map

f)

,- 34'6-

Y - T , Siu

from

{(~,7) c;: ZL(A(tO,P)XU' ,"l)E9C l - l(A(tO,P)xl.(,"l) ~= 67 on A( to " f) Xl(} to

Cl-1(A(tO,p)xVl,"l)

where

C is a constant independent of .;

(When?

is not

() exist for linear over t n - t~

linear over ([[tJ

f

c [tJ

n-flat, for a fixed

t

such that

,7 ,

to, and

f .

°,([ -linear 'f ' 1j ,

sufficiently strictly small but may not be and

C may depend on

to

and

f.

is not a zero-divisor of any stalk of OF , then

can be chosen to be independent of

fn

.)

If C

- 3'4 7-

Y,.T. Siu

§6.

Sheaf Systems The treatment given here is a simplified version of

the sheaf systems introduced by Forster-Knorr [3]. is a collection of open subsets

(6.1)

X.

of a complex space

A'"

U

For

A • v~ 0 v

U. (1 ••• nu. ~O

~11

means

1 $ iO a.". (i O'

with

(i O ' ••• , iJl)

Ua..

by

{iO' ••• , iv}

For

C

... ,

< iJ1 ilJ)

and let

~ k

G: A , denote

f\'" (jO' ••• , jll.) G: A , a. C f\ ,..

is defined as consisting of

a coherent analytic sheaf ~a.

i)

Ua.

on

such that Ya.a.

a. C ~

is the identity map of -9a.

is isomorphic to

(§a.' f~a.)

a. C ~

and

(O~a. I Ua.

is said to be free if each

for some

A morphism from a sheaf system (~, t~a.)

J pa.

0

Pa.· (§a.' 1j'~a.)

is defined as a collection

sheaf-homomorphisms e~ • Y~a. =

for

ct.

A sheaf system

system

a. G: A

for

a sheaf-homomorphism Yf\a.: §a.IU p ->§p

ii)

§a.

<

{jo' ••• , jp.l •

A sheaf system on £1

for

Av be the set of all

Let

(Ba. I U~)

ea.:

-ga. --> Wa.

for

a. C p.

to a sheaf

(Ba.)

of

such that The kernel of a mor-ph i sm,

an exact sequence of sheaf systems, and an epimorphism of

- 34 8 -

Y -T . Si u

sheaf systems are understood in the most obvious sense. Suppose 1,t of

t

t

X which refines Vl

1:*§ = (~a"

..,'11 t VL

on

For a sheaf system

~(atdU~t

where

"'t"(iO' ••• , i v) also denoted by ~I~'

7

Suppose

lUI.

~(~')-r(a'dU;'

equals

means

(T(i

.

~.

Vi

as a refinement of Q (1"'*

7 .

with

In such a

0; with the sheaf system a-*7 •

Suppose

Ui(l ~ i ~ k)

pact open subset of a Stein open subset space

is a relatively com-

'" Ui

of a complex k

~

"..., -> ... ->1(, m -> ... ->R.1 ->R.a ->;r

0

X

and

1

is

is a coherent analytic sheaf on

We can regard

case, we identify

-c*-§

o ) ' ••• , 1:(iv)) •

the unique index map a- and obtain

(6.2)

->

as follows.

J~ta': 9+a t ->W~t

ii)

1: ': {I, ••• , k I}

define a sheaf system

"'90, C/L

J ~ 'a tl

i) '¢fa,::o

.Q: •

by an index map

I

on

1U ) T ~a

is a collection of open subsets

Ui C Ul: (i) •

(1, ••• ,k} , i.e. .p... (.f1. '1 ~a'

k'

{Ui}i=l

is a coherent analytic sheaf on

Then there exists an exact sequence

of sheaf systems on

?Jl:

::0

lUi }~::Ol

such that each

U

U i::ol i

R.. m

.

is free.

- 34&-

Y -T. Siu

Proof.

,

Take Stein open subsets

"

, k V~ = !UiJi=l

~~

k = {UiJi=l.

,~"

(1 ~ i ~ k)

and let

suffices to

show that, for an} '. """Rf system

on

1Jl'

1t." ,

and

~

-§ =

as follows.

.J - ,->.g Itt'.

f) :

a O'

Fix a multi-index

J

struct a free sheaf system and a morphism

(a)

e

R/

0

(a)

0

=

(

(§a' '!f~a)

on

We construct them

It suffices to con(a)

.J a

0

(a),

-> §IVl

0

It

(.Ja , cr~a)

there exist a free sheaf system ;/ =

and.an epimorphism

N

U. C CU. C C U C CU. ~ ~ i ~

is surjective, because we can set ~ =

(a)

,cr~a 0 )

such that

$Jl a

(a )

on

~

(a )

BI¥ O 0

0

and obtain

O

There exists a sheaf-epimorphism on

;/(a o ) a

,

=

(a.O )

0

a

tr (a O )

~a

e(a O ) a (a ) O

ea

,

for

Define

for

a

a

et a

O

O

C a

PI U~' the identity map of (Ox

~a

o:(a o )

p

(OX1ua

C/; a

=

0

=

Yaa 0 (7 I Ua )

for

a

O

a

0

for

"o

for

et a

•

The construction is complete.

a

O

C a

for

,

a

O

cae ~

- 3 50 -

Y - T . Si u

(6.3)

Suppose ~. {Uilf=l

of a complex space on VL.

X

and.q = (§a.' 'Y'pa.) is a sheaf system

Introduce the cocha1n group

= (where

is a collection of open subsets

At

is as in (6.1))

IT

a.(;A - "1

r'(u

a.

J

-§ ) a.

and define the coboundary map

by

where

fl •

;:...

••• , Ji J

SUppose

...

7

_>R,m

is a coherent analytic sheaf on

X and

-> ... _>R. l _>R.0 ->'7 ->

is an exact sequence of sheaf systems on £n , where each ~m

is free.

Consider the following commutative diagram.

0

- 3'5 1-

Y - T . Si u

o

o

o

o

-> ...

-> ...

-> ...

-> ... ...

where the horizontal maps are coboundary maps and the vertical maps are defined by the morphisms of (*). OJ IT cl+JJ (Vi, e" ,

)I

An element ~

">R+ l.l ,~ ~

~ of CR(a)

R+v

c

(If, 1{,

~

j.

=0

is given by

by

.

(~l+~ ,~):=o with

Define

d : C1 (Vi) ->

Let

Cf.+ 1 (Vi)

- 3 52 -

Y - T . Si u

where

Define

by

(6.4)

With the notations of (6.)), if each

Proposition.

is Stein, then the map e*:

H~C' (11») -> Hl~. (Lt,1J)

Ui

in-

is an isomorphism for all £

duced by ()

(This proposition can easily be proved by a spectral sequence argument.

However we prefer to present a more elementary

proof, because it can be carried over to Proof. B

,.,

(a) (Surjectivityj.

Take l,

t~ case

with bounda , )

I

~ Z (Vl,?).

By Theorem

of Cart an-Dk a , one can construct, by induction on 11 , ~ "1.+)) ,v

r:

~

CJ.+)) (, .., ~i) V(

"'

(0

:i-

)J

< lD)

such that d.l,OE,i,O

,. ~

? Then

N

and 8!," l, •

1) •

- 353-

Y-T . Siu

(b)

(Injectivi ty) •

E,

Suppose :0

(SL+v,11 )::00

is mapped to 0 in Hi (c'(~, 7)) ~ c;: C2- I(Vl,'1) such that

c;: ZJ (t!) Then there exists

By Theorem B of Cartan-Oka, one can construct, by induction on v , (0 ~

v < 00)

such that d£_I,O 7,-1 , 0

t;

(-It ~l-I+)l,)I-l +

d£_I+J', 117R._I+1J,11 :0

5.t-2+ 1J,"J-17R.-2+ 11,))_1 ()I~l)

because

It follows that

n{ : satisfies

a? =

.

l, .

( 7£-1+)),11)1:00 )00 c;: Q. E. D.

Cl - l (Vl)

,

- 3S4-

Y -T . Siu

(6.5)

For a given strongly (p,q)-pseudoconvex-pseudoconcave

holomo~phic

map and for a given coherent analytic s heaf

on the domain space, we are going to use the results and and

§5

§6 to construct a sequence of complexes of Banach bun-

dles which Can be used to calculate certain direct images of the sheaf. Suppose

I,

are as in (5.2).

b ,.,

1 ",.

p, q , r, Ia,Q, ai' bi (i = 1,2) We arrive at such a situation by consider-

ing the graph of the strongly (p,q)-pseudoconvex-pseudoconcave holomorphic map and the trivial extension of the.coherent analytic sheaf to the product after its transplantation

to

the graph. Let

m ~ 1.

Choose

Choose finite collections

Vlj «Vlj + 1 « 7.Jl of Stein open subsets of i)

ii)

~

d.

I

(1

~ j

<

m)

such that ~

d +

n Supp7'cc.QxIUjICCQxxc j +1 (0

(.QXI J)

cj

N

j 1

~ j

< m)

there exists an exact sequence

... ->

II

~

-> ...

1 ->~

,.,

,..

-> ~0 -> '1->

of sheaf systems on Q)C 1.t where each

1?

,}

0

is free.

- 35 5-

Y-T . Siu

By (6.3) and (6.4), for

subset

'" of n

Q

,

1 ~ j ~ m and for any Stein open

we have a complex

(constructed from the co chain groups of

o ~ v < al)

whose 1. th

H't(Qx~j,l)

oJ

It IQ x 1tj ,

cohomology group is isomorphic to

(O~.e
By letting

Q

vary, we obtain a

complex of sheaves

""

on J2 whose cohomology sheaves are isomorphic to the direct

71nx

image of

/Vljl

under the projection.Q

lC

I~jl ->.Q .

Now we turn to the situation with bounds.

Fix

1

t < al)

and

Ua

Let

1tj

=

{Ui}~=l.

be as in (6.1).

i

~

C (Q x Vi., Jl) f

=

(fa)

C;(Q x z.tj ,

a ~ At.

(0

~

Define the subgroup

satisfies

for all

At

as follows.

J

belongs to

Let

Let

a

R)

~

j

~

m •

~(QxUj,Jt)

We say that an element

c;

At

c;

J.

C (Q x

oJ

Vlj , R)

i f and only i f

fa

c; r (Q x Ua ,(R/Vlj)cx)

- 356 - .

Y - T . oru

As in the case without bounds, we have a complex

As Q varies, we obtain a complex of sheaves

~ on ~,.

Since each

on"

is free, this complex of sheaves is

~

naturally isomorphic to

o -> ocE?) -> (9ci1:) -> ... -> VCEJ~) -> ... J J where V(E~) is the sheaf of germs of holomorphic sections ofa t'"

1

.

trivial Banach bundle E j on Iz , Let 1:'j be the i t h cohomology ,., dj ... sheaf of (#)j. Let 7rj : .Q.xXCj - ) Q be the natural projection map.

We have natural maps defined by restrictions:

').It. >'f'J

For

_> Rt Cn.) *;r "? -> ')/1. ,.,..J- 1 -> J

l. Cn . 1) * ..., ;r • J-

R

p ~ l < r - q - n , by the bumping techniques of Andre-

otti-Grauert [1] and by C5.2), one concludes that the maps RlCn j

): l - > 1.

9:tj

R£Cn j_l

).'7

L

-> ~j-l

satisfy certain conditions of surjectivity and injectivity. lliese conditions can be translated into relations between ~jI and

Rf Cn j )*1

.

llie coherence of

gated by working with

Wi.

RlCnj);:t will be investi-

lliese statements will be made

precise and presented in detail in Part II.

- 35.7-

Y-T.Siu

THE POWER SERIES METHOD

PART II

§7

Finite Generation with Bounds

(7.1)

In

(6.5) we constructed a sequence of complexes of

trivial Banach bundles and stated that the direct images, whose coherence we are interested in, are related to the cohomology sheaves of the complexes.

NOw we are going to con-

sider abstractly a sequence of complexes of holomorphic Banach bundles satisfying certain conditions and derive conclusions concerning the finite generation of the cohomology sheaves of the complexes. First we introduce some notations. trivial bundle with a Banach space ~(B)

For

When

(F,

Suppose

B is a

II-U F) as fiber.

be the sheaf of germs of holomorphic sections of to ~ A and

to

=

d ~N~

° ,~(B)(tO,d)

By identifying

Let B.

let

is simply denoted by

~(B)(d)

•

~(B)(tO,d) with its zeroth direct image

under the natural projection from .6. to

we can regard

V(B)(tO,d)

duced space A* .

as an analytic sheaf over the re-

(O(B) (to ,d)

is the sheaf of germs of holo-

morphic sections of a trivial bundle over A * whose fiber is

- 358 -

Y -T .Siu

TT

the direct sum of

d.foo

copies of

d.

1

1

For

F'.

with power series expansion

v

(t P to) f)l define

[r liB Denote by

B(to tdtf} f

with

0 d

tt t tf

..

the set of all

<; r(A(tOtflt O(B)(tOtdl)

[r liB, to ,d 'f

When

is simply denoted by

d .. (00 , ••• tOOl

°,dtf)

, sre

B(pl •

Now we introduce a formulation in terms of abstract

complexes of holomorphic Banach bundles.

••• -->

e:-

6 )1-1 l

1

a.

->

Let

6)1

Et:

a.

- > ~+l_>

lr~+l,a

"\I

1

)1+1 ••• - > ~-l_> ~+l-> Fu+l-> +1 )l

· 359 -

Y-T . Siu

be a commutative diagram of trivial Banach bundles and bundIe-homomorphisms on f:::. wi th al

~

a

~

a2

aI' a 2; that is, we have a Some statements concerning these com-

for s ome integers

sequence of complexes.

plexes which we will consider later are true only for al + c

~

a

~

ing only on

a2 - c n

where

c

is a positive integer depend-

(and other given numbers).

Such a restric-

tion will be clear from the proofs and will not be explicitly stated.

We will be interested in the behavior of these com-

plexes in a neighborhood of place A

O.

So we will sometimes re-

by some sui table A( fP)

For

t

o~

A and

n

d ~N*

• ~

0

let ~a[t ,d]

denote the

vt h cohomology sheaf of the complex

...-> (D(Fb,)/-1 )(t0 ,d ) -> When

to .. O· , ¥~[ to ,d]

when

d

>

O

~

(Q(~){t

.

,d)

->

1I+1

(Q(Ea

a

<

->"0

})

is simply denoted by 9?a[d] , and,

(0), ••• , 0)) , it is simply denoted by

For

0

) (t ,d)

'#a" •

let

~

0/

y

r~ ,~-l r~_1,~_2

For

When

to ~ A

d" (0),

and

d

~ N~ let

is simply denoted by

Y - T . Si u 11

~

.,)

(E) , (M) , (F)

Consider the following conditions

,

~

(B)n •

(Er

(Quasi- epimorphism with fuunds).

stant

C with the following property.

~~~(tO,d'f)

;""

with

6;;=0

There exists a conFbr

a < (:I ,

there exist

:

with

? such that

i) ii)

l'

iii)

~

0

,d,p

and

1f

a,(:I,t

0

,d,f

(Quasi-monomorphism with fuunds).

(M)

C with the following property.

stant

e; (;

a,(:I,t

•

~

0

(t , d , f) oJ ~

and

r(:la""

= 6(:1v-I ? ~

such that

i)

and

:

? (; Ft3

~l

0 ( t , d , p)

are linear over

There exists a conFor

a < (:I ,

satisfying

there exists

epa,(:I,t 0 ,d,p ( .."~

?)

r: ~

4: [t]

Fo"-l( t 0 ,d'f ) il.+l

-- 3,61- .

Y- T. Siu

l'

iii)

0 a.,P ,t ,d'f

( F).J

(

and

d ~Nn

is bilinear over It [t] •

) Finite-dimensionality along the Fi b ers.

Fbr

t

°

I~

~

there exists a commutative diagram of continuous

linear maps

°

~ 0 ...-> ~-l(tO,d'f) -> Eh(t ,chp) -> EttHI (t ,d,p) ->. ...

1

...->

1

F1>-l -> a.,t ,d,p

°

F-J a,t ,d,p

°

->

1

t

°

°

1

)1+1 ->... F 0 a,t ,d,p

1

"+1 0 1>-l( t ,d 'f) -> Eb.+l(t )I ...-> Eb.+l ,d,p) -> Ea.+l(t ,d,p) ->... where i)

11)

the composite vertical maps are

f'l'

°

a.,t ,d,p

-> Ff

a+l

(to,d,p) factors through a Hilbert

space iii) iv) v)

the middle row is a complex of Frechet spaces

"

. ° )<

dim H (F
IX)

° )- > H)/(F'a+l,t°,d,p ) is bijective

HY(F' a.,t ,d'f

-3 62 -

Y - T . Siu

~

(B)n (Finite Generation with Ebunds)a)

b)

°

~

Im(~

--->~

~,t

o c=

t

Let

t, (1 ) , ••• ,

(1

k)

f:J

11

m is finitely generated over n~tO

0)

~+l,t

for

~ i ~

~

A_

and let

A be the

0 - . Then, for f

~+l,t

there exists a constant If

~ . c;: ..~ (to 'f)

~ Ln '>..l ..,...,.,. ~+l,tO

with

C f

satisfying the following.

6~E,"

° such that the image of

A , then there exist

c;: r(A(tO'f)'

? c;: <;i(tO ,r) such that

of

sufficiently strictly small,

b e I ongs to

a (1) , •• _, a(k)

V O-submodule

n t

n([))

- 363- -

Y -T . Siu

(7.2)

Proposition.

,

( E))I

(M))/+l ,

)J

~

d '"

(CXl ,

°

"N~[iIJo

'Wa.,

III

~ (B)n •

to .. 0.

It suffices to prove the case

Proof.

\I

(B)n-l

Let

n G N

••• ,00,1)

*

°- 1 W + l ,0) H)l .. Im(~a. ,° Wa.+l,0) . 11+1 J}

H«+l

II

"7 Since Ha.+l

II

II

Im(~ (J,

:::

rt

--;>

--'Ii

is finitely generated over

... , = ° (1 ~

n_lJo'

t,(l) , with

o~ E,( i)

(for some i

over

f*

there exist

~

k)

p°)

such that they generate

n-l(f)0·

A ssume

=

···,Pn-l)

=

••• ,fn-l' fn)

°

f'n f.n < ~.

For

°

In this proof clearly we can assume without loss of generality that

0;

and

r:+l,r are norm-nonincreasing.

In other

proofs where we can also make such an assumption without loss of generality, it will not be explicitly mentioned.

Since

to= 0, we will drop to from all notations if no confusion can arise.

We break up the proof into four parts.

-364 - -

Y -T . Si u

(I)

First we prove the following.

f

Fbr

sufficiently strictly small, there exists

~ <;:

such that, i f

<. (p )

exists 9 <;: ~+ 1 (p*)

and

6~ E," a ,

C,

then there

satisfying

,. a (* ) 1/ 8

"11

110.+1'f * ,.< , n f)

By

"

a

in for

(B) n-l (1)

f

1lJ;llo. 'Io

is the power series expansion

where of ~

"11

C_

t

n

•

sufficiently strictly small there exist

f

<;: r(~(p*),

7

<;:

~~i(f*)

l;

<;:

~+ 1 (p

."

*

n(!J)

)

such that

"711~~i, r

independent of independent of

tn

tn

- 36.5-

Y- T.Siu

Let ~*..

(Xl

1:

~

>"=1 >..

t

(-!!)

>..-1

Pn

.

Then

Define

Then

Hence

M:>reover,

where

C"

f

is a constant depending on

Therefore (*) is proved. (II) Next we prove the following.

f .

It follows that

- 36 6 -

Y - T . S iu

For f

sufficiently strictly small, there exists

such tha t , i f there exists

c;

Y

>I

E

and 0o.l,=

Eo.(p)

~ * em ~ E~+l(P)

and

m~ 1,

p

then

such that

o~lX.+l (e m + r~+l a. ~em ~ 'A

a

D

)" -m

n) (t fn

A

'\ )

= a

(t)

is the power series sion of

E, in tn.

We are going to construct ease

m=1

y

Then 0ll+l L

em

by induction on

has been done in (*) of (1).

1:'"

or

or

a •

IJoe 11:+l,p We assume that

expan~

~ rlt+l,cx. Let

e

(

00 2: ~

h"'m

A

Suppose we

r~ve

,-m J + e m (fn)

y

,,; 1lE,II Cl,p

1s so small that v

1I"t11~+l'F ~

The

tn

< e + Ilemll:+l'f '"

fn

m.

2e •

Then

< or

e

+

fnD rr

e •

2 •

So

- 36 7-

Y -T. Siu

is independent of applying (*) of (I) to

1:

we obtain

,

'" <;;:

;V

~ (F* )

~+2

tn. such

that ~ ~a+2

,a+l (I"" +

~ 1: r a + 2,a+l *)

y

1I;tlla+2,p*

;f

=

o

fnCr/e •

Let

t

(X)

l,# =

We are going to apply

E

~ (

n)

A-m-l

A=m+l A fn

(M) " +1

to

For this purpose we need estimates on the norms of the two expressions in parentheses. First we are going to obtain an estimate on the norm of

~~ ~#.

by defining

Since

By

- 368-

Y-T . Siu

f'or

tn

=I

~+l(f*l.

0 , we Can regard

6~ E.# as an element of'

Because

it f'ollows that

e •

NOw we are going to obtain an estimate on the norm of'

(r~+2 ,a. c,# - r~+2 ,a.-+l"r* -

1-).

From the def'inition of'

~

(where we have defined

2e + e +

PDr De

"r ,

~

4e

- 369 -

Y -T .Siu

(;~) for

tn

f O}.

-1

(LO -

r~+l,o.C,m

- em)

It follows that ' \I

(;n)(r~+l'o.S#

- 1:*)

~

,

o.+l,p*

n

Pn

-0 4e •

Pn

Hence

By

(M)

where

)1+1

C'

,there exists

such that

is the constant coming from

(M}~+l .

This fin-

ishes the proof of (t) if we put

(III) Now we apply ~

~+ lover

lJ o

~

(B) n-l

• Let

to get the finite generation of

-3 7 0 -

Y - T . Si u

be the power series expansion of

em in

Since

tn

and

~ emO

v

by applying 11

Im (Ha + 2 ,)

ra+l,at,

(B)n_l

to the

:i-

n_lOO-submodule

-v

-11

-> (i)

11

+ r ,) + ~ 11 a 1 ,a m a + 1 'P*

of ~a+2,o generated by the images of

~+2)

(1 :i- i ~ k) , we can find, for

f

sufficiently

strictly small, (i)

(; r(A(f* l, nlD )

'7 m

(;

am

l;m (;

such that

~';}(f* l 11

*

~+2 (f

)

independent of independent of

tn

tn

- 37 1-

Y - T . Siu

f*

< =

" f n D + l)e Cf5(

117mll::~ ,p*

< =

C,o ( n

IIt; ml(+2,p*

~

" (f D + l)e , C_

I a (i) I m

where

C" p

it

'"

f

+ 1) e

n jO )l

(B)n_l •

follows that

IIt;ml(+2 'f* C

" r Dr

is the constant from

From (#)

where

jO

<

'"

"

C"' C_( fnD- + l)e

f

f

is a constant. m

By multiplying (#) by

we obtain

(~~)

and summing over

where

e"" a

(i)

III

E

a (i)

III

t

m=O

m

t

(...E.) Pn

n)

E ~ ( m-O mPn

m

m

m ,

- 3 72 -

Y -T .Siu

It follows that

lie11:+ 1 'f I a {i} I ~

r

II? Il a+ 2 , f -;

Ill; Ila +2 .f

~

f n Dpe

~

" C-(fD_+ 1)e f n f'

" fn D < cpt 1

>2

~

+ 1)e

" CC '" Cf{fnDf

...,

+ 1 )e

'" is a constant. C

where

1 }e •

By

~

(E)

,we can find

o

such that

-373-

Y -T . Siu A

where

~

C is the constant from

(E)·

•

Choose

so sma 11

D 'n

that

"( C fn D

2l" +

r

r

)

-BCC ,It" C_(~ D

r

fn

n

1 ~-2

+ 1)

f

Define Fi

,.

Let '¥ denote

(i)

a

(~)

11

1

~(~)

r a';3 ,a+27 +

'f(~)

,..., ~

qlo ••• o'J!

....

?

.

(). times).

Then

" (f D_ + 1) e + e • 2 C_

r

~

n

r

This is almost but not exactly the result we want concerning finite generation, because the equation is in of

Eh+l'

We are going to remedy it by using

Ea +

instead

3

(E)"

first

before using the argument to get the equation. (E)"

By . y

there exist

~(i) ~ Ell

a-2

(pO)

(1

~

i

A(i)_

° such that they. generate (We retain f Im(f-b:-l -> Ha_ 1) over ° and n-l °

with

°a_2 E..

"

~ k)

-

)I

I[)

k

.- 3 7.4-

Y-T , Siu

simply to avoid the introduction of more symbols and such a retention clearly does not result in any loss of 'gener al i t y . ) (E)"

· By app1 y~ng

to ~~

/\

,we can find

))

t, ~ Fu-2 (f)

/\

with

11

7~

~+l (f)

such that

11~1I:-2 ,p ~ Ce 11911:+ 1 , p ~ '"Ce

•

/\

t{i)

By repeating the preceding argument with t, , ~ of E"

{instead

~(i)), we obtain (for p sufficiently strictly small)

such that r

{where again

II

1\

J:

a.+1 ,a.-2 ">

" ' D f

Cp

..

are retained simply to avoid more sym-

bols and such a retention does not result in any loss of gen-

- 375-

Y-T. Siu

erality).

It follows that

~

Hence

is finitely generated over

~+l

nOO.

We have actu-

ally proved more than this, namely, we have shown that the finite generation is with bounds when generators are chosen in a certain way. (IV) We are going to prove the full strength of

,)

(B)n b)

by

invoking the existence of privileged polydiscs (in the sense of Grauert).

6~(i)

.. 0.

Suppose Let

~(il ~ ~(fo)

A be the

~

(1

k)

with

~:+1,0

gener-

and t h e finite generation

(E)"

By

~

n
~(l), ••• , ~(k) •

ated by

i

n_l
with

» Im(Hll_ l

° such that

6,)11-2 l; (il ""

-> -HL1).

Let

t -> Pill+l,O v / A

f: nOO be induced by the (

1 xI

,)

matrix

r"'+1,"'_21" '" '" ..,

(an element of

(l)

(v(i 1 l I· •

(1) )

being represented by a column i-vector).

There exist (after shrinking --

)l

, ••• , r"'+l '" 21" '" 1"'-'"

• I

veil) f,

pO)

( r (A (0 I::. P)

I

(1

~ i

~ m)

- 3 76-

Y-T .Siu

such that the germs of Ker f .

y(l) ••••• y(m)

There exist (after shrinking u i )

j

~

r (A( pO),

K(i)

~

<;i(

at

0

generate

pO)

n(9) (l ~ i ~ m, 1 ~ j

~

£)

pO)

such that

(1 ~ i ~ mj

Let 1': n(!)m ( y(i}) j

->

oJ,

on A(pO)

n

l~i~m, l~j~£·

Et: (f)

Now. take .; ~ image of .; By

be defined by the matrix

(III), for

in

'N~+ I ,0

wi th

6~ ~

belongs to

=

A.

° such that the Let

e ""

11.;/1:. f

p sufficiently strictly small, there exist b

i n

~

r(A(p), neD)

~

<;i(f)

such that

Ihi If

lin 11~;i.f

C-e *

r C_e * f

•

--37-7-

Y -T . Siu

where of

E.

is a constant depending on f Since the image v in 'N-a+l,o belongs to A, it follows that

c~

p

By (1.6), for

(Coker

sufficiently strictly small,

p ) -privileged

Grauert.

neighborhood of

Hence, for such

f '

A(p)

is a

in the sense of

0

there exist

(1

~ i

:i-

m)

such that

~ c .(vl(i), ••• , v(i») i""l a 1,

where

c~

is a .co ns t a n t depending on k

t ( i-I

f.

It follows that

c

~ c u(j»)r-; ~(i) + 611 - 1( ~ .4l:(j) + l j i a+l,a a+1 j""l J j

This concludes the proof of Notice that, if

tn

n) .

Q. E. D.

( B)" • n

is not a zero-divisor of

(Coker ~)O ' then all the constants involved in the proof are independent of

(7.3) (F) "

fn •

Observe that, by the open mapping theorem,

===*

)1 (B)O.

ltbreover, by the condition of factorization

through a Hilbert space given in

"

(F)

,in the statement of

- 378 -

Y - T . Siu

"

{B)O

we can choose

a

(i)

and

?

so that the map

is linear over ([ [t] • By induction on 11

(E)

, (M)HI, (Ff

~

( E)

on

(M)

n

HI

n

,

it follows from (7.2 ) that

===} (B)" • Under the assumption of n 11 (F) , we are going to show, by induction

,and

that the natural map £)

from

to

is an isomorphism. to ~ O.

that II

(F)

The case

~

( E)

trarily

follows

immediat~ly

follows from

~

(E)

•

from Suppose

for some

, t.,* is the image of some m ~ l.

-l, ~ 9f(l_l [dm.]O be ~

":I.'))

n = 0

The surjectivity of £)

•

e(~*) ~ 0

By

We can assume without loss of generality

m

l'l"~+l [d]O II m in P:t(l [d]O

is

Let

the ima ge of~.

Take arbiLet

Since the image of

C,

in

by induction hypothesis the image of ~

0

is

d m = (CD, ••• ,

y

c, ~ -aI-(l_l,O . n CD,m) ~~ * .

0

From the cohomology sequence

_ 37 9 -

Y-T .Siu

of the short exact sequence

°-> (O(~)o ->(Q(~)O -> (Q(()O(d m) -> a-

(where

o: is defined by multiplication by

"

in . Wa. ,0

th&t the image of t,

Since

Im(~a.oJ,0

" -> ata.+l ,0)

t~), it follows

m v belongs to tn'H a. ,0·

Hence

l)o'

is finitely generated over

it follows from the arbitrariness of From now on, whenever satisfied, we denote

0

':I./~ Im(~a.

°.

m, that ~* =

(E)" ,(M) v+l ,and '11"

--->wa.+l)

':1./".

by

~

(F/,

are

"is

~

in-

dependent of the choice of a. • (7.4)

Let

us investigate under which circumstances, in the

statement of

¥-p

(Brn , we can choose

a(i)

and

?

so that the

C, f--> ( a (1) , ••• , a (k ) , 7 ) is linear over

a: [t].

Looking at the proof of (7.2), one

easily sees that this is the case if

i)

~

(B) n- I

.

has the correspond1ng property of

[[tl, ••• , tn_l]-linearity "

Im(Ha._l

)j

-> ~-l)

-oJ

for the

of "a.-l,O •

n_IOO-submodule

- 3.80 -

V -T o Si u

Coker Cf

ii) For

1

t

~

let

n

~

o.

is locally free at

1, ••• , 1) '----y--J

n-t Denote

Im(~[d(t)] ->~+l[d(.e)]) by ~

.

~~[d(t)]

which by (7.3) is independent of the choice of

The above condition By induction on

n , condition ~

ural maps from 9f [d for

1 < l

~

ii)

n.

(.l

)]0

ii) ~

y

A = Ha + 1 • is satisfied if the nat-

is satisfied if (

to 9:1- [d £-1)]0

.

are surjective

(A by-product of this surjectivity condition

is that all the constants i n the proof of (7.2) are independent of

p , because of the last sentence of (7.2)).

From

the exact sequence

(where

0-

is defined 'by multiplication by

tt)' we obtain

the exact sequence

->~:(d(t)) ->~:(d(l)) ->1J~(d(1-1)) ->Pirl(d (t)) _> ••• Hence the surjectivity condition just mentioned is satisfied if p.j~ = 0

for

v<

p.

< ~ + n.

Of course, in general , this

- 381-

Y - T . Siu

last condition is not satisfied. (E)~, (M)~, (F)~

for certain

Under the assumption of

~'s, we are going to modify

the complexes so that the new complexes satisfy this last condition. (7.5)

Suppose

L

v

P ",,6x ([ ~

and -;

->L

-;-1

)1" v+l - > ->L ->L

•••

is a complex in which the maps of (holomorphic) bundle-homomorphisms.

Let

be a commutative diagram of bundle-homomorphisms such that for

~~ a

~

~: ' it suffices to have

Note that, to have define

a <

V as r aa ~~ a

l

• and then

Let the complex

l

be the mapping cone of the above commutat ive diagram; that is, Ny

Ea

and

'"

- 38 2 -

Y - T .Si u

is given by

Define

by

We are going to prove the following three statements (after a possible replacement of .6. by any open polydisc relatively compact in a)

The complexes comp.Lexas

b)

Ft:

The complexes

"'.

~ c)

--.J

satisfy

The complexes N.

Fu

satisfy

~).

Et:

satisfy

( (M)"

( ( F)"

Statement c) is clear.

and

satisfy ( E)"

( E{'

and

.

satisfy

(M)"

satisfy

(Ft ::::::::}

. .

(M/+ l

~

the

:::=::} the complexes

Let us prove b):

the complexes

Suppose

_

for some

l, ED f

,

nltion of

,

Y-T .Siu

°

1

N~

,

383 ~

(; ~- (t,d'f) N~ "'''-1 6a ,6 we have 13 ~

i)

6a

s; +

=

1 '

~

)I

r;aE. ..

6 - ~

f

iii)

6a a

a

~

..

~ rr~ .. 0"")1+ 1 a

It follows from

+ (-1)

13

cf

< 13.

a

)/+1 )1+1 ( -1 ) O""a f

.•

ii)

and

° (J

(Mt

tained from ii) ) .•

,)+1

I

13f

and i) , iii) that

of the complexes

" (~+ (-1) r13a "'>

~

,

t

°.

v+l ~ , Y 6 ( E, + (-1) o:a f ) a

By applying

From the defi-

~

,

0""a f )

t

E

to the equation (ob-

6

J,/-l~'

13

'-,

we obtain

such that

II

(-1) )l + 1 ()a" f , ,)

a,tO,d,p

So we have

I 'II" °

,l,

-1

13,t ,d'f

)

.

-

38 4 -

Y -T. Siu

The requirement on the estimation of the norm of the norms of

.; lB f

and

Now we prove a) , c,EDf ~

{

.;

,

Suppose

0 Ep(t ,d,p)

N~

6'"J 1; + (_l)H 10-;+ If

p

all+lf

E9 f

,

with '"

~" w lI> ~

f'

is clearly satisfied. a

<

p

0,

7,;(E,ED f)

i.e.

0

= 0

From the first equation, we obtain ~ c:?'+ 1 rp ,a-l a-I f

'"

y v c5 p (-1) C,

~+l • (M) of the complexes E t to this )1+1 "11+1 )1+1::\Y+l 6a_lCT"a_lf '" CT"a_l Q f = 0), we obtain

By applying property

equation, (since

such that r

y )1+ 1 ~ , a,a-lCT"a-I f = 6 > a~

It . follows that

By applying property

l,

+ (-1)

)1+ 1" , r pa t, ,we

(Et 0

btain

of the complexes

E(

to

-

385 -

Y- T . Siu

with

such that

'!hen

,

(f, +f,)

(9

f

c;:

0 Eh(t ,d'f)

"'~

with

'6:((E,'

+ l,) EB f)= 0

?$O c;: i;-l +1

l,

",11

r~+l,~( Sf)

(' r~+ 1 ,a (l, + ~)$ f) +

~11

6;~f(7$

The requirement on the estimation of the norms of and

? EB

(7.6)

0

by the norm of

Proposition.

(Et, (M))I+l, (F)v

C; $ f

Suppose, for

0) •

,

(l, + l, ) Q) f

is clearly satisfied.

p:;;:

v :i-

hold for the complexes

s , the properties

~.

Then there

exists a complex

of trivial vector bundles of finite rank on ~(fO)

pO c;: IR~

and A( fO)

(where

C A) in which the maps are holomorphic

bundle-homomorphisms and there exists a commutative diagram

-

386

-

Y -T .Siu

x: ()~

/

\

\}

~

0"'~+1

~ -> (+1

r~+1,a

of complex-homomorphisms on A{fO) of

-

for

p

satisfy

0-"

a

>J ~ s •

~

Proof.

such that the mapping

We are going to prove by descending induction on

that, for

p

~

p

~

s + 1 , there exist a complex

of trivial vector bundles of finite rank on some ~(fO ) a commutative diagram

such that the mapping cones

for

P

~

V~ s "

p

N.

pEb, of

0"""

pa

satisfy

and

- 387

-

Y-T . Si u

The case #L• a 0.

#

s

a

1

+

To go from the step

serve that, by (7.5), #E~ (F)#-l.

is trivial, because one Can set

By (7.2), #~ A:

has property

is finitely generated over shrinking c,(l) ,

to the step

has properties

(H\I(CO(#~)o)

Im

..

#

->

nmO.

# - 1 , we ob-

(E)#-~(M)#, and

(B)~-l.

H)I((9(#~+l)O))

One can find (after

pO)

... ,

whose images in

A generate

#_lL 0 #-1

)/

#

~

..

rr~

#-1 u

(#

~ ~ ~

rr~ (l

Let

be defined by ~(l), ••• , ~(k).

Define

#-11:' ( - 1 )·#-1 r au 1 1

all - 1

1l-1

Define

LV

# 'i/ #

nCOO.

A over

s)

-

388 -

Y-T.Si u

Then the complex

~_lL· and the map

requirement. Q.E.D.

a-. satisfy the a.

~-l

-

38 9 -

Y - T . Si u

§s.

Right Inverses of Coboundary Maps

As in § 7, suppose

v

... _~ ~-l _~ ~

(a

~

l

a

~

(2)

6a

~ <+1 _~

is a commutative diagram of trivial Banach

bundles and bundle-homomorphisms on A We use the same notations as in§7. i)

the complexes for

11 )

p ~

~

v :i-

(

satisfy

Fix

(E))I, (M))I+l, and

~

for

p ~

(B)~ and

(B)t

tional statement of
l

~ p + nand

hold with an addi-

So, for

p

sufficient-

~ .. p, p + 1 , we have maps

from

~;i(f)

v

a •

By the results of§7,

to

(Ft

p + n •

.

Im(~,O -~ ~a+l,OJ .. 0 all

6,)6)1-1 =0 • a a p. We assume that

with

linear over a:[ t] ~ ~ ra+l,a

such that

..

v-I t)l E,

6a+ l

11~:~lt:~,p ..

<

a )l

C

II~/la 'f

~

!l!a

-

390 -

Y -1' . S iu

where

C is a constant.

It follows that, for

l:, c;.

((p J

we have

By replacing LJ. by A( p)

small

p ,we assume that ~~ and ~~+l are defined f or

p.. (1, ••• ,1)

and, from now

maps for that particular

(8.2)

for some sufficiently strictly

on,2~

and

9?~+1 denote the

f·

We are going to define

such that (* ) on

(0((-1) •

Fbr

~(tO, p) C D. and

S c; ((to 'f)

with power series

expansion

define

where

~~

is regarded as an element of

~(l, ••• , 1 ) .

Be-

-

391 -

Y -T. Siu

cause of the norm estimates for

~g+l' ~g+l, we have

r.g~~(tO ,P) •

'fg;(;

To verify that this definition can g ive us a sheafhomomorphism, we have to prove its compatibility with re. I , 0 s tr-Lc tdo ns- Suppose ~(t ,P)CC ~(t 'f). Let

f,' (; r.g (t r , f' )

be the restriction of

z,

Then we have the

power series expansion

,

E,

=

Hence

It follows from the


~g+ i

'

~g+ 1 ,

6g

norm estimates that

equals the restriction of ~~ t, to

~(t

, ,

,r) .

It remains to verify the identity (*j.

~(tO'f) CC

Ll.., take

For

and their

_ 392 -

Y -T. Siu

Let

Then

E(E A 'I ~ and, by definition of

Since ~g+l' ~tl

1£ g

and (*) follows.

oA -f t )

,

both are linear over ([[t]

estimates, it follows that

Hence

) (t A A-T, ,. ~

and have norm

-

3 93 -

Y-T . Si u

12· (9.1)

Global Isomorphism

Eti

Suppose

is a sequence of complexes of trivial

Banach bundles as in that, for Im(~"

p:;'

oJ

§7.

II -> ~Cl+l)

II

..,

-> W- P+1) is an isomorphism.

Im (~p

to

Assume

a < P , the natural map from

and

~ S

s ~ p.

Fix two integers

Let (p :;.

... , ,.,

Assume that 9:jP, NP+ 1 ,

:1/5 -

1

s)

are coherent on

&"-1

Im(C9(~-l) ~>

Ker((O(~)

v :;.

•

LJ..

Let

(0«))

6Cl> (O(Eh+ l ))

and, for any open polydisc Q C LJ., let

H~(.Q) Suppose that there

exists, for every

Cl

J

a sheaf-homomor-

phism

such that P- l 6a+2

ep (l

=

We are going to prove the f ollowing two statements f or any open polydisc

n C A.

-

3 94 -

Y - T . Siu

Let

Consider the following two statements.

If(S'2, E:)

1))1

k

~

->

If (Sl, S:+V_P+2)

1 •

»

2)

))

If(Sl, 'tal - > ff(n,

)I

k

~

has zero image for

~a+)l-p+2) has zero image for

1 •

First, let us show that

==}

1)

» The commutative diagram with exact rows

S"a

0->

(* )

)l ~

~)la

->

1 .

,..~

for

2)

JJ

p

~

v < s

."

-> ,#->0

~

II

o - > Sa+v_p+2 ->~a+}1-p+2 - >

,)

'#->0

yields the commutative diagram with exact rows

If ($1,

e:)

->

ff(Q,

1

i

...

~:)

->

If (Q,

1 "

II

ff (Q., S a+1I-p+2) - > Hk(Q, ~C1+V-P+2) - > ff (Q,

Since

9-i~ is coherent, ff(Q,

!result follows.

W'll)

=

W")

0

for

k

~

1 •

vi') . The

-

399 -

Y-T. Siu

Next, we want to show that

1)

2)~ ~

~+1

for any

«,

The commutative diagram with exact rows

0->

->0

yields the commutative diagram with exact rows

If(.Q,(9(~))

->

1

1

k(Q,<9«+1I_P+3 H ))

->

~

Ihe result follows from the fact that 0-

"

e,

/0-

'"

-> ~+1 (n, ~la+1I-p+3)

1>+1 (S2,6a+1I -p+3 )

~(.Q, (!)(E~+)I_P+3)) .. 0

i)

and

together with the vanishing of

for

k;;;- 1 , implies that

for

p ~ y

for

p ~ ~ ~ sand

<

[k S;' 1)

•

and

factors through the map

Now we are ready to prove

0",

If+1 (Q, 9.~ a)

->

If(Q,B~+l)

sand

lip

2)11 ===} 2 )~

for

holds. 1),,1

<

'!he existence

~(.Q.,(ry( (+2)) .. 0 Since

for any

p ~ v

ii).

s.

).l

1)

II

,

==? 2)II

we have

1))/

'!he diagram (*) s

yields the commutative diagram with exact rows

-

3 96 -

Y - T . Si u

r(Q, ~~)

,..

(0,

By

l)s,

->

1

r(.Q,W S

)

1

II T

't~+s-P+2) ->

r(Q,

T is surjective and

S

'# ) -> Hl(Q, B~+s_p+2 ) i)

follo~s.

we have to distinguish between the case s

> p.

W-(Q,S~)

->

The case

tence of e~+l.

s· p Suppose

Tb prove

s = p

ii),

and the case

follows immediately from the exiss

>

p.

The diagram

(t)~;i

yields the commutatiYe diagram with exact rows

->

Suppose an element r(Q, *s).

Since

1-

of r(Q, ~~)

is mapped to

Then

factors through the map

it follows from

such that

t.

2)s_1

that there exists

0

in

-

39'7 -

Y -T. Siu

Hence

ii)

is proved.

(9.2)

SUppose

is a sequence of complexes of trivial

~

Banach bundles as in §7. that the complexes p

~ y ~

Max(s,p+n).

~

Fix two integers satisfy

(Et,

s ~ p • Assume l, (M)lI+ and (Ft for

By (7.6) (after replacing

~(PO)) there exists a complex

L::. by some

L' of trivial vector bundles

of finite rank on A and there exists a commutative diagram

of complex-homomorphisms on L::. such that the -mappi ng cones

~

for

of o-~

satisfy

p ~ y :? s .

By the results of

§e,

(after replacing ~

by some A(PO)) there exists a sheaf-hpmomorphism

such that

are as in (7.5).

-

39 8 -

Y -T . Siu

For any object derived from

E~ "

we put a

'"

on top

of its symbol to denote the corresponding object derived from

~.

For any open polydisc Q C A and for

d ~N~

let

For

~.6

to

and

d

~ ~~

,

'£~

to

~ b. and

induces a sheaf-homomorphism

8~(tO,d) from

Im((O(E;~-I) (to ,d) ->

~~(tO ,d): to

(0(~;~)(tO,d) such that

... 0 on 6~(t ,d).

By applying (9.1) to the complexes of bundles

~(~}(tO,d) , we obtain the following.

associated to

.... 0 .... I 0 ~p[ t ,d], ••• , Ws - [t ,d] to

(0(F&)(t O ,d))

i) ii)

""v

0

->

~ (SL,t ,d) ...,~

Ker ( Ha, (Q.,t

0

r(Q.,

for any open polydisc Q with Since

dn

f

<Xl

d

n

"!

<Xl

..." 0 JH t ,d])

,

then, for

p

~ v ~ s ,

is surjective

.. " 0 -> rtc, 'N [t ,d]))

,d)

C

d ~N~

b. for all

are coherent on

~ A and all d ~ ~~ with

If

",, ~ 0 Ker ( ~(Q,t,d)

C

A

->

and for all

-~

0

~+)l_p+3(n,t ,d)

to ~ A and all

•

~(~}(tO,d) is the mapping cone of

)

- .399 -

Y -T . Siu

and f(Q, (9(~) (to ,d))

is the mapping cone of

we have the following two long exact sequences:

°

°

11 ...,~ ->~a.[t ,d]->9rJ.a.[t ,d]

- > P\lHl((a(Ll (to ,d)) - > ->

9J:+ l [ to ,d]

_>

°,dl - > ""~fb. (Sl,t°,dl

I\z. (.Q,t j)

- > l+l(r(Q, (!J(L) (to ,d)))

-> ...

- > th+l(Q,tO ,dl

From the first long exact sequence and

(M)~+l

(p ~ v

< s) ,

it follows that 9>/-P(OCL)(tO,d)) ->~.P[tO,d] ->WP[tO,d]-> ~p+l((D(r:)(tO,d)) _ > ••• _>,#s-l[tO,d] '_

>

w,s-l[tO,d] ->W-S(O(Ll(tO,d)) ->9f.s[tO,d] is exact.

From the sharp form of the Five-Lemma, we conclude the following. If ~p[ to ,d], ••• , Ws [ to ,d] are coherent on 4 for all

i)

for

to G;: A p ~

JJ

and all

d G;:f\ll ~

°

with

d n"

CD

,

then

v y 0 ,d]) ~ s , ~(.Q,t ,d) - > f(Q,W[t

is

surjective il)

for

p

< v ~ s , Ker(~(.Q,tO,dl - > r(n,W[tO,d]))

C Ker(F{ (Q,tO ,d) - > ~+)I-P+3 (Q,tO ,dl)

for any open

-

.4 0 0 -

Y -T . Siu

polydisc Q C A d

n

=I

CD

•

and for all

t

o G;:

A and all

-

401 -

Y -T . Siu

§10.

Proof of Coherence

(10.1)

Suppose

is a sequence of complexes of trivial

~

Banach bundles ~s in §7. s ~ p + n.

Fix two integers

~

Assume that the complexes

(M))l+l , (F)'J

for

p

< ." = < s. =

a)

and for 8ny

rm({(Q,tO,d)

,6.

Assume

n

91[ to ,d] with

d

n

w~thout

for ~

1.

Ker

-> (

)l+ 1 H~

(S1,t-0 ,d) )

°

fh)l+ 1 (Q,t

P:i-'" <

s.

.d )

->

The case

p

~ v<

i)

s , a,

n n

° )

')1+ 1 (Q,t ,d) Ha,+l

that

=

~

° is

~

<

•

trivial.

d

~N~

Since coherence is a local property, by (9·2)

loss of generality we can assume the following for

d n ~ co •

with For

p

~,

is coher-

s , to (; b.., and

any open polydisc Q C.6 and for all d (; N~

~

The induction hypothesis states that

is coherent for ~ m.

~ )I

-> ~+l (Q,tO ,d))

We are going to prove by induction on ent on

p

pen polydisc .Q. C ,6. •

rm (~(Q,tO ,d)

°

C

(Et,

satisfy

-> ~+l(n.,tO,d))

Ker ( Ib.)l+1 (S1.,t ,d)

b)

0

such that

We use t h e notations of '§ 9 .

We assume the following two conditions for to (; ,6. , d (; N ~

p, s

s»<

surjective.

°

v s , Ha,(Sl,t ,d)

to (; ,6. and all

v -> r(S1., '9Ht

°

,d]j

is

_

~0 2

_

Y - T. Si u

ii)

For

p < y < s , Ker(~(Q,tO .d )

C Ker ( l\t~ ( Sl,t0 ,dl ->

-> r(Q:~/[tO,d]l)

~ (0) 'b.+)I-p+3 .Q.,t .d )

As in (7.3) we use the following notation.

Fbr

•

m ~ 1 , let

Suppose p ~ ~ < s and t n is not a zerom )1+1 'n divisor for t n '1+0 • If f < P in lR+ and p is sufficiently strictly small, then for to ~A(f) and A~~O (10.2)

Lemma.

r ( 0 A+ m) 1m l\t~ ( .6.( f , ),t,d

-> Het.;+ l (A(P),t0 ,d A)~)

is contained in

Proof.

Consider the following commutative diagram

~+l(~(P))

I

H: (.a.( p' »

which comes from the commutative diagram

-

403-

Y -T . Siu

°-~(Q(~+l) ...2....>(0(~+1) -~(!)(~+l)(tO,d>') -~ ° °-~

b AI

r~+l ,0.1r,..

(Q(()

2...> (D(~)

11\1

-~ (0(~) (to ,d>.+m) -~

°

where i)

I-

is the natural map

ii)

a

is defined by multiplication by

iii)

b

is defined by multiplication by

iv)

c

is defined by multiplication by

°>.+m

(t n - t n )

m (t n - to) n x (t n _ to) n

Let

~L\!+l

1 :

l'f"0

_

y

be defined by multiplication by

°

..... ::l/.)1+l >'f'

°

m• (t n - t) n

Let

.

Y v+l By (Bl +I (applied to A = 0), for r 0.+ I ,0. n f sufficiently strictly small, Ker J C Ker g • Since t n m ,#V+ I , i t follows that is not a zero-divisor of tn

be induced by

°

Ker

(* )

for

t~ =

°

When

t~ =

'f

C

Ker 1)'

° ,both

~ and 1jJ are isomor-

phisms and, hence, (*) trivially holds.

-

404 _

Y -T . Siu

One has

===> because

h = gf.

~

a

gf't

1.

a

hr

~

t

"te = a and

It follows that

The following lemma is in codimension

a

,

a Im

eC

Imo- •

strengthened form of (10.2)

Its proof is similar to that of (10.2).

Its consequence (10.5) will be needed only for the proof of the coherence of ~ p •

(10.3)

Lemma.

Suppose

s ~ p+2 , 1 ~ P,

d,t+l' ... , d n <;: ~,nCC.6. to ~ .Q. Fbr A <;:N* let

eA =

<

n ,

is an open polydisc, and

••• , d ) n

Then there exists m <;: No such that, for A <;: ~, Im (~(Q,tO ,e A+m) -> ~+3 (Q,tO ,e A)) is contained in

Im (~+3 (n,tO ,em) Proof.

-> ~+3 (Q,tO ,eA.))

Since ¥P+l[tO,em]

is coherent on

D., by consider-

ing the increasing sequence of subsheaves consisting of the kernels of the sheaf-homomorphisms 9:J-p+l[ to ,em] _> 'H P+ l [ to ,em] (t - t~)m as m varies, we 1 conclude that there exists m ~ ~O such that t - tJ is 1 not a zero-divisor of (t.f - t~.)m W- p + l [ to ,em]x for x <;

defined by multiplication by

n.

-

4.05 -

Y - T . Siu

Consider the following commutative diagram

°

p HG:+3(Sl, t

1

Clll o: p ,e ) -;> H~+3(.n, t

B

°~

°

"'l' p+l ,e ) ~ Ha.+3(Q, t

T

i

f

Cll ,e )

h

H~(Q,tO,eCO) ~ H~(n,tO,e~+m) ~ Hrlm.,tO,e

ClO

)

~

Hfl(Q,t O, ella .

which comes from the commutative diagram

.

°

o ~(O(E:u.+J)(t,eClD) -C4 (9(E.u+

I

I:-

b

3)

(0 t ,eClll) h.

r~+J,~ I

O~ (9(E~)(tO,eClD) ~(9(E~)(tO,e~+m)---7'O where i)

~

is the natural map

ii)

a

is defined by multiplication by

( t£

iii)

b

is defined by multiplication by

(tt

is defined by multiplication by

(t _ to) 1 'i

iV)

c

-

to) 1.

to) 1

Let

be defined by multiplication by

a

(t.( - tl, )

m

•

Let

1I,+m m

x

- 406 _

Y-T. Si u

g.•

fb:p+1 (,..,a,t p+l

be induced by

r a + 3,a

the choice of

m , Ker

°,elI) , -> J

p+1 (n,t U,ern )

~+3

.

By (10.1) ii) , Ker

f

C Ker'lf

.

J

C Ker g •

By

The conclusion follows

from repeating verbatim the last paragraph of the proof of (10.2) •

(10.4)

Lemma.

d.e+ l' ... , d

n

x <;: N*

Suppose

s" p+2 , 0 ;;;;:

<;: N , .Q C C A

1
is an open polydisc, and

to <;:Q.

lI), ~,dl+l' ••• , d n) <;: ~~ • Then L ' ~( for (f ••• , f.l) <;: N there exists (gl' ••• , g.t.) <;: l't l, such that, i f t, <;: ~ (Q, to and the image of C. in For

let e"'· (lI), •• q

.e,

9tp [ t O, elI) ]

t

°

belongs to

the image of ~ l ~ (t i

i"l

in

trivial.

(t

i

- t9)giS¥P[tO,elI)] ~

~+3.e (5"2, to ,elI))

t

°'

then

belongs to

°

P '0 )fi ~3R (,.., lI) ~"t ,e )

- ti

Proof.

f

1=1

We prove by induction on

1. •

By (10.3) there exists

The case

m <;:N O

.£ =

° is

such that

is contained in

Let

gi." f

t

+ m.

(gl' ••• , gl-1)

By induction hypothesis there exists

<;:~.t-l such that,

i f the image of

~

in

_ 407 -

9IlP[ to, e gl]

t

°

then the image

Y -T. Siu

belongs to

~

of

~

in

°

P ~+3 <.e-l) (Q,t ,e gl )

~

Observe that, if the image of

",p

then the image of l,

in ~p[tO,eOO] t

° gJ. ] ° belongs

in ~[t ,e

t

belongs to

° belongs

to

to

Hence

-1,=

in

of some

.

E (t. _

i=l

1.

f

t?) i ~. 1.

1.

-l,i ~ f\i+ 3 C€ -1) (n, t °,e gl ) .

for some l,i

.l-l

°

P ~+3l(Q,t ,e l,i ~ f\i+

the amage

C;

*

3t

~

)

equals the image in

(Q,tO,eOOj

s of <-,

in

By (10·3) the image of

(1 ~ i

P (t"'l ~+3i oJ"

<.£J.

°

00 t ,e)

° ~)

P ~+3£(Q,t ,e

It follows that

satisfies

-

40 8 -

Y-T.Siu ~

*

....

(l0.5)

1-1 a f. _ E (t . _ t.) l~. i=l

1

Lemma.

1

~

1

s ~ p+2 , then for

If

strictly small the following holds. f~~n

g~Nn

there exists

and the image of

t, 'in ¥p0 t

then the image of E,

in

p ~IR~

tO~t:.(fJ

Fbr

such that, i f

and

!, ~ H~(A(f))

nag· t.) l'9lo' i=l 1 1 t

belongs to

~+3n_2(L:!.(p))

sufficiently

E (t. -

belongs to

n a f. E (t. - t]..) 1~+3n_2(~(f))

i-I

1

Proof.

Since ~g+l

exists

m~No

t~ ,.,g+l.

is finitely generated over

such that

tn

nOO ' there

is not a zero-divisor for

gn" f n + m. By (10.4) (applied to £ - n - 1), there exists (gl' ••• , gn-l) ~ ~n-l such that i f Let

the image of E,

then the image

in

e;

~p[ to ,d gn] a t

of

E,

in

belongs to

p ( ~+3 (n-l) A(

f) ,ta,d gn)

to n-l a fi p ( a gn) E (t. - til ~+3 (n-lJ A(p) ,t .d • i=l ]. Observe that, if the image of ~

belongs to

belongs

- 409 -

Y - T . Si u

then the image of

~

WP[ to ,d

in

gn] t

0

belongs to

Hence,

-

p

0

(

for some c;i ~ I-\i+3 (n-l) 06((') ,t ,d

gn)

By (10.2), for

•

sufficiently strictly small, the image of f

~+3n_2(A(P),tO,d n)

~i

p

in

equals the image in

f

~+3n_2(o6(f),tO,d n)

of some

~i ~ ~+3n_2(.6(p))

t,* of l,

follows that the image

•

It

~+3n_2(A(('))

in

satisfies ~

*

n-l i=l

0 f.

_ E (t . _ t. ) 1

1

1~.

1

~

. coherent on 9/~ 1S

(10. 6)

Theorem.

Proof.

It suffices to prove that

11

~

>I

for

p

~

p < s •

is coh erent at

o.

We break up the proof into three parts. (Ii

We first show that

9J

~

is of finite type at

o.

-

410 -

Y - T . Si u

e

,

Take for

f

(* )

{

It


< f

"'

in 1R~

and

to <;: A(

l)

By

sufficiently strictly small,

-> H:+ I (A( f' n)

Im (~(A( fit))

generated r(A(

p' J ,

generates a finitely

n(O) -modu Le, (()

~+l

Since 11- 0 is finitely generated over nO" there exists N 'ALlI + 1 m <;:I~O such that t n is not a zero-divisor of t m • n ~O

f

"

Im ( Fb._l(A(

f

By (10.2), for 'J

(# ) {

tained in By (10.1) i),

sufficiently strictly small, rn

) ,t

0

'J

,d

Im ( lb:(A(

m+l)

y

-> F\t(A( P ) .e

"

It

0

1 ~ ,d ))

is con-

01

f )) -> F\t(.6( P ) ,t ,d ) )l

It

we have the surjectivity of

It follows from the coherence of

~

W [to ,dm+l]

and the

Theor em A of Cartan-Oka that

,

;.s surjective.

Since W~o Lemma,

t

From (#) and (t) we conclude that

is finitely gener a t ed over

nVO' by Nakayama's

-4 11-

Y - T . Si u

It follows from (*) that

-, ~lIA( f

(II)

is generated by a fi nite

~i

be induced by

pI).

Let

(l:i- i

< k) •

Next we prove that the relation sheaf

16

), ••• ,

We distinguish

-,

f ) is coherent at, 0.

';kIA(

v>

between the Case in IR~

,

~l' ••• , ~ k ~ ~ (

number of elements

~i ~r(A(f''')~W")

v

W IA( p )

p

Y· p.

and the case

The case

~

to ~A(P )

for some

<

p.

; that is

k

n~

t

°

for the reduced subspace

~ r(4("" ),

n
Let

"M

t

°

as well as the ideal sheaf

By taking the

{to]. a

at

i

th

A

to , we obtain

such that (1

:i-

i ~ k) •

It follows that

Since

,

°

Ea ·(';·)O=O. i=l ~ ~ t

partial sum of the power series of bi

f
Suppose

denote the maximal ideal of (n

Take

•

~ Rt

on

of

9/-"[ to ,d A+ m]

Cartan-Oka

is coherent on A , by the Theorem B of

- 41 2 -

Y - T . Si u

equals

k ~ b .~.

Hence the image of

i:&l

in

~ ~

I t ), 9+-oJ[t 0 ,d ;\+ m]) r (A(('

belongs to ;\ ( " ),9Ht II 0 ,d ;\+m) r(~(f" )'IWvO)I~(P ] • t

By

(10.1) i) and ii), the image of

(It

oJ

0

~+~-p+3 A(f ),t ,d

;\+m)

(It

II

~+II-p+3 .A(p ), t

0

,d

l

A+m)

(A(('")

(lOA)

)l

~ ~

in

,to ,d A+m)

By (10.2), for

ciently strictly small, the image of ~+)I-P+3 A(p ),t ,d

i:al

belongs to the image of

r(A(p"), 1N>,A ) O t in

k ~ b.~ .

k

~ b.~ .

i=l

~ ~

I

p

suffi-

in

belongs to the image of

r(A( p' j, """AO) ~+l(~( pi)) t

in

II

~+)I_P+3(A( f

k

~ b. t,. i"'l ~ ~

By k

y

in

~

), t

,d ) . I

Ib.+1I_P+3(A( f))

( B)n' f or

~ b·S· i"'l ~ ~

'OA

f

It follows that the image of belongs to

s uffici ently strictly s mall , th e image of equa l s th e i mag e of

k

~

i "'l

c .z,

1. 1.

in

-

413 -

Y -T . Si u

(1

:i

:i k) •

i

Since

and

it follows that

1t 0

C fWIIhO f!)kO + nifor(,6(,,), R) • tnt t I

t

Since

h

is arbitrary,

Hence

R

is coherent on ,6(P)

(IIIl

The Case

=

~

p.

The only difference between this

=

p , (10.1) ii)

case and the previous case is that, when

~

no longer implies that, for open polydisc

n. C

and

d G;;:N~

with

d

Ke r ( ~ (Q, to ,d)

(**) { When

f

!Xl

-> C

n = 1 , (**)

can assume that

n

,

r(.Q, W"[ to ,d] l)

"0" 0 Ker(%. (Q,t ,d) -> lb.+11_P+3(Q,t ,d)) .

is clearly satisfied for

n? 2

~, to (;: ~

Since

s?

~n

'Y = p.

~IP+l

, n

So we

is coherent

- 414 _

Y-T . Siu

on b,.

by (II).

Now we modify the argument of (II) to avoid

the use of (**) when

= p.

~

We pick up the argument of

(II) at the point where k

-

"A

~ r(A(f i.

L b.~. i=l J. 1 II

p

By (10.5), for I

A

ii)

I

A

->

A -> co

as

co

k E b .t,.

the image of

I

p

(B)~, for

k

i~lb/)i

in

i=l

1

J.

p

("

H
in

belongs to

•

sufficiently strictly small, the image of

Y

I

k L c . ~.

equals the image of

fh+ 3n-2(A(f))

~+3n-2(~(f'))

p 0W ) •

sufficiently strictly small, there exists

A' P (It r(~(f" )'N,\0)~+3n-2 A(p ))

By

t

A such that

depending on i)

AJoIy

i=l

J. J.

for some

,As in (II), we conclude that

1<Since

,

A

t

0

C

-> co as

trariness of

A'

!IN'v

t

k

0 n(0 0 t

k

+ n(f)

t

(

Or\,~(

pI ), 'n) II\"

•

A -> co , it follows from the arbi-

A that k

I

I!) Or(~(p i,f?)

n t

•

in

-

4 15 _

Y - T . Si u

Hence R is coherent on (10.7)

~(f

) •

Proof of Main Theorem.

So

Fbr every

one can find a proper holomorphic

~ S

map o: with finite fibers from an open neighborhood

So

into an open subset

G of

~ n.

We have

It is easy to see that an analytic sheaf ~

RO~*~ is coherent on

ent if and only if M is an

~S ,s -module for some

codh O

S,s

where

M

codh lD

S,s

U is coher-

on

G. MOreover, if

s ~ U , then

= codh lD

n a-(s)

M;;;- n , then

M

ncD0- ( s ) -modu.Le -

M is regarded naturally as an

particular, if

of

U

M is a flat

module (see (A.$)-(A .12) of the Appendix).

In

nWo-(s )-

Hence for the

proof of the Main Theorem we can assume without loss of generality that

S

=

A

and

'7

is

n-flat.

course of the proof, any replacement of

MO r eover , in the by

~

~ (f )

(with

P~IR~) does not result in any loss of generality. In (6.5) we have constructed a sequence of complexes ~

of trivial Banach bundles on A.

By (5.2 ) and the re-

sults of Andreotti-Grauert [1] these complexes

(El, By

(M)I+I, (Ft , and (l0.1) a), b)

(l0 . 6 ),

W"

is coherent on b. . for

p~ »

for p

~

~

)I

<

satisfy

< r - q - n,

r - q - n_ L

- 41 6 -

Y -T. Siu

By (7.3) and the

bumpin~

techniques of Andreotti-Grauert

1

it is easy to see that

< r-q-n , a* ~ a < a# ' and over for p < ~ < r-q-n , for

p

~ y

b# < b < b*.

MOre-

- 417 _

Y - T . Si u

PART III §ll.

APPLICATIONS

Coherent Sheaf Extension For the definition and properties of gap-sheaves

7 [n]

used here, refer to the Appendix. (11.1)

Theorem (Coherent Sheaf Extension on Ring Domains).

Suppose

~ a < b in JRN ,

0

is an open subset of ([n ,

D

D X GN(a,b)

is a coherent analytic sheaf on

'1.

'][ntl] ..

ytic sheaf Proof.

7'

Then .Q.U

"1

such that

extends uniquely to a coherent anal-

D x AN (b)

The uniqueness of

such that

...

?

'7 [ntl] = 7' .

follows from the extension

theory of sections of gap-sheaves (see (A-le) of the Appendix) •

...,

For the existence of

? , we

consider first the spe-

cial case where i) 11)

iii)

For

D is bounded and Stein codh? S? n + 3 ~

n: m~

is flat with respect to the natural projection

D X GN(a,b) ---> D • ~

sufficiently large, there exist

o<

a

<

a < a" < b

~

<

in

lR

b

in RN

"J

Y - T . Si u

such that , GN(a ' ,b)

For

CC

{z

<;:
~

{(t,z) <;: DxceNlex + £ < £/t/ 2 + Let

,

ex (l

=

" =

P

,

P"

(l

< PI < ex

a <;:

and

+

2"£

(l

+

£

P

- 2"

P

-

Iz .1 2m < p-£} • ~

e

,

P < P2 < P

£

N E

i=l

.

2m 11 z, ~

•

Let

x

{(t ,z) <;: Dxa:Nlex < cy(t,z) < P}

Xi

Xn{

n

~I x

rt

t

,

i =l

(l

£ltj2 +

~(t,z)

Fix

GN(a,b) •

> a sufficiently small, D x GN (a ' ,b , ) is contained

£

Take

CC

i

D •

linear funct i on

(i = 1,2)

n!X i For any f(t, z)

(i =1,2)

.

a a

(t , z ) <;: ~ - Xl ' there exists a such that

f(tO,zO)

=a

and

f(t,z)

in

- 41 9 -

Y - T . Siu

is nowhere zero on

Xl -

Let

fined by multiplication by Xl ' Supp Ker '8

and

e: "} -> '1

f -

Supp Coker

Since

e

f

Xz

on

be de-

is nowhere zero on

are subvarieties of the

Stein space

and hence are Stein-

From the cohomology sequence of the

short exact sequence

e

0-> Kere -> 7-> we conclude that 8 l". ., .

Ime

-> 0 ,

induces an isomorphism

2"7 0 (R1 n._)

t

->

( R1 n. 2 (Im 8) ) 0t

Consider the following exact sequence

1 2"7

(R n• .T) 0

t

coming from the short exact sequence

o ->

Im e

c-> "1 ->

We are going to prove that ~ show that diagram

Ker

7 ..

O.

Coker e

->

is surjective-

0 • It suffices to

Consider the following commutative

-

4~O -

Y -T . Siu

(R

e

9 ,

where

1 2

n;l)

t

0

92

->

e

and g is the restriction 2 are induced by 1 map- Since 82 .. 1 and C, is an isomorphism it suffices to show that Ker 9 2 .. 0 - Since is an isomorphism on Xl ,81

c.

e

is an isomorphism-

By applying the Main Theorem to

the map n: X -> D together with the function

sheaf Hence

7

on

Ker

e2

X, we conclude that and C,

0

=

g

po

and the

is an isomorphism-

is surjective-

Since

Supp CokerE

is Stein, the image of the natural map

(ROn~(Coker e)) generates

(Coker 9)

0

t

0

0

(t ,z )

-> (Coker e) 0 0

(t ,z )

over

n+N({) 0

0

(t ,z )

Since

t,

is surjective, by Nakayama's Lemma, the image of the natural map

•

generates

"1

neighborhood phism

(J

0

0

By letting

(t ,z )

:n+P

f

U

of

->

to

t

on

in

zO

vary, for some open

D we can find a sheaf-epimor-

-

421 -

Y -T. Siu

some open neighborhood

U"

to

in

->

Ker

of

7 ,

Ker c: instead of

By applying the Same argument to

U'

for

we can find a

sheaf-epimorphism

-r : n+~

0-

on

< 1(t,z) < P" } •

{(t,z) ~ U" xa: Nj" a

or extends to a sheaf-homomorphism

By Hartogs' Theorem,

T: on

f (t,z) G

U• x

a: N

..

(

I

l'(t , z) < {3'}.

Then

Coker~ /

)[n+l]

/O[n+Z]Cokerl:

, '1 I U x GN (a' ,b)

extends

n+~q -> n+Nif

fl

"J I" U)C

and, hence, extends

(see (A.IS) of the Appendix).

GN (a, b ' )

By the arbitrariness of

to

and the uniqueness of extension, the special Case follows. Fbr the general case we use induction on be the set of points of

D x GN(a,b)

where

n.

Let

S

codh"J:i- n + Z •

Let TI Let

TZ n-flat.

..

{(t,z)

~

be the set of points of Take

a < a"

?

sjdim(t,Z)s n({t} x GN(a,b)) N

DXG (a,b)

where

I} •

'1

is not

< b < b in IR N • Let

By applying the special Case to

'1 I U X GN (a ' ,b , )

for bounded

- 422' -

Y-T , Siu

Stein open subsets

"11 (D-A) X aN(a,b) sheaf

'1

on

U of

D-A, we conclude that

can be extended to a coherent analytic

(D-A) x AN(b)

satisfying

7[n+l]

= :;.

Since

dim S :f n , rank ~l Tl U T2 < n (cf (A. 1)) of the Appendix). Since A = f! when n = 0 , the case n = 0 is proved. Take arbitrarily

t o ~ A.

After a coordinates transformation, we

and '( > 0

to = 0 and there exist 0 < a < ~ in ~ n in IR - l such that Lln- l (1') x al(a,~) is dis-

joint from

A.

can assume that

By induction hypothesis, the sheaf on

which agrees with

and agrees with

?

1\

7

on

on

can be extended to a coherent analytic sheaf 7* on Lln- l (,t) x Al (~ ) x d (b) satisfying (,:1*) [n+ 1] = 1*. The general case now follows from the arbitrariness of uniqueness of extension.

to

and the

-

.4 2 .3 -

Y -T. Siu

§12 • (12.1)

Blow-downs A holomarphic map n: X ---> S is said to be strongly

I-pseudo convex ~:

cf

if there exist a

X ---> (~,c*) C (~,ooi

function

and a real number

< c*

c

such

that i) ii)

n/[f:i- c}

'f is strongly l-pseudoconvex on

(When the additional condition c < c*

c < c*

is proper for

[SO:i- c}

[Cf> c#} •

= [1' < c}-

for

is added, this definition agrees with a special case

of strongly Fbr

(p,q)-pseudoconvex-pseudoconcave maps.)

:r <;:

r(X,lOx)

image of the germ of

f

x <;: X let

and at

x

:rex)

denote the

under the natural map

a: .

Ox ,x -> <9x ,x/"~X ,x

We are going to prove the following result concerning blowing down.

If'

n: X ---> S is strongly

is Stein, then

X

S

is holomorphically convex (that is, for

every discrete sequence f <;: r(X,(!)X)

I-pseudo convex and

such that

[x~}

f(x~)

in

->

have the holornorphic convexity of

X there exists 00

as

)l

->

00).

Once we

X, we can blow down X

by the Reduction Theorem of Remmert (whose generalization to the unreduced case can be proved in a way analogous to the reduced case [30J).

-

424 -

Y -T . Siu

The result on blowing down will be proved by using the finite generation of

(Rln;Oxl s

for

c# < c < c*.

s <;: Sand

For

such a finite generation, it suffices to consider the case S = A and

where

s '" o.

Strictly speaking this finite

generation does not follow from the Main Theorem, because in general is not

n-flat.

However, this can be obtained

from the argument used in proving the Main Theorem. proof of the Main Theorem, the

~

sequence of complexes for

p

~ ~

< r-q-n.

Ox

is needed to get a

~latness

satisfying

In the

(E{, (Ml~+l, (F)

Such a sequence of complexes is needed

for getting right inverses of coboundary maps (§8) and global isomorphisms (99) which, in turn, are essential for proving the coherence of the direct image sheaves under consideration. However, when only the finite generation of the stalks of the direct image sheaves under consideration is needed, it can be proved directly by the arguments of §7 without using the sequence of complexes, 'provided that n-flat on

{~

< a#}

needed and, when eration of

.

i

is

Of course, some modifications are

» '" p , one can only obtain the finite gen-

(R)l (n~i/J)o

As before, we replace

a* < a < a# by the graph of

for X

finite generation by induction on

n.

and

b# < b < b* •

n •

We prove the

By replacing

X by

X~ with a* < a < a# and b# < b < b* ' we can assume without loss of generality that there exists tn

m <;:N O such that

is not a zero-divisor for any stalk of

ing the induction hypothesis to exact sequence

? It~ "]

t~t.

BYapply-

and considering the

-

42 5 _

Y - T . Siu

we Can without loss of generality replace assume that

m = O.

7

by

Use the notations of (6.5).

get the finite generation, one need only replace

t~? and

Now, to ~(f)

by

and use the parenthetical statement at the end of (5.2). note that the case of the strongly spends to the case where no flatness of ~X

p = 1

We

l-pseudoconvex map corre-

and

{ 1 < 8fI} = RJ.

Hence

is needed for the finite generation of

(Rln;COx}s •

(12.2)

Lemma.

If

n: X ---> S is a strongly

holomorphic map and

S is a single point, then

l-pseudoconvex X is holo-

morphically convex. Proof. in

Take

{T > c}.

{x~}.

c# < c < c* Let.J

and take a discrete sequence

be the ideal-sheaf of the subvariety

The exact sequence

yields the commutative diagram with exact rows r(X,
{x~}

7 rtx, ~5) -> Hl(X,5) -> HI (X,

0-1

~

HI (Xc ,9)

1

---> Hl(X c ,COX} 'l:

- 426 -

Y -T . Siu

S

Since

XC ,

= CD

X on

1:

is an isomorphism.

By the results

of Andreotti-Grauert [1], o: is an isomorphism. that

?

f'{x)/ )

->

is surjective. (J)

f~

r(X,OX)

such that

•

Suppose

Lemma·

(12·3 )

There exists

It follows

convex

map-

k~N

such that

Then for

n: X

-> S is a strongly I-pseudo-

s c;: S and

c# < c < c*

there exists

(ROn;~)s -> (ROn; (lOX/.w.s,s(OX)) s has the same image as

Proof. U of

Use induction on s

in

f(s) ,. 0 (R

l

and

n;1Js

m ~NO

ii)

dimsS.

S, there exists fs

For some open neighborhood

f c;: ,(U,lOS)

such that

is not a zero-divisor of ~S,s

is finitely generated over

Since

~S,s ' there exists

such that ·

fx

(when

f

r(n-l(u),lOx»

is naturally regarded as an element of is not a zero-divisor of

xc;:xcnn-l(U) • The commutative diagram

~~ for

- 427 -

Y - T . Siu

o -> ?m+l(9 _> 19. _> 19 /?m+lCQ -> 0 X X X X

II

1

-> (Ox ->

CDx/f~

1 o ->

f~

-> a

yields the following commutative diagram

We are going to show that

ba

~

O.

Consider the following

commutative diagram

where i)

~

and

ii)

e

is induced by the inclusion map ~'9x C->

P are defined by multiplication by

~l

tOx

- 428 _

Y -T. Siu

iii} iv} By

0-

is defined by multiplication by

~

~

is defined by multiplication by

f.

the choice of

m ,

~

is an isomorphism and

Ker a C KerO-.

It follows that b(Ker c}

C b (Ker 'ta-e~ -l) Hence

ba = 0 • By

It

follows that

o.

b (Ker b) 1m E, = 1m 7 .

induction hypothesis, there exists

kEN such that

and

have the same image.

(12.4 )

Lemma.

Then

Suppose

k

n: x - > S is a strongly

convex holomorphic map. s <;;: S of

n-l(s}

K in

.

yhoose

exists

k <;;:N

,

and

K

I-pseudo-

is a compact subset

Then there exists an open neighborhood

X such that

Proof.

satisfies the requirement.

(G,VxIG)

c# < c < c*

G of

is holomorphically convex. such that

K C Xc.

There

satisfying the condition of (12.3).

(n-l(s)'~x/~~,s~XJ is holomorphically convex·

By

(12.2).

There exist

-

429 -

Y - T . Siu

and an that

0

pen neighborhood

K in

n-1( s) tI XC

such

K is contained in

By the choice of in

U of

k , for some open neighborhood

D of

s

S, there exist

such that

fi

and

gi

have the Same image in

r(n-l(s),~x/~s,s~x) • Let W be an open neighborhood of K in

XC

such that

open neighborhood o'f

rt

~ () n -1 (s) CU. Q of

s

in

There exists a Stein

D such that the restriction

to

is a proper map into

Q.

It follows that

is a holomorphically convex neighborhood of (12.5)

Lemma.

Suppose

n:

K in

X. Q.E.D.

X-> S is a strongly I-pseudo-

convex holomorphic map, S is Stein, and

c# < c < c*.

X ;s Stein if and only if for every compact subset

Then

K of XC

there exists a strongly plurisubharmonic function on an open neighborhood of

K.

-

43lJ

-

Y-T . Siu

E!:2.2.f. Only the "if" part requires a proof.

S is the coun-

table union of relatively compact Stein open subsets that

Sk CC Sk+l

has dense image. for each

k.

sucl

and the restriction map

It suffices to show that

Yk

Let

tion function on

Sk

Sk.

is Stein

n-l(Sk)

be a strongly plurisubharmonic Take

c# < e < b < a < c*.

exhau~

By

assumption there exists a strongly plurisubharmonic function

e

n -l( Sk+l)

n

Xa •

..2 Choose a nonnegative C" function a ~ on X whose support is contained in X and which is identically 1 on Xb • There exists a if function ~ on

on

(_ro,c*)

such that Supp o: C (c#, c* )

i)

the first and second derivatives of

ii)

are iii)

0-(;\)

(J

are

~ 0

and

> 0 on (e,c*) __> ro as ;\ -> c* froa the left.

For some positive number

A, the function

is a strongly plurisubharmonic exhaustion function on n -I( Sk).

(12.6)

Lemma.

Q. E. D•

Suppose

D is an open subset of ([n ,

[V J.. J.J. { I i s a locally finite open covering of D and ~

L.

J.

-

4 31

-

Y -T. Siu

{i

~ II

in

Ui

rf

is a

nonnegative funct ion on

such that all first-order derivatives of

on the zero-set of

l:: "to i ~ I ~

>

0

an open subset of

a: N

DxG

Then there exists

Al ~

x G.

U

i

G-->

f: D x

and let

i ~ I

Let

Suppose

be

G is

D

be a compact subset

K

be the natural projection.

1\: lR -->

lR and a function

~ l::

Let 1)1

(Ii (1 ~ I) is a strongly pluri-

and

subharmonic function on

van ish

~i

D•

on

D.

a strongly plurisubharmonic function on

of

D with support

("t • •

~

fie

IR

~

Mr. ~ + B{Y· f)

K

is strongly plurisubharmonic on some open neighborhood of

Proof.

For a

rf

x ~

Q and

and for

function

on an open subset .Q

h

a ~~m ,let m l::

(

L(h;x,a)

denote

m

l::

-

ah

i=ldz ~. Let in

"" -r.~ '"

<e n+ N

a ~ S

'1:'.

~

•

D

f

denote

~J

J

~

a(h;x,a)

a: m

a2h) (x)a.a_ .

i,j=ldZ .dZ'. and let

of

and

Y''" '" 'Y'

D

(x/a

i

.

s

and let

f

be the unit sphere

It suffices to show that for fixed

there exist

,

A (x s a ] ~

IR

and

,

x ~ K

B (. ,x,a) ~

lR

and

such tha~

- 4 32 -

if

,

A? A (x,a)

and

Y-T . Siu

,

B? B. (A,x,a) , then

Direct computation shows that

].

+

The first bracketed term is first

n

components of

ed term is

o.

a

> 0 when A > 0

are all zero, the second bracket-

When the first

all zero, there exists

n

components of

a

are not

B* (.) ~ lR such that the second

> 0 when B? B*(Al.

bracketed term is

When the

The third bracket-

ed term is at least as great as

where the only nonzero terms are those with O(
J

to.

Therefore there exists

> 0 when A? A*

Q.E.D.

~i(X)

t

0

and

A* ~ lR · such that

- 43 3 -

Y - T . Siu

Lemma.

(12.7J

Suppose

n: X--> Sis a strongly

convex holomorphic map and

S is Stein.

If every point of

n-l(U)

S has an open neighborhood such that

l-oseudo·

is Stein, then

X is Stein. Proof.

Let

K be an arbitrary compact subset of

By

X.

(12.5), it suffices to prove the existence of a strongly plurisubharmonic function on some open neighborhood of embedding an open neighborhood of

n(K}

~n , we can assume that

Cover

S

a

(n.

By

K.

as a subspace of

n(K)

number of open balls

Bi such that n-l(Bi) t ron ' coo f C hoose a nonnega ti ve unc

by a finite is Stein.

For

Bi i on with compact support such that the first-order derivatives of each

'C

i

i

,

vanish whenever

Since every point of

'C

i

vanishes and

'C

ET. > 0 i

~

on

K has an open neighborhood

n(K) • D which

can be embedded into (n+N whose equals

composite

nlD,

by a (nonproper) holomorphic map with the natural projection (n+N -->
by using the fact that avery strongly plurisub-

harmonic function on a subspace can be extended locally to a strongly plurisubharmonic function on the ambient space, we conclude from (12.6) that, if

)V

(respectively

strongly plurisubharmonic function on n- l( Bi)), then

S

~i)

is a

(respectively

is strongly plurisubharmonic on an open neighborhood of for some

A,B> 0 •

K

_ 4 34 _

Y - T . Siu

(12.8)

Theorem.

If

n: X ---> S

convex holomorphic map and

S is

is a strongly

I-pseudo-

X is holo-

Stein, then

morphically convex.

By (12.4), every point

Proof.

neighborhood

s

S

admits an open

U such that

is holomorphically convex for some

7:

of

cH < c < c*.

G ---> R be the Remmert quotient of i)

Let

G (that is,

R is Stein

?

ii) iii)

is a proper surjective holomorphic map

R07*
iv )

?- l (x )

is connected for every

x

I~

R) •

Since no compact irreducible positive-dimensional subvariety of

G can intersect

{~>

cHl

by virtue of the maximum

principle for strongly plurisubharmonic functions, Gn {

rr>

piece

cH 1 biholomorphically onto its image.

n-l(U)

n {CJl>

cHl

and

7

R through

?

mapS

Hence we can and obtain a

'" R. n induces a strongly l-pseudoconvex ,.., holomprphic map R ---> U. Since there exists a strongly c~mplex

space

plurisubharmonic function on

the uniqueness of Remmert quotients, as piece together all the Remmert quotients form a complex space

,

,

convex map n : X

,

X

---> S.

R is

R, by (12.4),

n

s

Stein.

By

varies, we can R together and

induces a strongly Since, for every

s

~

l-pseudoS ,

-

435

-

Y - T . Siu

'1

(n }- (U)

=

N

,

R is Stein, by (12.7)

X

there is a pro per holomorphic map afollows that

(12.9)

X.

v

Then for

~

)I

?

S

Y

->

HY(n-l(U),"j}

r(U,R,)n*"J)

any Stein open subset

,

Proof.

,

Letn, X

,~

U of

J.t

~ 0 ,

S.

be as in the proof of (12.8). RJ.to;

"J

theo~em

is coherent on

is an isomorphism for any Stein open subset Since

maps

(j"

y

~

c# < c < c*

is an isomorphism for

a- is proper, by Grauert's direct image per case), for

,it

is a coherent analytic

R)l n* 1--> R n; '] is an isomorphicm for

iii}

,

X

1

R n* "} is coherent on

ii)

for

X to

n ; X --> S is a stronglY

Suppose

I-pseudoconvex holomorphic map and

i)

from

Since

X is holomorphically convex.

Theorem.

sheaf on

is Stein.

Since

(for the pro-

X'

and

W of

X

,

{ f > c# J biholomorphically onto its image,

1

By applying Grauert's direct image theorem (for the proper

case} to

I

lI

n' Supp R 0-* 1, it follows that, for

RO(n'}*(Rv~*1)

is coherent and

»~

1 ,

-

436 -

Y-T . Siu

for every Stein open subset

U of

S.

It follows that

-

437 -

Y- T. Siu

§13.

Relative Exceptional Sets

(13·1)

Suppose

variety

A of

above

n: X ---> S is a holomorphic map.

A sub-

is said to be proper nowhere discrete

X

n/A is proper and every fiber of n/A is pos-

S if

itive-dimensional at any of its points.

A subvariety

X which is proper nowhere discrete above exceptional relative to

S

A. of

S is said to be

if there exists a commutative

diagram of holomorphic maps

i

-> Y

X

s such that i)

~

ii)

is proper

every fiber of ~1!(A)

iii)

~

maps

o .

iv)

R ~*(DX

has dimension

X-A biholomorphically onto

~ 0

y- (A)

= COy •

The following result on relative exceptional sets is a consequence of (12.8). (13.2) and

Theorem.

Suppose

A is a subvariety of

crete above

S.

Then

and only if for every

n: X --> S is a holomorphic map X which is proper nowhere dis-

A is exceptional relative to s

~

S if

S there exist an open neighborhood

- 438 -

Y -T . Siu

U of

s

n-l(U) and

and an open neighborhood such that

Atln-l(U)

nlW:

w--->

W of

Atln-l(U)

U is strongly

I-pseudoconvex

is maximum among all subvarieties of

which are proper nowhere discrete above

U.

in

W

-

1 $9 -

Y -T. Siu

§14.

Projectivity Criterion

(14.1)

Suppose

n: X --> S is a proper holomorphic map and

p: V --> X is a holomorphic vector bundle. be weakly negative relative to exist an open neighborhood hood

U of

D --> U is strongly Let

Jk/Jk+" 1

as an

Ox-sheaf.

Suppose

herent analytic sheaf on

such that

N

J

V and

with

be the

and consider

X

It is easy to see that, where equals (Q((L*)k)

where

V

L* is

p: V ---> X is a weakly negative X and

kO ~NO '}? 1 and

S.

such that

is zero on

k? kO

K for

The case where

If

"1

is a co-

K is a compact subset of

S , then there exists

R~n*(7@~jk/jk+lJ

S is a single point was proved by

The proof of the general case is completely

analogous to that of the special case. coherence of

S .t here

L.

Theorem.

Grauert [7].

~

I-pseudoconvex.

holomorphic vector bundle relative to

Proof.

s

and an open neighbor-

Vln-l(U)

We identify

N •

is a line bundle, Jk/Jk+l the dual of

s

N be the zero-section of

ideal-sheaf of

(14.2)

S if for every

D of the zero-section of

no plD:

V is said to

R')1 [rt

0

pi DJ* [p*'])

It fol lows fro m the

and the fact that

- HO _

Y-T. Siu

is a subsheaf of and

U,D

(14.3)

*

R~ (n. p I D)*(p "]) ,where

are as in (14.1).

Theorem.

phic line bundle

)I

~

r~ 1, k \:.1'0'

Q.E.D.

If there exists a weakly negative holomor-

p: L

--->

X relative to

S

and

S

is

Stein, then for every relatively compact open subset T of S there exists a holomorphic embedding o: of n- l (T) into PNx T

tion

--

Proof.

such that the composite of

PN x T ---> Let

T

equals

Let 52

Stein open neighborhood of points

x,y

of

°->/Wo2X ,x'OIl.

101k

X.

~

in

be a relatively compact S.

Take two distinct

Consider the two exact sequences

->W.X .xfl)£k ->

= I#..X ,x {\ I\MX,y •

where """'"X ,x,y

and the natural projec-

n.

-

l. = (0 (L* J .

~

By

(ANY

/2

X ,x AMX ,x )~.c

(14.2), for

k

k

Rln*('MI ,x,yfl}
ROn*£k

->

ROn*

ROn*(w.,x,x~Lk) -> are both surjective on Q.

(((Ox/"MX'X,y)~

°,

sufficient-

ly large, both ish on Q.

->

Van-

£,k)

ROn* ((........x,x/......i,x)@.,(,k) Since

Q

is Stein, it follows

Y -T. Si u

that there exist enough holomorphic sections of n-1(Q)

to construc t an embedding ~ .

Q. E. D.

L* over

-

4 42

-

Y - T . Siu

§15.

Extension of Complex Spaces

(15.1 )

Theorem. (O[n+l] = ~X and X

S is a Stein space of dimension

~

n •

n: X ---> S is a holomorphic map and

Suppose

X ---> (a*,b*) C

~:

X is a complex space with

Suppose

function such that.

is a strongly olurisubharmonic

(~ro,m)

nl la

~ ~ ~ b}

is proper for

X Then there exist uniquely a Stein space ""

a* < a < b < b*.

and a holomorphic map

n: X--->

S such that

i) (9~n+l] .. (9X ii)

branch of iii}

iv)

in

the restriction of

~~l]

=

n

to

"" X- l!f>a}

First we prove the case

is proper for

n = O.

It follows

Ox that, outside a subvariety of dimension

X, codh (Ox ~ 3.

such that let

X

intersects every

n = ~Ix

Proof (sketch). from

X which

X is an open subset of

codh (Ox ~ 3

X~ .. Ic < f < d}.

ideal-sheaf on

We can choose on Let

I
j

~0

a* < a < b < b < b* For

a* ~ c < d ~ b*

be an arbitrary coherent

X such that ~ • ~x

the following commutative diagram

,

on

X~'.

Consider

-

443 -

Y -T Siu

Hl(X~,,1) ->

a-I

VI ;;

~(X~' ,,1) -> coming from

o ->S ->

Ox ->

lDxlJ ->

0 •

cr is an isomorphism. b' , 1: is an isomorphism. It follows = CD X on Xa , is an epimorphism. From the arbitrariness of

By the results of Andreotti-Grauert [1],

Since

J

that

e

S

we conclude that there exist a holomorphic map

such that, for some

a < a i)

,

< b f

,

ii) iii)

X:'

c

V of

in IR N and some

biholomorphically onto a sub-

GN(a,Pl

f-l(~N(a})

x~, C f-l(G N(p ,n)

By (11.1), V 6.N(P )


Cl(GN(a,p))

maps

space

0

.

can be extended to a complex subspace

(9~1] =([)r;.

satisfying

By gluinp;

V by means of f , we can piece together N b C l (G (a, l')) U X to form a complex space

*

X

bl

is Stein.

X

satisfies

(0~xl] = (!)i

~

V of

f-l(GN(a,fl)) ,..., V and

X.

and

By (12·5),

and extends

b X

*

bl

By

-

44 4 -

Y -T . Siu

the property of gap-sheaves (cf. [A.16] of the Appendix), any complex space satisfying these three conditions is isomorphic to a

X and

< b" <

the ·i somor phi s m is unique.

b,

(instead of

X).

,By

Choose arbitrarily

repeating the preceding argument with

b ), we obtain a complex space extends Xb* , in particular

X

Since

bIt

'" X.

and is therefore uniquely isomorphic to trariness of of

b" ,we conclude that

A

b"

X

(instead of

X

extends

b X * b'

From the arbi-

X is an open subset

'"X. n >

Now, consider the case

o.

The uniqueness of

X

is a consequence of the properties of gap-sheaves (cf. [A.16] of the Appendix).

Because of (12.7), it suffices to prove

the existence of ly representing

X locally (with respect to

S).

By local-

S as an analytic cover, we can assume withS = A.

out loss of generality that special Case where

Ox

the finite generation of

is

First consider the

By

n-flat and codh ~X ~ n+) 1 b 11'1 (R (na)*~X)s

for

a* < a < b < b*

s ~ S , we use the methods of §12 and of the case

and

to get the local existence of

n

=0

""

X.

Fbr the general case, as in the case of sheaf extension, we use a ring domain to avoid the bad set and appeal to induction on Rand Y ~ i)

on

n. MOre precisely, we choose 0 < 1 and a* < a < b < b*, such that

IRt

n-I(An- I("() x Gl(~,~))

codh lOX ~ n+)

n X~

~X

is

~

<

~

n-flat and

in

-

·445 -

Y-T. Siu ii)

the extension

n: X-> An-l(n x Gl(
n-l(An-l('t) x Gl(
satisfies

codh

of

lOX ~

n+2 •

Now we apply the induction hypothesis to the holomorphic map

which is induced by the composites of the natural projection cp-l(r) x d (~) -> A n- l (r) with n and with ~ •

- 446 -

Y -T. Siu

APPENDIX This appendix contains materials on homological

c ~di­

mension, flatness, and gap-sheaves which are used in these lecture notes.

For more details, see [28].

(A.l)

M is a finitely generated module over a

Suppose

local ring f

... , f k

~

I\Nv

.E fiM for

1

l,

/~';'l

Pi

~"l

Same length. R

An

(R,~)

M-seguence is a sequence

such that

< = J. ;i. k

.

fj

is not a zero-divisor of

All maximal

M-sequences have the

Define the homological codimension of

(denoted by

codhRM or simply by

length of all maximal regarded as over

S

codh Ml as the common

M-sequences.

morphism of local rings, then When

Mover

If

S ---> R is an epi-

cOdhSM = codhRM when

M is

R is regular of dimens ion

codh M agrees with the maximum of

n-l

n ,

such that there

exists an exact sequence

o ->

It

- > ... ->

Suppose , space

X.

Sk(7l

codh

J.

;i. k •

->

lO -> M -> 0

•

is a coherent analytic sheaf on a complex

We define

codh"J x

Let

II

->

as the function codh», 1... I:IX,x X

den ote the set of points of

X where

-

447 -

Y - T . Si u

(A.2)

Lemma (Frenkel ).

,

~

a < b

D i s a Ste in doma in i n

is a non empty Stein subdomai n of

D

o

Suppos e

N

R

in

.

then, f or

I

1

~ ~

D.

< N- l ,

This lemma is proved by Laurent series ex pansi on. tails, s ee [1, pp.217-219].

If

For

de~

As a corollary, we hav e the

f ollowing.

(A·3i

Proposition (Scheja).

dimension

in a complex space

~d

analytic sheaf on

o

< r-d

~ ).I

for

0

~

.

Suppose

X with

He nce

v < r-d- l

~

X and

H (X,"] )

--->

.

?

is a coherent

Then ¥~A"J = 0 for H (X-A,?) is bij ective

codh"l ;;;- r .

A is a subvariety of

~

and in j ective for

)I

= r-d-l •

We can assume without l oss of g ener a l i ty that an

0

pen subs et of

cI: n.

When

"J =

n([)

and

X is

A i s regular, i t

is a direct consequence of Frenke l's lemma, and, when the

,

singular set

A

of

A is nonempty, it f ollows from the

long exact sequence

When

"J

r n'!)

,

we use a local finite free resolution of

"J .

Now we define relative gap-sheaves with respect t o a subvariety.

Suppose

A is a subvariety of a complex space

-

448 _

Y -T. Siu

1 C -1

X and

the sheaf

are coherent analytic sheaves on

7 [A]1 U

X.

Define

by the presheaf

...-> {s~r(u,.~)I(sIU-A) ~r(U-A,7)}·

The following is a consequence of the Nullstellensatz. (A.4)

Proposition.

If.5

is the ideal sheaf of

A , then

Q)

1[A1 = U ("l:J k ) -1

k=l

1

and is therefore coherent, where

is the subsheaf of ~ s ~~x

whose stalk at

such that J~s C

tx

Proposition.

(A·5 )

Proof.

x

is the set

•

We can assume without loss of generaltiy that

?

7

is

By taking a local finite

defined on an open subset of ~n. free resolution of

< m=

is a subvariety of dimension

and considering the rank of the matrix

defining the extreme left sheaf-homomorphism of the resolution, we see eas ily that

Sm(?)

dimension estimate, the Case ing

,Ia[ {x}]?

for

from induction on

is a subvariety.

m=

x ~ So (':1).

a

For the

is obtained by consider-

The general case follows

m and considering the quotient of

7

by

the subsheaf generated by a holomorphic function whose germ at some

iJ C 1.

x ~ Sm(7)

is not a zero-divisor of 'x

Now we define the

dt h

relative gap-sheaf.

Suppose

are coherent analytic sh eaves on a complex space

Define the subsheaf "}[dJ.~Of

1-

by the presheaf

X.

-

I

A of

Proposition.

coherent and Proof.

for some subvariety

U of dimension

"1[dJ~="1lSd(-§/7)J~

~

dJ •

Hence '][dJ~ is

dim Supp(7[dJgI11 ~ d •

Lemma.

"1 C.fj

Suppose

on a complex space

X

and

mary submodule o f ~x di mx

Proof.

Supp..gl1 ~

x

P •

Jk-lj C "J

are coherent analytic sheaves

c;: X such that

whose radical d

and

There exists

k

c;:

~

P

7x

is of dimension

d.

J

U of on

x

in

X

there

U whose stalk at pkg C ~ x

such that

.

x

Hence

on some open neighborhood and

Let

Y = su PP('7[d_ l J9 1:fl

Y.

Since

and l e t

!

dim Y < d , there exists

be the ideal sheaf of f

c;:

Jx-P.

Nullstellensatz,

f or s ome

is a pri-

(1[d_l J§ lx = ?x •

For some open ne ighborhood

exists a coherent ideal sheaf is

c;: r(U-A,':1)

Follows from (A.3l and (A.5).

(A.7 l

Then

-

Y -T. Si u

U 1--> {S c;: r(U ,$) (s] U-A)

(A. 6)

449

1. c;:

~.

Sin ce

f

Cl

P , i t f ollows t hat

By the

-

4 50

-

Y - T . Si u

(A.B)

Proposition.

on a complex space

is a coherent a nal yt i c sheaf

X, x (;. X , a nd

is not a

zero~divisor

dim x V(f)

n

V(f )

?

Suppose

7x

of

Supp O[k] 7 < k

f (;. r( X,(!)X) ·

some

Suppose

Supp s

fx

s (;. re U,?)

borhood of

i f and only if f or a ll

k (;. N* ' wher e

i s a zero-d ivisor.

with

x.

Let

°

, where Sx f k = dim x Supp s

.

Then U is

fo r ° an open ne ighf x sx

Then

C V(f) () Supp e lk] 1 . Suppose

f

is not a zero-div isor.

x

the kernel of the sheaf-homomorphism multiplication by U of

y~U.

If

By considering

7 --->7

defined by

f , we conc lude that, f or s ome op en neigh-

x , fy dimxV(f )

borhood

is not a zero-divisor of

n Supp 0l k]?=

for some open subset

W of

U

k

7y

f or some

k

7y

a non-zero-divisor of

Proposition.

on a complex space Supp Oed]

':1

Suppose X.

equals the

Equivalently, f or

for

for

,

then,

,

which, because of the Nullstellensatz, contradicts

if

fx

= Supp
Proof.

(A·9)

Then

fy

being

y (;. u •

7

Then the

is a coherent analyt ic sheaf d-dimensional component of

d-dimensi onal component of

x (;. X , dim x Supp O[d ]1 < d dim x Sd ('1) < d •

Sd (7) •

i f and only

-451_

Y-T . Siu

Proof.

We prove the equivalent statement.

The "if" part

follows from

For the "only if" part, we assume that

dim Supp O[d]7
and dimx Sd(t) = d. X - Supp Old] "] such that ing

f ~ r(U,~)

UnSd (7 )

and

d.

After replac- .

does not contain any 0lk] "J for any

"Jy

not a zero-divisor of Sd ("1) n U

(A.IO)

=

dim ~+l ('1) ~ k

k;;;-

(A.12)

7x

°.

said t o be

t

> d

y~U.

By (A.$) , f y

°

is

Hence dim U n Sd ("]) = d

.

i f and only i f

k < d •

f ~ r(X,~X) , f is not a zero-dix i f and only i f dimx (Supp lOx/fC9x) Sk ("]) < k

Suppose

spaces and "]

0ld]"]=

42.£

Corollary.

visor vf

for

k

contains

k-dimensional

Sd_l (7/f"1) n U , contradicting

Corollary.

(A.II)

V(fj: = Supp (OX/fOX

such that

V(f)

un Supp

branch of

ule.

=

U by a smaller open subset, we Can assume that there

exists

for

dim U" Sd ('I)

For

n

rt :

X

- >

Y is a holomorphic map of complex

is a coherent analytic sheaf on

n-flat at

is said t o be

x~X

if

tx

is a flat

n- flat on (or at) a

X

.

7-

is

Oy,n (X) -mod-

subset G of

X

- 452 -

Y- T. Si u

7

if

is

n-flat at every point of

When t

r

to

Y

= 4:n , '7 is n-flat at x if and only if

is not a zero-divisor for

j

(t~, ••• , t~) (A.l) j

n(xj •

=

sheaf on a complex space that

7x

Let

Proof.

X and

is a coherent analytic n ; X -> <en

be the set of all points of

Z

is not

and the rank of

Hence

"JjjEl(t._t?j'Z, where x i=l 1. 1. x

Proposition. Suppose "]

phic map.

G.

n-flat. nlZ

is

Then
.

Z

is a holomorxG:X

such

is a subvariety of

X

By (A. H) ,

Z

x G: Sk (1)

is a subvariety of such that

it follows that

X•

Let

Tk

rank x n I Sk (7') < n •

rank nlZ < n •

coherent analytic sheaf on a complex space of

?

Since

Q. E. D.

Now we define absolute gap-sheaves. absolute gap-sheaf 7[d]

be the set of all co

Z C V Tk " k=O

Suppose X.

"1

is a t h The d

is defined by the pre sheaf

U t---> ind lim r( U-A,"1) A G:Ol.d(Uj where

~d(U)

of dimension

is the directed set of all subvarieties of ~

d •

U

-

4 53 -

Y-T. Siu

(A.14)

Proposition.

Suppose

sheaf on a complex space

7

X.

is a coherent analytic

Then the following three con-

ditions are eguivalent. i) 7[d]

is a coherent on

ii)

dim Supp O[d+I]7

iii)

dim Sd+l (1) ~ d •

Proof.

d •

The equivalence of ii) and iii) is (A.9J. To show iii)

open subset of ~n (* )

~

X.

o

i), we can assume that

~

X is an

and there exist exact sequences

-> K _> n[/n-d-2 -> ... -> n(9

PI

Po

-> n(9

->7->

0

(** )

-> on

X.

Let

on

X, let

Il"Iql v

n

qo

-> n<0

A = Sd+ I (1)·

R~1 be the

~/n(f<:, (0) nl:.! n

->

0

For a coherent analytic sheaf By (A.3) and (*),

By applying

~nO (., nO) to

(**),

we obtain an exact sequence qo

o -> J( -> n(0 on

X-A, because

ql

-> n([)

1-

~th direct image of ~lx-A under

the inclusion map X-A C-> X. 1[d] ::::: R~? ~ R~-d-2.k •

->

->

K is locally free on X-A.

It follows

-

4;34 -

Y - T. Siu

from (A. 3) that

is coherent. To show

i)

~

ii ), we suppose

dim Supp O[d+lJ,]= d+l

and are going to derive a contradiction. open

I-disc.

Let

n

be the unit

Without loss of generality we Can aSsume the

following. X

Let

A = nd x 0

and let

'1

=

nn

be the zero th direct image of

O[d+lJ7Ind+lx O-A under the inclusion map nd+1 x O-A C--> nd+1 x O.

'1 ~ is coherent.

Since (O[d+l]7) [AJ(7[dJ)

Since the sheaf-homomorphism ~ ---> ~ de-

fined by multiplication by

t d+ I

Nakayama's Lemma that 10 = 0 in nd+lx O-A approaching 0

.

is Let Since

0

,

it follows from co

{x» lV=l

be a sequence

nd+lxo_A

is Stein,

there exists

such that

Sx

f 0 f or al l

V.

It f ol l ows tha t

s

defi nes

-

4 5 5-

Y -T.Siu

a non-zero element of 10 ' contradi cting 1 0 Proposition.

(A.15)

Suppose

'7

0

is a coherent analytic

sheaf on a complex space X. Then th e natural s heaf- homomorph ism"] > 7 Ld ] i s an isomor phis m i f and only i f dim 1k+2(7) ~ k Proof. d

=

~

d

i)

-1

The "if" part.

is trivial.

U - (Ansd+l(']))

Us e induction on

Suppose

in an open subset

Since

k < d •

for a ll

d.

The case

A is a subvarie ty of d imension

U of

X

Since

codh' ~ d+2

on

, i t follows from (A.3) that

dim Sd+l(7)

~

d-l , by i nduc t i on hypothesis

~ r(U-(Ansd+l(':t)), "]).

r(U,"]) Hence "] ~ "J[d] • ii)

The "only if" part.

conclude that Hence

Old] 7' = 0 •

0[d+l]7= O. Sd+l(?l

k > d+1.

x

f ~r( U,OX )

such that

Supp 0X/f~

k-dimensional branch of Let

-§

= "]If"J.

and dim un Sk+2 (7)

for

U of

but contains no

U n Supp O[k] '] for any O[d ]1 = 0

By (A.14 ), dim Supp O[d+l]1 ~ d.

x ~ X there exists

For

for some open neighborhood contains

From t he defi niti on of "J [d] , we

k < d •

Q. E. D.

~

dim 1k+l (~)

~

k

Then

-

456 -

Y -T . Siu

(A.16)

Proposition.

domain in q:n If "}

,

and

a

Suppose

,

D

~

< b

a

D is a D •

is a nonempty open subset of

Dx~N(b)

is a coherent analytic sheaf on

1[n-l]

in (RN

such that

';I, then the restriction map

:z

is an isomorphism. Proof.

For any open subset

U of

is injective, because the support s

of

from

Ker a.

D, the restriction map

V of any nonzero element

would be a subvariety of

U x aN(a,b)

and, by considering

UX 6,N (b)

disjoint

({xJ x AN(b)) (\ V for

x ~ U , we conclude that dim V ~ n, contradicting In particular,

e

is injective.

e

We are going to prove the surjectivity of duction on

n.

s t;. Im

such that

s!QXaN(a,bJ

r(Q x ~N (b)

element

Let.Q

x

,1).

,7).

e•

Consider first the special case where D x ~ (b).

by in-

s t;. r(D x aN(a,bl) U (D' x AN(b})

Take

We have to show that

on

O[nJo;= 0.

codh

1?

n+l

be the largest open subset of

D

extends to an element of

To show that Q

D

Let

D such that

x t;. P

of the boundary of Q

nonempty open polydiscs in

is closed in

There exists an exact sequence

in

D

,

take an

pep

and

,

be

P C Q.

-

4 57

-

Y -T. Siu

Pl

... -> on

n+ Nl[)

->

·..PO

n+ Nt}

-> "1 ->

0

It follows from

o that

P

C

Q •

Hence.Q

is closed in

Now consider the general case.

IRN • Let

Let

n: D xllN(bj

->

D and Q Take

a

<

= b

N-l)

D • I

<

b

in

D be the natural projection.

A = n(Sn(1l n(DxAN(b')))

s/(D-A)xGN(a,blj

v<

(l :;i.

By the special case,

can be extended to

Sl ~ r(D-A)XllN(b '

),7) .

To finish the proof, it suffices to show that, for any given x ~ A , there exists an open neighborhood such that

s/UxGN(a,b)

r( U x llN(b) ,"]).

Since

U of

x

in

D

extends to an element of dim Sn (1) :;i. n-2 , we can assume with-

out loss of generality that x = 0 and there exist 0 < t 2 in IR n - 2 and 0 < (1 < 13 in lR such that n-2 n-2 6 ('() x t:;P (13) C D and 6 (¥) x (a ,13) is disjoint from

if

A·

By induction hypothesis, the restriction

rc.b.n - 2 (n x .6.2 (13 ) x AN(bj ,'])

r((.Dr.n - 2 (n x i is surjective.

map

->

(13) x aN(a,b») U (An-2 (n x Hence the element of

if (a,f3)

x AN(bl)

,i)

- 4 58 -

Y-T . Siu

with

s'

(Ad7)

on

.6,n-2(t)

Lemma.

X.

(-§,'])) [n] = ~(!)

X

(~, "])

sheaves

•

X

Proof.

can be extended t o an

7, -fj are coherent analytic If "1 = 7 [n] , then

Suppose

on a complex space

(1Jr"..(9

xcf(a.,~)X6N(b')

Because of the local nature, we can assume that there

exists an exact sequence

Ot -> (0~ ->.fj -> a on

X.

Hence

is exact on lJ

'?k1tC(9 ({!)X,"J)

X

of

'7.

X.

The result follows from the fact that

is isomorphic to the direct sum of Q.E.D.

Proposl.. t ao . n-

( A• I 0d )

an open subset of on

DXaN(a,b)

a: n

a =<

Su ppose

:t i

, and

such that

every sheaf-isoJllOrphism

1i

aj

=

,

aj

< a '< b a =

lRN

l.'n

, D l.' S

is a coherent analytic sheaf (i=1,2).

= ':li [ n ]

71 -> '72

uniquely to a sheaf-isomorphism

l!££f.

copies

V

on

Then

D x GN(a' ,b)

71 -> "12

£!!

extends

D x GN(a,b) •

We can assume without loss of generality that for

2 ~ j ~ n.

By (A.17) and (A.16), both re-

- 4 59

-

Y -T. Siu

striction maps

are bijective.

-

460-

Y - T . Siu

REFERENCES

[1]

A. Andreotti and H. Grauert, "Theoremes de finitude

pour La cohomologie des aspace s complexes", Bull. Soc. Math. France [2]

90

(1962), 193-259.

A. Ibuady, "Le probH:!me des modules pour les sous-

espaces analytiques compacts d'un espace analytique donne", Ann. Inst . Fourier (Grenoble)

16

(1966),

1-95· [3]

O. Forster and K. Knorr, "Ein Beweis des Grauertschen Bi1dgarbensatz nach Ideen von B. Ma1grange", Manuscripta Math.

[4]

i

(1971), 19-44.

, "Re1ativ-analytische R8.ume und die Koharenz von Bildgarben", Invent. Math.

16

(1972) ,

113-160. [5]

J. Frisch and G. Guenot, "Prolongement de faisceaux

analytiques coher-ent s'", Invent. Math. 3 21-343. [6]

7

=

(1969) ,

H. Grauert, EinTheorem der ana1ytischen Garbentheorie und die MDdu1ra.ume komp1exer Strukturen, I.H.E.S. No.

i

(1960). Berichtigung. I . H. E. S. No. 16 (1963),

35-36. [7]

, "Uber MDdifikationen und exzeptionel1e ana1ytische Mengen", Math. Ann.

146

(1962), 331-368.

-

4 61 -

Y - T , Siu

[8]

R. C. Gunning and H. Rossi, Analytic FUnctions of Several Complex Variables, Englewood Cliffs, N.J.: Prentice-Hall, 1965.

[9]

C. Houzel, "Espaces analytiques relatifs et theoreme de finitude", Preprint, Nice, 1972.

[10]

R. Kiehl, "Variationen iiber- den Kohar-enz aat z von Grauert", Pre print , Frankfurt, 1970.

[11]

, "Relativ analytische Raume", Invent. Math. 16

(1972),40-112.

[12]

and J. -L. Verdier, "Ein einfacher Beweis des Koharenzsatzes von Grauert", Math. Ann'.

195

(1971), 24-50. [13]

K. Knorr, "Der Grauertsche Projektionssatz", Invent. Math.

[14]

12

(1971), 118-172.

_ _ _ _ _ , "Noch ein Theorem der analytischen Garbentheorie", Preprint, Regensburg, 1970.

[15]

and M. Schneider, "Relativexzeptionelle analytische Mengen", Math. Ann.

[16]

(1971), 23 8-254.

H.-S. Ling, "Extending families of pseudo con cave complex spaces", Math. Ann.

[17]

193

204

(1973), 13-48.

A. Markoe and H. Rossi, "Families of strongly pseudoconvex manifolds", Proc. Coof. Several Complex Variab l e s (Park City, Ut a h , 1970) pp. 182-207, Lecture No t e s in Math. Spri ng er 197 1.

184, Be r l i n -He i d e l be rg - New York :

- 4 62

-

Y -T. Siu

[18]

R. Narasimhan, Grauert's Theorem on Direct Images of Coherent Sheaves, seminaire de Mathematiques Superieures, '1969, MOntreal: Universite de MOntreal, 1971.

[19]

J.-P. Ramis and G. Ruget, "Residue et dualite", Preprint, Strasbourg, 1973.

[20]

H. Rossi, "Attaching analytic spaces to an analytic space along a pseudo concave boundary", Proc. Conf. Complex Analysis (Minneapolis 1964), pp. 242-256, Berlin-Heidelberg-New York: Springer, 1965.

[21J

P. Siegfried, "Un theoreme de finitude pour les morphismes

[22]

Y.-T. Siu, "Extending coherent analytic sheaves", Ann. of Math.

[23 ]

q-convexes', Preprint, Geneva, 1972.

90

(1969), 108-143.

_____ , " A Hartogs type extension theorem for coherent analytic sheaves", Ann. of Math •

.21

(1971),

166-188. [24]

, "A pseudo concave generalization of Grauert's direct image theorem I,II", Ann. Scuola Norm. Sup. Pisa

[25 ]

24

(1970), 278-330; 439-489·

_____ , "The

l-convex generalization of Grauert's

direct image theorem", Math. Ann. [26]

190 (1971), 203- 214.

, "A pseudoconvex-pseudoconcave generalization of Grauert's direct image theorem", Ann. Scuola Norm. Sup. Pisa

25

(1972), 649-664.

- 463 -

Y - T . Siu

[27J

Y.-T. Siu, Techniques of Extension of Analytic Objects, University of Paris VII Lecture Notes, 1972 (to appear in the Marcel Dekker Lecture Notes Series).

[28]

and G. Trautmann, Gap-sheaves and Extension of Coherent Analytic Subsheaves, Lecture Notes in Math. 172, Ber l i n- He i de l berg- New York: Springer 1971.

[29]

G. Tr a u t ma nn , "Ein Kontinuitatssatz fUr die Fortsetzung koharenter analytischer Garben", Arch. Math.

18

(1967), 188-196. [30]

K.-W. Wiegmann, "Uber Quotienten holomorph-knonvexer komplexer Raume", Math. Z. 97 (1967), 2.51-2.58.

Department of Mathematics Yale University New Haven, Connecticut 06.520 U.S.A.