COMPLEX MANIFOLDS JAMES MORROW KUNIHIKO I(ODAIRA
AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island ~'t~~~
~.! "BtiiYi;
,••,
2000 Mathematics Subject Classification. Primary 32Qxx.
Library of Congress Cataloging-in-Publication Data Morrow, James A., 1941Complex manifolds I James Morrow, Kunihiko Kodaira. p. cm. Originally published: New York: Holt, Rinehart and Winston, 1971. Includes bibliographical references and index. ISBN 0-8218-4055-X (alk. paper) 1. Complex manifolds. I. Kodaira, Kunihiko, 1915- II. Title.
QA331.M82 2005 515'.946--dc22
20051
© 1971 held by the American Mathematical Society. Reprinted with errata by the American Mathematical Society, 2006 Printed in the United States of America. @) The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321
11 10 09 08 07 06
Preface
The study of algebraic curves and surfaces is very classical. Included among the principal investigators are Riemann, Picard, Lefschetz, Enriques, Severi, and Zariski. Beginning in the late 1940s, the study of abstract (not necessarily algebraic) complex manifolds began to interest many mathematicians. The restricted class of Kahler manifolds called Hodge manifolds turned out to be algebraic. The proof of this fact is sometimes called the Kodaira embedding theorem, and its proof relies on the use of the vanishing theorems for certain cohomology groups on Kahler manifolds with positive lines fundles proved somewhat earlier by Kodaira. This theorem is analogous to the theorem of Riemann that a compact Riemann surface is algebraic. This book is a revision and organization of a set of notes taken from the lectures of Kodaira at Stanford University in 1965-1966. One of the main points was to give the original proof of the Kodaira embedding theorem. There is a generalization of this theorem by Grauert. Its proof is not included here. Beginning in the mid-1950s Kodaira and Spencer began the study of deformations of complex manifolds. A great deal of this book is devoted to the study of deformations. Included are the semicontinuity theorems and the local completeness theorem of Kuranishi. There has also been a great deal accomplished on the classification of complex surfaces (complex dimension 2). That material is not included here. The outline is roughly as follows. Chapter 1 includes some of the basic ideas such as surgery, quadric transformations, infinitesimal deformations, deformations. In Chapter 2, sheaf cohomology is defined and some of the completeness theorems are proved by power series methods. The de Rham and Dolbeault theorems are also proved. In Chapter 3 Kahler manifolds are studied and the vanishing and embedding theorems are proved. In Chapter 4 the theory of elliptic partial differential equations is used to study the semi-continuity theorems and Kuranishi's theorem. It will help the reader if he knows some algebraic topology. Some results from elliptic partial differential equations are used for which complete references are given. The sheaf theory is self-contained. We wish to thank the publisher for patience shown to the authors and Nancy Monroe for her excellent typing. James A. Morrow Kunihiko Kodaira
Seattle, Washington January 1971
v
Contents v
Preface
Chapter 1.
Definitions and Examples of Complex Manifolds 1. Holomorphic Functions 2. Complex Manifolds and Pseudo group Structures 3. Some Examples of Construction (or Description) of Compact Complex Manifolds 4. Analytic Families; Deformations
11 18
Sheaves and Cohomology Germs of Functions Cohomology Groups Infinitesimal Deformations Exact Sequences Vector Bundles A Theorem of Dolbeault (A fine resolution of lP)
27 27 30 35 56 62 73
Chapter 2. 1. 2. 3. 4. 5. 6.
Chapter 3. Geometry of Complex Manifolds 1. Hermitian Metrics; Kahler Structures 2. Norms and Dual Forms 3. Norms for Holomorphic Vector Bundles 4. Applications of Results on Elliptic Operators 5. Covariant Differentiation on Kahler Manifolds 6. Curvatures on Kahler Manifolds 7. Vanishing Theorems 8. Hodge Manifolds Chapter 4. 1. 2. 3. 4.
Applications of Elliptic Partial Differential Equations to Deformations Infinitesimal Deformations An Existence Theorem for Deformations I. (No Obstructions) An Existence Theorem for Deformations II. (Kuranishi's Theorem) Stability Theorem
Bibliography Index Errata
1 1 7
83 83 92 100 102 106 116 125 134
147 147 155 165 173 186 189 193
vii
Complex Manifolds
[1] Definitions and Examples of Complex Manifolds I.
Holomorphic Functions
The facts of this section must be well known to the reader. We review them briefly. DEFINITION 1.1. A complex-valued function J{z) defined on a connected open domain W £: C" is called h%morphic, if for each a = (a., ... , all) E W, J(z) can be represented as a convergent power series +00
L
ck , ..• kn(Zl
alt' ... (ZII
-
-
alit"
k,~O.kn~O
in some neighborhood of a.
LC
If p(z) = k ... kn (Zl - a1)k, •.• (Z/I - allt n converges at z = w, then p(z) converges for any Z such that IZk - akl < IWk - akl for 1 ~ k ~ n. REMARK.
Proof We may assume a = O. Then there is a constant C > 0 such that for all coefficients Ckl".k n ,
Ick,· .. k
n
Wk'1 ... wknl /I < - C•
Hence (1)
If
Izdwil < 1 for 1 ~ i
~
n, (1) gives Q.E.D.
1
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
2
We have the following picture:
Figure I
n is the region {zllz;l <
Iw;l i For convenience, we let
~
n}.
P(a, r) = {zllzv - avl < rv , v = 1,' ", n}. Sometimes we call P(a, r) a polydisc or polycylinder. A complex-valued function/(z) = l(x1 + iYl' ... , Xn + iYn), where i = 1 can be considered as a function of 2n real variables. Then:
J-
DEFINITION 1.2. A complex-valued function of n complex variables is continuous or differentiable if it is continuous or differentiable when considered as a function of 2n real variables. We have: THEOREM J.l. (Osgood) If l(z)=/(ZI' ... , zn) is a continuous function on a domain W £ en, and if 1 is holomorphic with respect to each Zk when the other variables z j are fixed, then 1 is holomorphic in W. Proof Take any a E Wand choose r so that P(a, r) ~ W. We use the Cauchy integral theorem for the representation for z E P(a, r)
f ... )- 1 f ,zn - - .
f( ZI' ... ,zn) -- _1.
f( W1,Z2'
and so on.
21tl
IWI-ad:',
21t1
Iwz-azl:'z
f(w 1, z2, "', zn) d
WI'
WI -
ZI
f(W I ,W2 ,Z3"",zn)d W2
- Z2
W2'
l.
3
HOLOMORPHIC FUNCTIONS
Substituting we get
Weare assuming
Iz.w. -- a.a. 1<1. Hence the series 1 w. - z.
1
= (w. - a.) + (a. - z.) =
=(
1
)
w. - a.
[
1
]
1
1 - (z. - a./w. - a.) w. - a.
f (z.w. -- a.)k a.
k~O
converges absolutely in Pea, r). Integrating term by term we get 00
fez) =
2: Ckl ... kn(zt -
alt l
.00
(Zn - an)kn,
(2)
n~O
where 0..
Ckl
_
kn
-
(_I)n f 0
2nl
f
000
IWI-ad~rl "olwn-anl=rn
f(W l
.00,
(WI - al)
k
I
n)dW
w
+1
.00
l
0.0
dW n
(W n - an)
k
n
+1 •
Then
where M = sup{lf(w)llw E pea, r)}o It follows that the representation (2) for is valid for ZE Pea, r) and hence the theorem is true.
f(z)
We now introduce the Cauchy-Riemann equations. Letf(z) be a differentiable function on domain n £: en. DEFINITION
1.3. The operators
af
a/az., a/oz., 1 (a f
0
I
~
af)
ax. - l oY. ' af 1 (a f . Of) oz. = 2: ax. + I ay. '
0'1. = 2:
where z. = x.
+ iy. as usual.
v ~ n are defined by
4
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS Let J(z) =
u(x, y) + iL'(x, y). Then aJ = ~ [au + i av + i(OU + i av)] af 2 ax ox oy ay = ~ [au _ av + i(av + au)].
2 ax
°
ay
So, aJ/az = if and only if au/ax Riemann equations).
ax
ay
= all/ay and al'/ax = - au/ay (the Cauchy-
REMARK. If aJ/az = 0, then df/dx = af/az, where The following calculation verifies this:
aJ az
=
dJ/dx = au/ax + i(avjax).
~ [au + i av _ i(OU + i av)] 2 ax ax ay ay
=~ [au + i av + i(av _ i au )]. 2 ax ax ax ax THEOREM 1.2. Let f(z) be a (continuously) differentiable function on the open set n £; en. Thenf(z) is holomorphic if and only if af/az. = 0, i :::;; v :::;; n. Proof This follows easily from Osgood's theorem and the classical fact for functions of one complex variable. We need another simple calculation. From now on differentiable will mean having continuous derivatives of all orders (ceo).
PROPOSITION 1.1. Suppose few) = f(w\, "', wm) and giz) I:::;; A. :::;; mare differentiable and such that the domain off contains the range of (g\, .. " gJ = g. Thenf[g\(z), "', gm(z)] is differentiable and if w;.(z) = g;.(z),
aJ = az.
f ;.=
(aJ aw;. + aJ aw;.) az. aw;. OZ. '
law;.
(3)
(4) Proof All statements follow trivially from the chain rule of calculus. For punishment we calculate (3). Let IV). = U). + iv;. = g.(z). Then
f
aJ[g5z)] = aJ a~;. + aJ a~;.. az. ;.=\au).az. ov).oz.
I.
5
HOLOMORPHIC FUNCTIONS
Making the substitutions,
uA =
1
2(g A + 9).),
we get
. of) 09 +-1 (Of -+1- A} 2 au).
oz.'
Ov).
which gives (2). COROLLARY 1. If f(w) is holomorphic in wand if w = g(z) = [gl(z), ... , gm(z)] where each g;.(z) is holomorphic in z, thenf[g(z)] is holomorphic in z.
COROLLARY
2. The set
(90 of all functions holomorphic on
n forms a ring.
In order to study complex manifolds we must consider holomorphic en and letfbe a map from U into em,
maps. Let U be a domain in
DEFINITION 1.4.
f is holomorphic if each fA is holomorphic. The matrix ofl
ofm
OZI
OZI
ofl
ofm
OZn
OZn
=
(:~:L=I . . . Y=
m
1, ... , n
is called the Jacobian matrix. If m = n, the determinant, det(of)./ozv) is called the Jacobian. Writing out the real and imaginary parts w). = UA + iVA = fA' z. = Xv + ;y .. we have 2n functions U)., VA of 2n real variables xv, Yv' We write briefly det [
O(UI> VI> " ' , Un' Vn)]
o(X I , YI"", Xn , Yn)
o(u, V)
=--.-. o(x, y)
6 REMARK.
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
If/ is holomorphic, o(u, v)/o(x, Y) = Idet(o/.doz')l z ~ O.
Proof We write it out for n = 2 and leave the general case to the reader. We use the Cauchy-Riemann equations and set aVA = ouA/ox. = ovA/oYv, b,;. = OVA/OX. = - ou;./oYv. Then
OUI OVI oU z oV z oX I oX I oX I OXI
-
OU I
OVI °YI OYI
OU z OV 2 = OYI °YI
all
b ll
al2
biz
-bll
all
-b 12
a 12
a21
b 21
a22
b22
-b 21
aZI
-b 22
a zz
We perform the following sequence of operations: Multiply column 2 by i and add it to column I; do the same with columns 4 and 3. Then multiply row 1 by i and subtract it from row 2; do the same with rows 3 and 4. Making use of the fact that B.;. = O/A/OZV = a.;. + ib v ;', we get gil
o(u, v) gZI --= 0 o(x, Y) 0
glz * * gzz * * 0 gIl gl2 0 gZI gll
= Idet(g.;.)I Z
by interchanging columns 2 and 3 and rows 2 and 3.
Q.E.D.
THEOREM 1.3. (Inverse Mapping Theorem) Let/: U -. en be a holomorphic map. Ifdet(o//l/oz.)lz=a -=1= 0, then for a sufficiently small neighborhood N of a,fis a bijective map N -+ /(N);f(N) is open and/-II/(N) is holomorphic
on/eN). Proof The remark gives o(u, v)/o(x, Y) -=1= 0 at a. We then use the inverse mapping theorem for differentiable (real variable) functions to conclude that /(N) is open, / is bijective, and /- I is differentiable on /(N). Set cp(w) = /-1(11'); then z/l = 'P/l[f(z)]. Computing,
o = O~I' = oz.
f ;'=1
o'Pl' O~A OW;. OZ.
+ O~I' o~;. OW;. oz.
But det(oJ;./oz.) = det(o/A/oz.) -=1= O. So by linear algebra, oCPI'/ow;. = 0 and cP =/-1 is holomorphic. Q.E.D.
2.
COMPLEX MANIFOLDS AND PSEUDOGROUP STRUCTURES
7
COROLLARY. (Implicit Mapping Theorem) Let f;., A = I, ... , m be holomorphic on V s; en. Let rank (8f;./8z.) = r at each point z of V and suppose in fact that det(of;./oz');'$r i= O. If fla) = 0 for A. :::;; m for some a E V, then in a small neighborhood of a, the simultaneous equations,
have unique holomorphic solutions
A:::;; r. For more details in this section one may consult Dieudonne (1960).
2.
Complex Manifolds and Pseudogroup Structures
We assume given a paracompact Hausdorff space X which will also generally be assumed connected. We want to define what we mean by a complex structure on X (or structure of a complex manifold) which will be an obvious generalization of the concept of a Riemann surface. First we want to assume X is locally homeomorphic to a piece of IC". DEFINITION 2.1. By a local complex coordinate on X we mean a topological homeomorphism z:p - z(p) E IC" ofa domain Us; X. z(p) = [Zl(p), ... , z"(p)] are the local coordinates of X. DEFINITION 2.2. By a system of local complex analytic coordinates on X we mean a collection {z j} jEt (for some index set I) of local complex coordinates Zj: Vj such that:
en
(1)
X=UU j
•
JET
(2) The maps fjk: Zk(P) - Zj(p) are biholomorphic [that is, Zj 0 Z;1 = fjk and r;/ = Zk zjt are holomorphic maps from Zk(V j n Vk) onto z}Vj n Uk)] for each pair of indices (j, k) with U j n Uk i: fjJ. 0
DEFINITION 2.3. Two systems {Zj}jt/, {I1';J"<J' are equivalent if the maps zip) - 1I',,(p) are biholomorphic when and where defined. DEFINITION 2.4. By a complex structure on X we mean an equivalence class of systems of local complex (analytic) coordinates on X. Bya complex manifold M we mean a paracompact Hausdorff space X together with a complex structure defined on X.
8
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
EXAMPLE. Complex projective space IPn. This is constructed from {OJ by identifying (p '" q)p = (pO, pI, "', pn) and q = (qO, "', qn) if and only if p). = cq). for some nonzero c E e, for 0 :-:::; .A. :-:::; n. Then IP n = en + I {OJ j '" is a compact Hausdorff space and one can construct a system of complex coordinates as follows: We let Vj = {p E IPnlpj::f. OJ. Then {VJj~n is an . 0 f It"". ITl>n 0 h open covenng n V j th e map Zj -- (Zj ' , Zjj - I , Zjj+ I , . . . , Zjn) , were = p)./pj gives a local coordinate on V j ; in fact, Zj(V) = en. Then fjk: Zk -4 Zj is given by z~ = zt/zf for A. ::f. k, z~ = I/zf. (One simply multiplies by pkjpj.) Thus we see that {V j ' Zj} is a complex analytic system defining a complex structure on IPn.
en+ 1 -
° ...
z/
Generalizing this procedure we introduce the idea of a pseudogroup structure. All spaces will be Hausdorff in what follows. 2.5. A local homeomorphism f between two spaces X and Y is a homeomorphism of an open set V in X to an open set f( V) in Y. One has a similar definition of local diffeomorphism. A local homeomorphism (diffeomorphism) of X is such a map with X = Y. Let 9 be a domain of IR nor en. Letf and 9 be local diffeomorphisms of 9. If open W s; 9, fl W denotes f restricted to W which is the restriction off to domain (f) n W. If W is some open set such that 9 is defined on Wand W nf(V)::f.
I(U)
Figure 2
DEFINITION 2.6. A pseudogroup of transformations diffeomorphisms of 9 such that
In
9 is a set r of local
(I) fEf=>f-IEf. (2) fE r, 9 E r => 9 a fE r where defined. (3) fEr => fl WE r for any open W s; 9. (4) The identity map id E r. (5) (completeness) Let f be any local diffeomorphism of 9. If [) andfl Vj E r for each}, thenfE r.
= u
Vj
2.
COMPLEX MANIFOLDS AND PSEUDOGROUP STRUCTURES
9
DEFINITION 2.7. Let f (a pseudogroup on 9) and X (a paracompact Hausdorff space) be given. By a system of local f-coordinates we mean a set {z j} j d of local topological homeomorphisms Z j of X into 9 such that Zj 0 Z;:l E f whenever it is defined. {w ...} and {Zj} are equivalent (f-equivalent) if 11'). 0 zjl E f when defined. A f-structure on X is an equivalence class of systems of local f-coordinates on X. A f-manifold is a paracompact Hausdorff space X together with a f-structure on X.
EXAMPLES I. 9 = en, fe = (all local biholomorphic maps of en). Then a fe-structure is a complex structure, and a fe-manifold is a complex manifold. 2. 9 = IR", fd = (all local diffeomorphisms of IR"). Then a fd-structure is a differentiable structure and a fd-manifold is a differentiable manifold. 3. Let f be the set of a local diffeomorphism / of 1R2" satisfying the following condition. The matrix (c)..) will be defined to be
0
-1
0 0
-\
0
0 0
0 1
-1
0
where the blocks (? - b) occur on the diagonal and the rest of the entries are zeros. If x = (Xl, "', X2n) E 1R 2",f(x) = [.ft(X), ... ,f2"(X)] then the derivatives of / should satisfy
A system satisfying Example I is called a Hamiltonian dynamical system, and such an / is a canonical trans/ormation. In this case a f-structure is called a canonical structure. 4. Let f = (local affine transformations of IR"). These transformations have the form
j'(x) =
L" a~ XV + b\ v~
,
where the a~, b). are constants and the matrix (a~) is nonsingular. [n this case a f -structure is called flat affine structure. If pseudogroup structures f, and f 2 are such that f, c f 2' then every system of local f, coordinates is a system of local f2 coordinates, and f, equivalence implies f 2 equivalence. Hence, every f I-structure determines a
10
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
r 2-structure. By assumption r c r d for all r. So every r -structure on X determines a differentiable structure on X and every r-manifold is a differentiable structure on X and every r-manifold is a differentiable manifold. The r-structure M is defined on the differentiable manifold X. The problem of determining the r-structures on a given differentiable manifold M for given r is one of the most important (and difficult) problems in geometry. It is known, for example, that if X is a compact orientable differentiable surface (real dimension 2), then the only complex structures on X are those of the classical Riemann surfaces. In case X = S2 (as a differentiable manifold), then X = [pI complex analytically (this is a classical fact). If the underlying differentiable manifold X is diffeomorphic to [pn, then one conjectures that X = [pn complex analytically [see Hirzebruch and Kodaira (1957)], and Kodaira and Spencer (1958). If S211 is the sphere with its usual differentiable structure, it can be shown [Borel and Serre (\ 953) and Wu (1952)] that S211 for n f= 1,3 has no complex structure
[S
2n -_ {(Xl' . . . , X2n+l) 12n~ if--2I Xi'2
(Xl'
•••
,X2n+1
)
E
IR
2/1 + I }] .
For S2 there is the usual complex structure. It has been recently proved by A. Adler (1969) that S6 has no complex structure. As a final example, let M be a compact surface and let r+ be the pseudogroup of all local affine transformations,
v = 1,2 such that
We have: THEOREM 2.1. [Benzecri (1959)] If a r+-structure exists on M, then the genus of M is I. If M is not a torus, then M cannot be covered by any system {(xj, X])} of local coordinates such that lax;/ax~1 is constant on Vj n Uk for each pair of indices (j, k). The proof will not be given here. We continue with the definitions. Let M be a complex manifold, Wan open set in M, and {Zj} a coordinate system. Then a mappingj: W -+ C l is holomorphic (difjercl1tiahle, and so on) if I zj I is holomorphic (d(fJerentiable, and so on) for each j where defined. Let N be another complex manifold with coordinates {1I' .d and I: W -+ N. Then I is holomorphic (differentiable, and so on) if II'A a I 0 zj I is holomorphic where defined. 0
DEFINITION 2.8. A subset S £; M of a complex manifold is a (complex) analytic subvariety if, for each S E S, there are holomorphic functions IA{P) defined on a neighborhood {; 3 S, I ~ A~ r, such that S n U = {p Ilip) = 0,
3.
CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS
11
I ~ A ~ r}. Then f). = 0, 1 ~ A ~ n, are the local equations defining 5 at s. The subvariety 5 is called a submanifold if 5 is defined at each s E 5 by local equationsf). = 0 such that
l
Of;.(P)] . d d f ran k - = r. IS In epen ent 0 s.
ozi(p)
Suppose det(af)./ozj)1
<;).sr
=F O. Then letting
lS;\'S;r
w;(p) = fip),
for A = 1, ... , r
w;(p) = z;(p),
forA=r+ 1,"',/1,
we have a local coordinate wi = (w), ... , wi» such that 5: w) = 11'] = ... = wj = 0 (is defined by). Let (1(p) = IV'/)'(p) = zj+"'(p) for p E 5 n Uj • Then 5 is a complex manifold with local coordinates {C}. We want to introduce meromorphic functions on a complex manifold. They should be those functions which are locally quotients of holomorphic functions. More precisely: DEFINITION 2.9. A meromorphic function f on M is a complex-valued function defined outside of some proper subvariety 5 of M (5 =F M) and such that given q E M, there is a neighborhood U of q in M and local holomorphic functions g, h on U such thatf(p) = g(p)/h(p) for p E U - 5.
EXAMPLES Any holomorphic mapj: M ..... 1P 1 = C u {co}, [5 =f-I(co)]. 2. In C 2,f(Zt, Z2} = ZI/Z2 or f(zt, Z2) = P(ZI' Z2)/Q(Zt, Z2), where P and Q are polynomials. 1.
3. Some Examples of Construction (or Description) of Compact Complex Manifolds First we have submanifolds of known manifolds (IP", IP n x lPn, and so on). Let IP n have homogeneous coordinates (Co, "', (n). Let f;.(O, 1 ~ A ~ m be homogeneous polynomials and define M = {( IfiO = 0, I ~ A ~ m}. Suchan M is called a projective algebraic (or simply algebraic) variety. If the rank of (of)./oc), is independent of ( E M, then M is a complex manifold. These are exactly the classical algebraic (projective algebraic) manifolds. In some cases the equationsf). = 0 give some easily read information about M. For instance, if f is homogeneous of degree d, then Md = {( Ifm = O} is called a hypersurface in IPn of order d. If at least one of (of/oC})m =F 0, I ~ A ~ n, for each CE M d , then M d is nonsingular.
12
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
EXAMPLES 1. Md £: !p 2 a nonsingular plane curve of order d is a Riemann surface of genus g = td(d - 3) + I. 2. A nonsingular M d £: !p 3 • M d is simply connected and the Euler number X(M d ) = d(d 2 - 4d + 6). [The formulas in Examples I and 2 can be obtained from Hirzebruch (1962), p. 91, Equation (5). They are well-known classical formulas. The simple connectivity is also well known and it follows from the Lefschetz theorems on hyperplane sections-see Milnor (1963), p.41.J 3. Let M £: !p 3 be defined by
M = {(I
'1'2 -'0'3 = '0'2 - 'i = O,'~ '1'3 = 0,
-
O}.
We claim that M is complex analytically homeomorphic to pl. One can easily check that the map J.t:!p 1 --+!p3 defined by J.t(t) = (t5, t~tI' toti, Ii) where t = (to, td E !pI, is a biholomorphic map of !pI onto M. We remark that in the cases of complex or differentiable structures, submanifolds give many examples; but for general r-structures one does not usually get sub r-structures. Second we get quotient spaces. DEFINITION 3.1. An analytic automorphism of M is a biholomorphic map of M onto M. The set of all analytic automorphisms of M forms a group g with respect to composition. Let G £: g be a subgroup. DEFINITION 3.2. G is called a properly discontinuous group of analytic automorphisms of M if for any pair of compact subsets KI , K2 £: M, the set {g E G IgKI n K2 =t= 4J} is finite. DEFINITION 3.3. points.
G has no fixed points if for all g
E
G, g =t= 1, g has no fixed
THEOREM 3.1. If G is properly discontinuous and has no fixed point, then the quotient space MIG is a complex manifold in an obvious natural manner.
Proof We shall assume that M is connected (or a countable union of connected manifolds) and paracompact. Hence, M is CT-compact (a countable union of compact sets). Let MIG = {G p I p EM}, where Gp = {g(p) I pEG} are the orbits of P E M. As notation set MIG = M*, Gp = P*. We shall show that given q E M we can choose a neighborhood U of q such that PI' P2 E U, PI =t= P2 gives p~ =t= p~. In fact, there is U 3 q such that gU n U = 4J for all g E G, g =t= 1. M is locally compact so let VI :::> V 2 :::> V 3 ••. be a base of rela-
3.
13
CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS
tively compact neighborhoods at q. Then Fm = {g IgVm n Vm =f. cp} is a finite subset of G and Fm;2 Fm +S' ;2 •.•. If 3gmE Fm, gm =f. 1 for all m, then since each Fm is finite, n Fm 3 g, 9 =f. 1. Therefore, gVm n Vm =f. cp, for all m and Vrn -+ q, gives g(q) = q, contradicting the nonexistence of fixed points. Hence we cover M with open sets Vj such that p" P2 E Vj implies pi =f. pi and thus, Vj ~ V; = {p* I p E V j } is 1 - 1. We give V; the complex structure that Vj has. That is, if Zj: P -+ zip) is a local coordinate on V j ' then zj: p* -+ zj(p*) = ziP) gives a local coordinate on M*. The system {zj} then defines a complex structure on M* and the topology of M* is just the quotient topology for the map M -+ M*. Q.E.D. EXAMPLES Complex tori. Let M = Take 2n vectors {w" ... , W 2n }, n ••• , w kn ) E C so that the Wj are linearly independent over ~. Let
cn.
1. (Wk"
Wk
=
2n
G = {g I g: z -+ g(z) = z
+
L mkWk , mk E l}.
k='
Tn = en jG is a (complex) torus of complex dimension n. Let n = 1 and arrange it so that WI = I, W2 = w, where the imaginary part of W is positive. Then T=C'jG.
w
Figure 3 Z I
exp 2"j
We have a map C - - C*, z -+ w = e "z where C* = {z I z =f. OJ. If we first take g(z) = z + m,w + m 2 and then exponentiate, we get e 27ti (z+m l w). So exp 21ti 0 9 = ami. exp 21ti where a = eZ"iw and g(z) = z + m,w + m2' and o < lal < 1 since Im(w) > O. Looking a little closer we see we have the diagram If'
exp 2"i
IV--C
*
·c--c 1 ." ,., 1:-' which commutes. Hence, if we let G* = {g* I g*: w -+ am l1', mEl}, we see T = C/G = C*jG*.
14
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
Figure 4
Hop! manifolds. Let W = eN - {O} and G = {gin I m E Z, g(wl' .. " = «(XIIV I , " ' , (XNWN), where < l(Xvl < I}. Then WIG is a compact com2.
°
wN ) plex manifold since it is easy to see that G is properly discontinuous and has no fixed points on W. It is also easy to see that WIG is diffeomorphic to Sl x S2N-I. 3.
Let M be the algebraic surface (complex dimension 2) defined:
M
= {(
I (g + n + (~ + (~
=
o} £
[FD3.
Let
G = {gml m = 0, 1,2,3,4 where g«(o,"', (3) = (p(o, p2(1' p3(2' p4(3) and p
= e 2"i/5}.
Then 9 is a biholomorphic map [FD3 ~ [FD3 and g5 = 1. Consider the fixed and the points of gm on [FD3. They satisfy (0 = v ~ 3), (pln(v+l) - c) (v = fixed points are (1, 0,0, 0), (0, I, 0, 0), (0, 0, I, 0), and (0, 0, 0, I). These points are not on M so there are no fixed points on M and MIG is a complex manifold. We saw before that M is simply connected and X(M) = d(d 2 4d + 6) where d = 5. Therefore, the Euler number of M is 55. Then the fundamental group 1t 1 (MIG) ~ G and x(MIG) = II. 4. Last we have the classical examples of Riemann surfaces and their universal covering surfaces. If S is a compact Riemann surface of genus 9 ~ 2, the universal covering surface of S is the unit disk D = {2 E ell 121 < I}. Then S = DIG where each element of G is an automorphism of D and hence of the form
°
g(2)
.
2 -
(X
= el8 - - , (X;: -
I
I(XI < I.
3.
CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS
15
Finally we consider surgeries. Given a complex manifold M and a compact submanifold (subvariety) ScM, suppose we also have a neighborhood W::::> S and manifolds S* c W* with W* a neighborhood of S*. Suppose I: W* - S* ~ W - S is a biholomorphic map onto W - S. Then we can replace W by W* and obtain a new manifold M* = (M - W) u W*. More precisely, M* = (M - S) u W* where each point z* E W* - S* is identified with z = I(z*).
-
J
[-ry
Figure 5
EXAMPLE I. Hirzebruch (1951) Let M = pi X pl. In homogeneous coordinates, pi = {(I ( = «(0' (I)); = {C u {(Xl in inhomogeneous coordinates, (= (d(o E C u roo}. M = pi X pi = {(z, 01 z E pi, (E pi} contains S = {OJ X pi and W = D X pi where D = {zllzl < E} is a neighborhood of Sin M. Let W* = D X pi' = {(z, (*)Iz ED, (* E pi'} and S* = {OJ x P*. Fix an integer m > 0 and define I: W* - S* ~ W - S as follows:
I(z, (*) ~ (z, 0 = [z,«(*/z'")J
where 0 < Izl <
E.
Then 1 is biholomorphic on W* - S* and let M! = (M - S) u W* where 0 = (z, (*) if (* = zmC 0 < Izl < E.
(z,
REMARK.
M and M! are topologically different if m is odd.
Proof (for m = 1). M = pi X pi is homeomorphic to S2 x S2. We show that the homology intersection properties of M and Mi are distinct, hence, proving that they are topologically different. A base for HiM, Z) is given by {SI' S2} where SI = {OJ X pi, S2 = pi X {OJ. Hence, any 2-cycle C is homologous (,...,) to aS I + bS 2 , a, b E Z. The intersection multiplicity I(C, C) = /(aS I + bS 2 , aS I + bS 2) = a 2[(SI' SI) + b 2[(S2' S2) + 2abl(SI' S2)' Since SI' S2 occur as fibres in pi x pi, [(SI, Sd = [(S2, S2) = O. Hence, ICC, C)
= 2ab == 0 (mod 2).
(I)
16
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
In M we have the following picture:
w'
w M
/
r-- V
s
s,
f--
.
Figure 6
where Ac is the submanifold of Mi defined by ( = c and (* = zc with the coordinates explained before. Then Ac is a 2-cycle and Ao '" Ac. Hence /(Ao, Ao) = /(Ao, Ac) = I. Since for any 2-cycle Z on M, /(Z, Z) == 0 (mod 2) we see M #- Mi. REMARKS
M! #- M:(m #- n) as complex manifolds. M~m = M topologically. 3. Mim+i = Mi topologically. These facts are proved in Hirzebruch (1951). 1. 2.
EXAMPLE 2. (Logarithmic Transformation) LetM = T x Pi,T = CjG, G = {mw + n I m, n E 7L, 1m w > O} where Tis a torus of complex dimension 1. For any (E C, we denote the class in CjG = T by [(]. We perform surgery on M as follows: Let S = {O} x T, W= D x T where pi = C U {co} and OED= {zECllzl <e}. T
w
s Figure 7
Then set W* = D x T = {z, [(*] I zED, [(*] T. Define/: W* - S*~ W - S as follows: /: (z, [(*])
where 0 <
Izi < e.
~
{z, [(*
E
T} and S*
+ (1/2ni) log z]},
= {O}
x T£ D x
3. (z,
CONSTRUCTION OF COMPACT COMPLEX MANIFOLDS
17
Then f is biholomorphic and we can form M* = (M - S) u En) = (z, [C*J) if [n = [C* + (l/2ni) log zJ, 0 < Izi < e.
W*, where
REMARK. For the first Betti numbers b l we have b 2 (M) = bz(T) = 2, but bl (M*) = 1. In fact, M* is topologically homeomorphic to S3 x Sl. Proof H 2 (M, Z) ;;; 7L Ef) 7L is clear by the Kunneth theorem. To study M*, first we notice that M - W = (pI - D) x T is homeomorphic to ~ x T where ~ is a closed disk, and T is homeomorphic to Sl x Sl. If C= x + yw, we can identify [n with (x, y), where x + I is identified with x, y + I with y, where x and yare real (E IR). Therefore W* = D X Sl X Sl, M - W = ~ X Sl X SI. Since we are only interested for the moment in the topological type of M* we may as well assume that D is the unit disk and that the identification in the definition of surgery takes place on the boundary of D = {e i8 1 0 :::;; () :::;; 2n}. Then we identify (w, x*, y*) and (w, x,y) if x = x* + «(}/2rr), y = y*. Hence, M* = B X Sl where B is a circle bundle over 8 2 ; and in fact, we easily see by the transition function that this is the Hopf bundle S3 -+ 8 2 • Hence B = S3. This proves that M* = S3 X SI; b l (M*) = I follows.
EXAMPLE 3. We mention also the classical quadric transformation (blowing up, a-process). First we discuss the case where M has complex dimension 2. Let S = p be any point on M, and let S* = !pI be a copy of the Riemann sphere. We define M* = (M - p) u !pI as follows: Choose a coordinate patch W = {(Zl' zz)llz11 < e, IZzl < e} in a neighborhood of p so that Zl(P) = zz(p) = O. We define a submanifold W* of W x !pI as follows:
W* = {(Zl' Zz; C1' Cz) E W x
!pI 1 ZI (z
- Zz Cl = O},
where (C1 Cz) are homogeneous coordinates on !pl. W* is a submanifold since (0J/OZ1) = (2' (of/ozz) = - Cl iff = Zl(Z - Zz Cl' and hence [(Of/OZ1)' (of/ozz)J "# (0, 0). Let f*: W* -+ W be the restriction of the projection map W x !pI -+ W to W*. Then W* 2 0 X 1fD1 = S*, f* : S* -+ P = (0, 0), and f*: W* - S* -+ W - pis biholomorphic. The first two statements are obvious. For the proof of the last, let (Zl' zz; (I' Cz) rt S*. Then at least one of Zi =F 0 and hence (Cl' Cz) is determined by (Zl' zz) f* -1: (Zl, Zz) -+ (Zl' Zz; Zl' zz). By surgery we obtain M* = (M - p) u !pl. We make the following definition: DEFINITION 3.4. The quadric transformation Qp with center p is the manifold Qp(M) = M*. REMARK.
QPm'" Qp,(!P 2) can be complicated! For example, Qp6 ... Qp,(!p Z) = {C 1(~
+ (~ + (~ + C~ = O}
£; !p 3 ,
18
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
For manifolds M of dimension
~
2 we proceed analogously. If
dime M = n, let (ZI' "', zn) be coordinates centered at pep = (0, "',0)]. If W = {(ZI' "', z.)llz",1 < I $ r:J. $ n}, we set W* = {(z, 0 I Zl C - ZV C;.. = O,i $ ..1., v $ n} ~ W x \pn-I. Again W* is a manifold, projection onto W defines a biholomorphic map W* - \pn-l -+ W - p, by (z, 0 -+ z. We form M* = (M - p) u W* = (M - p) u \pn-I and call M* = QiM) the quadric transform of M with center p. E,
4.
Analytic Families; Deformations
Consider a torus Tw = CjG, G = {mw + n I m, n E 71. 1m w -+ O}. We have a family of tori depending on the parameter w. Many examples of compact complex manifolds depend on parameters built into their definitions. We also have the examples of hypersurfaces of degree d in \P n• Each such surface Md= {ClJCO =O} is defined by a functionJof the form J=Lko+"'+kn=d ako ... k.. (to ... C~n. In a sense to be made precise M d depends" analytically" on the coefficients ako ... kn off We make the following definition: 4.1. Let B be a (connected) complex manifold and let {M t I t E B} be a set of compact complex manifolds depending on t E B. We say that M t depends holomorphically (or complex analytically) on I and that {M t I t E B} forms a complex analytic Jamily if there is a complex manifold ..It and a holomorphic map (jj onto B such that DEFINITION
= M t for each lED, and the rank of the Jacobian of (jj is equal to the complex dimension of D at each point of ..It. (I)
(jj-l(t)
(2)
We note that (2) implies M t is a complex submanifold of ..It. Now for some examples. As before, we denote Tw = CjG, G
= {n + mw I n, m E 71., 1m w > O}.
Let D={wIIm w>O}cC. Let t#={gmnlgmn:(w,z)-+(w,z+mw+n)}. Then '!J is a properly discontinuous group of transformations on B x C without fixed point. Hence, ..It = B x Cj'!J is a complex manifold. The projection map B x C -+ D induces a holomorphic map ..It ~ B, and ir I(W) = Tw' It is easy to see that the Jacobian condition is satisfied so {Tw 1 wED} forms a complex analytic family. But suppose we proceed as follows: Again Tw = CjG and the map C -+ CjG is written Z -+ [z]. Let D = unit disk = {tli/l < l}. On D x Tw consider the group '!J = {t, g} where 9 :(t, [z]) -+ (-t, [z + 1]) is of order 2.
4.
ANALYTIC FAMILIES; DEFORMATIONS
19
Then ~ is properly discontinuous and has no fixed points so D x Tw/~ is a complex manifold. Let 11:: D x Tw ---+ D be defined by (t, [z]) ---+ t = t 2 • Then the diagram
commutes so 11: defines a holomorphic map on .4t. The Jacobian condition is not satisfied by Jr, since (at/at) = 2t = 0 at t = O. We notice that Jr-l(t) = Tw if t i= 0, but Jr -1(0) = T*, a torus of period ro/2. DEFINITION 4.2. Let M, N be compact complex manifolds. M is a deformation of N if there is a complex analytic family such that M, N ~ {M t It E B}, that is, M to = M, M tt = N. We have the following sequence of problems to guide our work: PRO BLE M.
Determine all complex structures on a given X.
PROBLEM.
Determine all deformations of a given compact manifold
M. PROBLEM. of a given M.
(easier?)
Determine all "sufficiently small" deformations
DEFINITION 4.3. We say that all sufficiently small deformations have a certain property f!jJ if, for any complex analytic family {M t I t E B} such that M to = M, we can find a neighborhood N, to ENe B such that M t has f!jJ for each tEN. By standard techniques in differential topology we prove the following theorem: THEOREM 4.1. Let M t be a complex analytic family of complex manifolds M t • Then M t and M to are diffeomorphic for any t, to E B. Proof The reader will notice that we really only use the differentiability of the map Jr:.!It ---+ B, analyticity is not needed. In fact, we prove: Let A be a differentiable family of compact differentiable manifolds such that the differentiable map 11:: .4t ---+ B has maximal rank (A and B are differentiable manifolds). Then M t is diffeomorphic to M to '
20
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
First we construct a Coo vector field 0 on a neighborhood of M to in .I( such that n induces n*(0) = a/as, where s is a member of a coordinate system (s, x 2 , ••• ,xm) in a neighborhood of the point to E B chosen as follows:
Figure 8
We connectt o and tby an embedded arcy: (-e,1 + e) -+ {yes) Is E( - e,1 + e)}. A compactness argument shows that we can assume that t and to lie in the same coordinate patch and since y is an embedding we can find a chart with coordinate (s, t 2 , ••• , t m) around (o(to = (0, ···,0), t = (s, 0, ···,0». Because of the rank condition, n- 1 (y)=n- I {(s,O,···,0)I-e<s
Li
d ds xj(r)
= 0j[x(r)],
(1)
where 0j is the a-component of 0 in the coordinate patch ilIi j ' with initial conditions xj(O) = ylX, where (0, y2, ... , yn) is some point close to (0, ... , 0). If S is small enough and Iyl is small enough, Equation (1) has a unique solution xj(r, y) on some small interval. By compactness, we can assume that M ta C U jilli j ' a finite union of such patches, and that in each ilIi j ' (1) is satisfied for Irl < Jl where Jl is independent of j. If xj(t, y) is such a solution, let Xj = fik(Xk) and define ef[r,hlO, y)] uniquely on ilIi j n ilIik by (2)
Then dxj(r, y) =
dr
L oxj oef[r,jklO, y)] (I
oxf
or
'
4.
ANALYTIC FAMILIES; DEFORMATIONS
21
and by the uniqueness of the solution to (I) (3)
Equation (3) implies that x(-r, y), 1.1 < fl, y E MIO is a well-defined differentiable map defined on M ta for each " 1,1 < jJ., and x(O, y) = y. Let cP.(y) = xC"~ y); then CPo = id (on Mia)' It is also easy to check that n[cprCy)] = yeo) since n.(0) = dlds. Hence, CPT maps M ta into MY(T) (for small ,), We can repeat this argument for My(r) and define "'.: My(r) -+ My(r+v) and by uniqueness get '" _ r 0 CPr = id, CPr 0 ' " _ Tl = id. Since everything is differentiable, the theorem is proved. Q.E.D. REMARK. This argument is very old. For a treatment from the point of view of Morse theory, see Milnor (1963). Sometimes this theorem is attributed to Ehresmann (1947). We consider some more examples of complex analytic families. The dependence of the complex structure of M I on t E B can be complicated as we shall see. EXAMPLE I. Consider again the family of tori {T", I (V E H} where H = {w 11m w > O} and T", = CjG, G = {mw + n I m, n E l}. From the classical theory of Riemann surfaces we see that T", and T",. are conformally equivalent if Wi = (aw + blew + d) where a, b, e, dEl, and ad - be = l. Let <§ be the group of transformations acting on H which have the form
aw ew
+b + d'
w-.+---
a, b, e, dEl,
ad - be = 1.
Then it is easily seen that <§ is properly discontinuous on H. A fundamental region :F for <§(ug:F = H, g:F n :F = l/J if 9 # id) is given by the shaded region in the figure below, hence T", # T",., if w # Wi and w, Wi E:F.
Figure 9
22
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
The elliptic modular function J defines a conformal map J: HjC§ --. C. So Tw = Tw' if J(w) = J(w'). EXAMPLE 2. (n-dimensional tori) We give an outline of some of the facts. A torus Tn = en jG where G = (L;~ I m j W jim j E Z} where WI' ••• , W 2N are complex n-vectors linearly independent over R (a) We can replace {Wj} by any other linearly independent basis of G. That is, 2n
w}), =
L a jk Wu ,
(4)
k=1
where ajk E 71., det (ajk) = 1 are also permissible generators of the lattice (group) G. (b) We may also introduce new coordinates in en so z), --. 2)" where n
2;, =
L Z"Yv)" v= I
Yv),
E
C,
det(yv),)
=1=
O.
Then,
(5) The resulting change from Equations (4) and (5) becomes
(6) We may assume that Wn + I ' ... , W2n are C-linearly independent. Hence by some change of coordinates (y.),), we can obtain
." w) (lOll In (Yv),) = Wnl
.. ,
W
2nn
(7) I
where I is the n x n identity matrix. So we may assume (Wi) =
(~),
where 0 = (Wi) 1 ::; i, j::; n and 1=
(~ ...~). (c)
We can also break (ajk) into pieces:
Then (4) takes the form
wi)' = (a jk)(~) =
(gD,
o~
= AO + B, O 2=
co + D.
4. If one assumes that
23
ANALYTIC FAMILIES; DEFORMATIONS
n; is invertible, then (gD(n;)-l n' =
(An
=
(~')
where
+ B)(Cn + D)-l,
(8)
det(~ ~) = l. The following treatment will be a bit sketchy; for more details consult Kodaira-Spencer II (1958). The fact that Wl' " ' , W n , (1, 0, "', 0), (0, ... , 0, 1) are real linearly independent implies (0, 1, 0,
°,), ...
which is the same as (2ir det [Im(w p ,)] =i' 0. Consider the space H = {n det(lm n) > o} [some sort of a generalization of 1m W > in Example (1)]. Let CfJ = the set of all transformations
°
n -+ (An + B)(Cn + D)-l = n', where
(~ ~) E SL(n, Z), the invertible integral matrices of determinant + 1.
This group does not really act on H since it is possible for cn + D to be singular; one should consult Kodaira-Spencer for more details. H should be extended to something more general on which SL(n, l) acts. In any case,
Tn = Tn-,
ifn'
= gn, 9 E r§.
We would like to form H/r§. But it turns out that CfJ is not discontinuous. In fact, for any open set U c H, there is a point n E U such that {gn I9 E CfJ} n U is infinite. Hence, the topologial space H/CfJ with the quotient topology is not Hausdorff and hence certainly not even a topological manifold by the usual definition. We next give some examples of families {M t It E B} such that M t = M for t =i' to and M to =i' M. EXAMPLE 3. A Hopf surface is a compact complex manifold of complex dimension two which has W = 1[2 - {(O, O)} as universal covering surface. More precisely, the Hopf surface M t is defined by M t = WI Gt where Gt =
{gm I mEl} and g: (Zl' where
°< lal < I and t
LEMMA 4.1.
{M t It
E
Z2) -+
E
(az 1
+ tz 2 , az 2 ),
that is, (::)
-+
(~ :)(:~),
C. Then M t is a compact complex manifold.
C} is a complex analytic family.
24
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
Proof
= {M t 1 t E C} = (: x W/f, where f = {I'm 1m E~}, and
v(i;) ~ (~ ~ DGJ
Q.E.D.
We claim (I) (2)
M t = Ml (complex analytically) for t Mo -=1= MI'
Proof of (1).
-=1=
O.
We make the following change of coordinates:
Then the equation
implies that Ml = M t when t Proof of (2).
-=1=
O.
First we prove a special case of Hartog's lemma.
LEMMA 4.2. Any holomorphic function defined on W = (:2 - {CO, O)} can be extended to a unique holomorphic function on (:2. Proof Let f(zl' define the function
Z2)
be the function on W. Pick a number r > 0, and
F(zl'
1 1
few,
Z2)
'j dw, 2m Iwl=r w - ZI
Z2) = - .
for IZII < rand Z 2 arbitrary. Then F(zl' Z 2) is an analytic function in its cylinder of definition which is a neighborhood of (0, 0). If we can prove f = F where both are defined, we will be finished. We know that few, Z2) is holomorphic if Z2 -=1= o. So Cauchy's theorem gives for IZll < r, Fix ZI' 0 < IZII < r. Then F(zl' in Z2; therefore,
Z2)
= f(zl'
Z2)
for
Z2
Z2 -=1=
=I-
o.
o. Both are analytic
F(zl' 0) = f(z(, 0). Hence they agree where defined, proving the lemma. Now let us suppose M, = Mo. I =r= o. Then there is a biholomorphic map --+ Mo· Wis the universal covering manifold of M, and M o , sofinduces
f: M t
4.
25
ANALYTIC FAMILIES; DEFORMATIONS
a map I: W -+ W which is biholomorphic, such that
W~W
commutes. It follows that Gr = 1-' Go! Hence for generator g, of G"
g, =1-'g"5'!
(9)
Write the map I in coordinates as
I(z, , Z2) = Chez"~ Z2)'/2(Z" Z2)]. Then by Hartog's lemma extend liz" Z2) to a holomorphic function F;. (z" Z2) on C 2 • Then F maps C 2 into C 2 [F = (F" F2)J, and F(O) = O. For if not, extend 1-' to F which satisfies F'[F(z)] = z on Wand by continuity, F[F(O)J = O. But if F(O) '" 0, F[F(O)] = 1-' [F(O)J '" O. This contradiction gives the result. Now expand F;.,
F;.(z, , Z2) = F;.,z,
+ F).2Z2 + F;'JzT + F)..Z,Z2 + ....
We know that/[gr(z)J = g"5 1 [fez)] so
( 0)
F[grCz)J = ~ a
±,
F(z).
Rewriting this gives
F,(az, +lz 2 ,az 2 ) =a±'F,(z"Z2), F2(az, + tz 2 , O:Z2) = a±'F,(z" Z2)' Expanding these and taking the linear terms yields
( FI1 F , 2
I)
F'2)(a = F22 0 a
(IX
0
O)±' a
This can only happen when t = O. Hence M, '" Mo.
Q.E.D.
EXAMPLE 4. Ruled Surfaces (examples of surgery) will be !P' bundles over !P'. Let !P' = {( I( E C U {oo}} coordinates). M(m) = V, x !P' U V 2 x.!P' where V, u V 2 = !P' - {O}, and identification takes place as follows Let (z" (,) E V, x p', (Z2' ~2) E V 2 X pl. Then
REMARK.
M(III) ' " M(I)
Our ruled surfaces (nonhomogeneous V 2 = p', V, = C, (recall Section 3):
for m '" {' (not to be proved now).
26
DEFINITIONS AND EXAMPLES OF COMPLEX MANIFOLDS
THEOREM 4.2. M«() is a deformation of M(m) if m - t == 0 (mod 2). Assume that m > f. Then there is a complex analytic family {M tit E C} such that Mo = M(m) and M t = M(f) for t '# 0.
= VI
where (ZI, (I) <->(Z2, (2) if ZI = l/z 2 , (I = Z;(2 + IZ~ where k = 1(111 - t). Then it is easy to see that {M tIE C} is a complex analytic family and that M 0 = M(m). Suppose t '# O. Introduce new coordinates on the first [PI by
Proof
Define M, as follows: M,
C1-On the second
V2
X [pI
k, ZI
Ie
I
I -
(linear fractional transformation).
[pI,
r'
':.2
Then, using
X [pI U
ZIZ2
= I, and
(I
'2 +
= I~m-kv "'"2
1,2
12'
= Z~'(2 + tz~, we get r' m- 2/.. ... , ':.1=Z2 ~2'
Hence, in the new coordinates,
Z1Z2
= I,
for
PROBLEM.
t
,~
=
i= O.
z~(;; so
Q.E.D.
Findapairofcomplexanalyticfamilies {Mtlltl < 1},{Ntlltl < l}
such that (a) (b)
Mo '# No,
(c)
Nt = Mo
M,=No
for for
1'#0, 1'# O.
(not complex analytically homeomorphic)
There are no known examples of this type.
[2] Sheaves and Cohomology I.
Germs of Functions
Let M be a complex (or differentiable) manifold. A local holomorphic (differentiable) function is a holomorphic (differentiable) function defined on an open subset U s; M. We write D(f) for the domain of f Let p E M and suppose given local functions f, 9 such that D(f) n D(g) 3 p. We say that f and 9 are equivalent at p if fez) = g(z) for z E W S; D(f) n D(g), Wa neighborhood of p. By a germ of a function at p we mean an equivalence class of local functions at p. Denote by fp the germ off at p, (!J p the set of germs of all holomorphic functions at p, and:?) p the set of germs of all differentiable functions at p. The definitions
afp
+ flgp = (af + flg)p
a, fl
E
C,
fp' gp = (fg)p, are well defined, hence,
(!) p'
:?) p become linear spaces over C. We also define,
(!) =
U (!)p,:?) = U :?)p. peM
peM
We put a topology on (!J and:?) as follows: Take any cp E (!J (or :?); then cp E (!J p (or ~ p) for some p. Take any holomorphic (differentiable) f with fp = cp and define a neighborhood of cp as follows:
where p E Us; M, U is an open set in D(f). It is easy to see that the system of neighborhoods iJ7J(cp;f, U) defines a topology on (!) (or :?).
EXAMPLE. (!) on the complex plane C. Let p E C. Then if f and 9 are holmorphic at p we have expansions valid in some neighborhood of p, 00
I(z)
= L Ik(Z -
00
p)k, g(z)
= L gk(Z -
k=O
p)\
k=O
so f and 9 are equivalent at p if and only if fk
9k for all k. Hence, the germ at = ring of convergent power series. And an element cp E (!) p can be represented by cp = fp = {p; fo ,fl' ... } where limk_oo Ifkll/k < + 00 and the radius of convergence is r(cp) = II lim. =
p is represented by a convergent power series;
27
(!J p
28
SHEAVES AND COHOMOLOGY
We define ~(q>;
{t/!I t/!
s) =
= f q, Iq - pi
< s where 0 < s < r(q>)}.
In terms of our representation we calculate 00
fez) =
I
OQ
fk(Z - p)k =
k=O
I
fm(Z _ q
+ q _ p)m
",=0
Hence
~(q>; s) = {"'It/! =(q; go, ... , gk' ... ), I(q gk
=
m~J;)fm(q -
-
p)1 < s
p)m-k}.
We note that t/I E If//(q>; e) means that t/I is a direct analytic continuation of q>. The case of E0 on IR is not so simple. If q> = fp where f is a Coo function at p, In
j(x) =
I
fk(x - p)k
+ O(x
_ p)m.
k=O
But f is not determined by the fk'S since there exist Coo functions f which are not identically zero, but which have all derivatives zero at some point. Define w: (!) (or E0) -. M by w«(!)p) = p. (1) wis a local homeomorphism (that is, there exists ~ such that w: ~(q>;J, U) -> U is a homeomorphism). (2) w-l(p) = (!)p (or E0 p) (obvious). (3) The module operations on w-l(p) are continuous (that is, IXq> + IN PROPOSITION
1.1.
depends continuously on q>, t/I).
Proof (1) ~(q>;J, U) = {fqlqE U} and w:!q ->q is certainly 1- 1. It is obvious that wis continuous. To show that w-1 is continuous, let ~(w; g, V) be a neighborhood of t/! = f . We want to find a neighborhood W of q so • that w = W-I(W) E ~(t/!; gY) for wE W. We know that gq = t/I = f q, so f and 9 are equivalent at q. Hence, f = 9 in some neighborhood N of q. Let W = N n V. Thenfw =gw on W, so fw E ~(t/I; g, V) for WE W. This proves that the w- 1 is continuous. (3) Let q>=fp,t/!=gp. Then IXq>+f3t/!=(rx/+f3g)p. Let ~(IXq>+f3t/!; h, U) be a neighborhood of IXq> + f3t/!. Then IXq> + f3t/! = hp = (IXf +f3g)p so
r
1.
29
GERMS OF FUNCTIONS
h =af + fJg in some neighborhood V£ U of p. Then if (J E cf/1(cp;f, V), , E cf/1(l/J; g, V), we have a(J
+ f3,
+ fJgq = (af + fJg)q
= afq
= hq E cf/1(acp Since cf/1(acp
+ f3l/J; h, V)
£ cf/1(acp
+ f3l/J; h,
V).
+ fJl/J; h, U) we are done.
Q.E.D.
We now give a formal definition. Let X be a paracompact Hausdorff space. A sheaf!/ over X is a topological space with a map w: X onto X such that
DEFINITION
!/ --t
1.1.
(1) wis a local homeomorphism [that is, each point s E !/ has a neigh. borhood cf/1 such that w: i1lt --t w( cf/1) C X is a homeomorphism onto an open neighborhood of w(s)]. (2) w-lex), x E X is an R-module where R = 7l., IR, 1[:, or principal ideal ring. (3) The module operation (s, t) --t as + f3t is continuous on W-I(X) where a, f3 E R.
(The reader can easily generalize this definition, but for our purposes it suffices.) The set!/x = W-I(X) is called the stalk of!/ over x.
EXAMPLES. (1)
(2) (3) (4)
(of sheaves)
on a complex manifold. on a differentiable manifold. The sheaf over X of germs of continuous (IR. or I[: valued) functions. The sheaf over X of germs of constant functions. (!)
~
In Example (4) !/ = X x I[: with the following topology: Let s = (x, z); then cf/1(s) = {(y, z) lyE U, z fixed}. If r --t f(r) is a continuous map into !/ of I = {ria < r < b}, thenf(J) = {(y, z) Iz fixed andy = w(j(r»rE l}. In other words we give X x I[: the product topology where X has its given topology and I[: has the discrete topology. DEFINITION 1.2. Let U be a subset (usually open) of X. By a section (] of !/ over U we mean a continuous map x --t (](x) such that w(](x) = x. Suppose X = M, a complex (or differentiable) manifold; and suppose !/ = (!) (or ~). If fez) is a holomorphic (or differentiable) function on U, then (]: p --t fp, P E U is a section.
30
SHEAVES AND COHOMOLOGY
1.2. Let a: V -+!J? be a section (!J? as above). Then a determines a holomorphic (or differentiable function) 1= I(z) on V such that
PROPOSITION
u(p) = Ip.
Proof a(p) E (1)p (or £2,,). Hence there is a holomorphic (or differentiable) g(z) defined on some neighborhood of p so that a(p) = gp' Since g depends on p we write, g(z) = gCP)(z). Define I as follows: I(p) = g(")(p). Then I is obviously well defined. Then (1)
I(p) is a holomorphic (differentiable) function on U.
Proof Take Wa neighborhood of p, W ~ V. Let I1IJ = l1IJ[a(p); g("), W] {(g("\ Iq E W}. Since a is continuous, for any small neighborhood N of p, N ~ W, we have u(N) ~ illt. Hence u(q) = (g(P\. But we also know u(q) = (g(q»q. Thus, (g(q»q = (g(P\, and g(q)(z) = gC")(z) for z in a small neighborhood V of q, V c N. But I(q) = g(ql(q) = g(Pl(q) for q E V. So I(z) = g(P)(z) for z E V and g(P) holomorphic (or differentiable) in V implies that I is also. =
(2)
By definition u(p) = (g(P»p = Ip for each p E V.
Q.E. D.
Hence we have the maps: local holomorphic (differentiable) functions ~ germs ~ sections = holomorphic (or differentiable) functions. f( V, !J?) will denote the R-module consisting of all sections of !J? over U. We remark that f( V, (1) are all holomorphic functions over V and f( V, f:&) are all differentiable functions over U. Let {V ).11 ::;; A ::;; n} be a finite family of open sets in X such that n V). =1= 4>. Let u). E f( V). ,!J?) and IX). E R. Then L IX). a). E f(V,!J?) where V = n V).. Let W be an open set and u E f( V,!J?) for some open set V. Then x -+ O'(x), x E W n V defines a section of f( W n V, !J?). We denote this section by rwa and call it the restriction of 0' to W n U.
2.
Cohomology Groups
Let X be a Hausdorff paracompact space and let !J? be a sheaf over X. Fix a locally finite covering ()li = {V j } of X. A O-cochain CO on X is a set CO = {a j } of sections 0') E f( Vj ,). A I-cochain C l = {O'jk} is a set of sections
2.
q
COHOMOLOGY GROUPS
31
Uk ,.9') such that O"jk = - O"kj (skew-symmetric). A q-cochain is a set of sections O"jo .. ·jk E qUja II ... II Ujk ,9') which are skew-symmetric in the indices Jo ... Jk. Let C(JU) be the R-module of all q-cochains. We define a map C(JU)~Cq+l(JU), the coboundary map as follows: For O-cochains, JCo = {'rjk} = {O"k -O"j} where CO = {O"k}; for l-cochains C l = {O"jd, JC l = {rjk(} where 'rjk( = O"k( -O"j( + O"jk = O"jk + O"u + O"(j. In general, Jcq = {rjo ... j.+ ,} if Cq = {O"jo ... j.}, where O"jk E
Uj
II
cq = {O"ja ••• Jk}
+ ( -l)q+ l(Jjo ... j. = L (_l)kO"jo··· j~ ... jq+,' where
A
(1)
means "omit."
We denote the q-cocycles by zq(JU) = {C q I JC q =O}.
The q-cohomology group (with respect to JU) is Hq(fI) = Well, 9') = zq(Olf)/JCq-l(Olf).
(2)
We should remark that JC is always skew-symmetric and JJ = 0 so that JCq-l(Olf) £; zq(JU) and Equation (2) makes sense. The qth cohomology group of X with coefficients ill the sheaf 9' is defined to be Hq(X, 9') = lim Hq(Olf, 9'). Vfi
This limiting process will now be explained. We say that the open covering "f" = {VJAeA of XisarefillementofJU = {Uj}jeJ ifthereisamaps:A -+ Jsuch that V). c: Us ().) = Uj ().) , where we setJ()") = s(),.). We define a homomorphism n~~
n~:
: (q(OU) -+ Cq("f"),
{O"jo ... j.} -+
{r Ao ···;.• },
where
(3) It is easy to check that
(4) so that n~ maps Zq(Ulf) into zq("f") and JCq-1elf) into JCq-l("f"). Hence n~ induces a homomorphian n~: Hq(~71) -> Hq("f"). LEMMA 2. J.
n$: Hq(:Jlf) -> W("f")
is independent of the choice of map
s : A -> J in the definition of refinement.
32
SHEA YES AND COHOMOLOGY
Proof
First some notation: fix indices ao, ... , aq E A. Let V = V,~ n'" n V,~, V t
= V,
~
.A.
n'" n V,~ n'" n V,~,
/'-.... /'-.... Uj( = UJ(ao) n ... n U naj ) n Ug(a J) n ... Ug(at) n .,. n Ug(a q) ,
and Ui
=
UJ(a,) n ... n UJ(a j ) n Ug(a J) n ... n Ug(a q )
,
where/, 9 : A -+ J are two refining maps. Define a function (kU)A""A q by q
(kuh, ... Aq
=
L (-1 y- I rv ocrJU,)' .. J(Ap)g(A p)"
.g()'q)
(5)
p~O
Let us call the maps n~, defined by f and g,J*, and g*. We claim that the following equation holds: (6)
The function kcr is not necessarily skew-symmetric in its indices; so we skew-symmetrize
-r~t ... ). = (k'-r)A, .. , A =~, L sgn(AI111 q. q
q
Next we use (6) to see that [(Dk'
+ k'D)U]aO"'a q =
(g*cr - j*cr)ao"'aq '
Hence, if Dcr = 0, Dk' cr = g*cr - f*u E DCq - I (1""). Hence,f* and g* induce the same map, Hq(o/i) --+ W(1""). Therefore we prove (6). The reader can easily check the following calculations:
=
t (- 1/ [ttl (
+
crJ(ao) ... J(ajg(aj) ... g(a;) ... g(aq)
i=O
t (-
i~(+
(Dkcr)ao'" aq =
-1)ir v t
rv
t~O
1)i - I rVI cr J(ao) ... j(;;)J(aj)g(aJ)'" g(a q)]
I
L (-1)(+
j
rv UJ(ao) '"
j«
J(aJ)g(aJ)'"~)''' g(aq)
+ 'L ( -
1)(+j+1 rv
= L (-
l)i+t rv cr J(.o) '"
crf(ao) .. , J(at} ... f(aj)g(a J ) ••• g(aq) .
(7)
j>t
Similarly, (kDcr)ao'" a.
j(;;) .. ,J(aj)g(aj}
., g(a q)
lsj
(8)
33
COHOMOLOGY GROUPS
2. Equations (7) and (8) give
q
- L ry
(1/(ao) "'/(aj)g(aj) '" g(aq)
j=O
= ry (1 g(ao) ," g(aq)
-
rv (1/(ao) " '/(aq) ,
(9)
Q.E.D.
proving Equation (6).
Knowing that the map n~ depends only on r1IL and "1"', we proceed to the definition of the limit. We write r1IL < "fr if "fr is a locally-finite refinement of r1IL. Then < is a partial order and given r1IL, "I'" there is "fr so that r1IL < "fr and "I'" < "fr. Hence the set of all locally finite coverings of X forms a directed set with respect to <, and the following equations can be verified (using Lemma 2.1) : n~ = id,
n~ = n~ DEFINITION 2.1.
0
n~,
if r1IL < "I'" < "/fl.
Hq(X,!/) = lim Hq(r1IL, !/). Oft
REMARK.
We recall the definition of the limit lim. We say that g, hE Hq tIJi
(r1IL, !/) are equivalent if there exists "fr> r1IL such that n~ 9 = the equivalence class of 9 by
n!, h.
Denote
g. Let
Hq(r1IL, !/) = {g I9 E Hq(lJlt, !/)}. The map 9 ~
g defines a homomorphism Il tlJi , n~:
Hq(r1IL, !/) ~ Hq("I"', !/),
and n~ induces a homomorphism TI~, n~
TI~ is injective.
LEMMA 2.2.
Proof
o and g =
: Hq(lJlt, !/) ~ HQ("I"', !/).
TI~g
O.
= 0 if and only if n~
0
n~ 9 = 0 for some W. So n~ 9 =
Q.E.D.
Hence, identifying HQ«()71,!/) with n~HQ(r1IL, !/), we may consider Hq(r1IL, !/) c HQ("I"', !/) provided that r1IL < "1"'. Then by definition,
W(X, !/) =
UHq(r1IL, !/), tIJi
34
SHEA VES AND COHOMOLOGY
and nUll: Hq(uu, f/)
--+
Hq(uu, f/)
£;
Hq(X, f/) is a homomorphism of
Hq(dI!, f/) into Hq(X, f/).
PROPOSITION
Proof
2.1.
HO(X, f/)
= reX, f/).
By definition C- I = 0 so H°(OlI, f/) = ZO(UU, f/).
= [0' 10' = {O'j},O'j E rcUj,
ZO(UU, f/)
f/),
DO'
= OJ.
But (j0' = 0 means O'iz) - O'k(Z) = 0 on Uj n Uk' Hence O'(z) E reX, f/), defined by O'(z) = O'/z) when Z E Uj , is meaningful. This proves HO(dI!, f/) = reX, f/) and implies HO(X, f/) = f/). Q.E.D.
rex,
PROPOSITION
COROLLARY.
2.2.
Ho/I: Hl(UU, f/)
H'(X, f/)
--+
HI(X, f/) is injective.
= U Hl(UU, f/). 0/1
Proof (of the proposition). Suppose hE H'(Ulf, f/) = Zl(Ulf)jDCO(i1l/). Then h = {O'ld, O'jk Ere Uj n Uk' f/) where O'ij + O'jk + O'kj = O. We want to show that no/lh = 0 implies h = O. no/lh = 0 means Ii = 0 and this is true if and only if n~h = 0 for some "Y, "Y > Ulf. Let "If! = {Wo.1 Wi;. = U j n Vl}' Then "If! is a locally finite refinement of "Y and n~h = n::;:. 0 n~h = O. Also "fII > Ulf since "fII jl c U j and we can use the maps(iA) = i in the definition of refinement. Then we have where
= 'iljll = "Wi" n Wj,. O'ij. Then n~ h = 0 implies {, jljll} = D{, il}, that is, 'iljll = 'jll - ' i l ' Since 'Wil = rW,,,nWi,,O'jj = 0, we obtain 'ill = 'U on W il n Will' U i = UlWil , and ' i = 'ilL on Will defines an element, jEre U i' f/). Then the equation 0' ij = ' j 'i implies h = O. Q.E.D. '(il)(jll)
Consequently, in order to describe an element of H'(X, f/), it is sufficient to give an element of Hl(Ulf, f/) for some Ulf.
EXAMPLE. dime HI(M, (1) Proof
Let M
=
{(Zl' z2)llzti < 1, IZ21 < 1, (Zl' Z2) i= (0, O)}. Then
= + 00.
Set
= {(ZI' Z2) I (Zl' Z2) EM, Zl i= OJ, U 2 = {(z" Z2) I (z" zz) E M, Z2 i= O}. UI
3.
35
INFINITESIMAL DEFORMA nONS
I n this case M = VI U U 2 so chose as covering Ill! = {V I, U 2}' Then HI(Ill!, &) = ZI(Oll, &)/tJCo(lll!, (9) where ZI(Ill!, (9) = {0"121 0"12 E nV I n V 2 , &)} ,CO(<J/, (9) = {T IT = (T I, T2), TILE n V Il' &)}, and tJCO(<J/, &) = {T2 - Til TIL E r(VIl' (9)}. We note that V, n U 2 = {(ZI' a Laurent expansion for 0"12
Z2)
+00
O"dz) =
L m=-co
10< IZII < 1,0 < IZ21 < 1}, so we have +00
L
"=-00
amnZ~Z2'
'I IS holomorphic on VI = {(ZI' Z2) 10 < IztI < I, IZ21 < I} so T,(Z) = L'~~-<XlL:=obmnz'rz~. Similarly for '2' T2(Z) = L~=oL:=<Xl-<Xlcmnz7z2' and T2 - TI = Lm?;Oorn?;O amn Z7 z2' Then H'(Ill!) ~ {0"121 0"12 = L;;;l-<Xl L;: - <Xl amnz'rz;}. Hence dim H'(OlI, f/') = +00 and since HI(<J/, f/') S; H'(X, f/'), dim HI(X, f/') = + 00. Q.E.D.
2.3. If H'(Vj,f/') =0 for all H'(X, f/') where <J/ = {V j }.
PROPOSITION
VjEIll!, then H'(<J/,f/')~
Proof We already know that H'(Ill!, f/') s; H'(X, f/'). Hence we only need to show the following. Let 1/ = {V).} be any locally finite covering. HI(Oll) -+ Let 1f/ = {Wj).1 Wj). = U j n V;.}. Then it suffices to show that H'("If/") is surjective. Take a l-cocycle {O"P,kv} of Hl(1f/) where O"Ujll + O"jllkv + O"kvi). = O. Then {O"WIl} for each fixed i is a I-cocycle on the covering {Wi).} of Ui' Since HI(U;, f/') = O,HI({Wi).},f/') s; HI(V;, f/')givesHl({Wi)'}, f/') = 0 for each i. This implies the existence of Ti). E r( Wi)., f/') such that 0" WIl = Till Ti).. Let, be the O-cochain {Ti).} on "If/". Then {O";)'kJ = {O"i).kv} - (;, defines a I-cocycle on "If/" which defines the same cohomology class in HI ("If/") as 0". From the definition of T we see that O";).ill = O. So O"i).ill + O"illkv + O"~vi)' = 0 yields O"illkv = O"i)'kv' Similarly, 0" i)'k. = O";llkw' Hence, O"ik = O"i)'kv = O"ivkv' and O"ik E r( Vi n Uk, f/'). Now we have found O"ik so that n~'(O"ik) = O";)'k., and {O"i)'kv} is cohomologous to {O"mv}' Hence n~ is surjective. Q.E.D.
n:- :
3.
Infinitesimal Deformations
Using cohomology groups we will give an answer to the following problem: Let .A = {M r/t E B} be a complex analytic family of compact complex manifolds Mr and let t = (tl, "', tn) be a local coordinate on B. The problem is to define (aMr/at·). For this we define the sheaf of germs of holomorphic vector fields. Let M be a complex manifold and let Wbe an open subset of M. Let <J/ = {Vj, z)
36
SHEAVES AND COHOMOLOGY
be a covering of M with coordinates patches with coordinates p --+ Zj(p) = [zf{p), ... , z7(p)]. A holomorphic vector field 0 on W is given by a family of holomorphic functions {OJ} on W 11 U j where
a L (}j(p)-ex a= OZj n
() =
I
on W 11 Uj • These functions should behave as follows: On W
Uk,
0
n
() =
11
L O~(p) OZk
-{J •
{J=I
We want
so the transition equation (1) should be satisfied on W 11 Uj 11 Uk. Thus we have a definition of local holomorphic vector fields and we can define germs of local holomorphic vector fields. As notation we denote by E> the sheaf over M of germs of holomorphic vector fields. (Later we shall give a formal definition of the holomorphic tangent bundle of a complex manifold.) Next we want to define the infinitesimal deformation (aM rlotJ. First we consider the case B = {tlltl < r} ~ C. vi! is a complex manifold and w: vi( --+ B is a holomorphic map satisfying the usual conditions
Mr=w-I(t); (2) the rank of the Jacobian of w = 1 = dim B. We can find an B > 0 small enough so that w-I(Ll), Ll = {tlltl < e} looks as follows: (1)
J
w-I(Ll) =
UlJlI j= 1
j
(a union of a finite number of open sets).
On each OIJ j there should be a coordinate system
p --+ [zJ(p), ... , zj(p), t(p)], where t(p) = w(p) and such that OIJ j = {pllzj(p)1 < Bj. It(p)1 < B}. We write p = (Zj' t) = (zJ, ... , z'j, t). This construction is possible because rank =I These charts are bolomorphically related so
w
zj(p) = fjk[Z~(P)' ... , z~(p), t(p)] on Uj
11
Uk. Let Urj
=
Mr
11
IJlI j , It I <
B.
= fjk(Zk, t)
Then set
{(zJ ... , zj, t) IIzjl <
Bj }
=
U rj ,
3.
INFINITESIMAL DEFORMAnONS
37
J, ... ,
so we can use {(z zj)llzjl < eJ } as coordinates on Utj. The transformation zj = fjk(Zk' t) depends on t. Consider p E t5Ii i n O!i j n t51ik • Then p = (Zi' t) = (Zj' t) = (Zk, t). So zf = f'U,(Zk' t) = frj(zj, t) = fiJfjk(Z) , t], where jjk = (fA, ... Jik). We set
(). ( Jk
p,
t)
= ~ afjk(Zk, t) ~. L.-
:It u
0<=1
;:) UZj
(l
Obviously (}jk(t) E r( Utj n Utk , 0 t) where 0 t is the sheaf of germs of holomorphic vector fields on M t •
Proof Before beginning the proof we remark that (}ij + (}jk + (}ki = 0 and (}ij = -(}ji is equivalent to (}ik = (}ij + (}jk. To prove the lemma we differentiate the transition equation to get affk = an~ at at
+
t
affj af~k fJ = 1 az~ at
Then
Q.E.D. *DEFINITION 3.1. (dMtldl) = {(}ik(P, I)} E H1(M" 0,). We have made several choices in this definition and we must justify them.
PROPOSITION
3.1.
(dM tldl) is independent of the choice of local coordinate
covering {zj}.
Proof Let {'t"".J be a locally finite refinement of {O!i j } such that are coordinates on 't"";. where 't"";. = {«(t, 1)1 "~I < eA.'
«(~,
I)
It I < e}.
Since {'t"";.} is a refinement of {t5Ii j } we have a map s: A.->J such that 't"";. s; t51is (;'). We also have holomorphic transition functions ({J A. I' where (~=({J;.i'v,t)
on
't"").n't""".
Then the cocycle defined by this covering is
38
SHEA YES AND COHOMOLOGY
As before s induces a map s*: {Ojd
O;,v(t) =
r;r;. rdrv ("\
~
{Olv}' where
M,[ejk(t)], j = SeA), k = s(v).
We must show that {'l;,v} is cohomologous to {G;,v}; that is, there exists a cochain {O/t)} such that
'l;'B(t) - O;,it) = Oy(t) - O;.(t). Since "f/"). s iJ{t j> j = seA), there is a holomorphic 9 j such that zj = gj(';" t) on The following equalities are clear:
"f/";,.
gj[q>;,.(,., t), t] = gj(C, t) = zj =
/'}k(Zk> t)
= /'}k[gk(C, t), t] on "f/"). n "f/" •• Differentiating we obtain
I
ogi oq>~v + agi = I a/}k agf + a/jk . a,~ at ot oze at at
(2)
Then (2) implies [multiplying by (a/ozj)]
0)
" azi ( ag~ "a/}k a " azj ( a ) aq>t L.. oze azj . at + L.. at azj = L.. o'~ azj
"ogj a
at + L.. at azj'
(3)
Hence,
'l;'v
[_a_J 0
[_0_]
= " ag~(\.) + "L.. iJg~().) "L.. p +).v, at OZS(;') at aZS(v)
(4)
on "f/";, n "f/" v' Therefore if we let
O;.(t) =
L ag~().) r~l' at
oZso.)
Q.E.D. So we see that the infinitesimal deformation, dM,/dt E H 1(M" 0,} is determined uniquely by the family vii = {M, I t E B} and is thus well defined. If we introduce new coordinates on B. t = t(s) so that t'(s) ¥ 0 then the relation (5)
is obvious. Now to return to the more general case, let {M, 11 E B} be a family where B is now a general connected complex manifold. Let Ll be a coordinate neighborhood around bE B and let (t 1, ..•• t m ) be local coordinates. Then
3.
INFINITESIMAL DEFORMAnONS
39
we may assume l:!. so chosen that w-I(l:!.) = U/'ltj' a union of finitely many coordinate neighborhoods on each of which there are coordinates (z), "', zj, tl, "', t m), where rJlt j = {(Zj' t)llzjl < Bj' tEl:!.}. Again we have transition functions fjk
zj = fjizk' tl, "', tm ) on rJlt j n rJlt k . DEFINITION 3.1. {8 jk I.(t)} where
(aMr/at') E HI(Mr' 0 r)
8.
Jklv
is the
cohomology
class
of
(t)= ~ afjizk' t)(~) ~t'
L...
«=1
~ « • uZj
u
If (a/at) denotes the tangent vector
a
m
a
at = V~I c. at
V '
then we define
aMr = ~ c aMr at V~I' at' . We make the following definition: DEFINITION 3.2. .A = {M r 1 t E B} is locally trivial (complex analytically) if each point b E B has a neighborhood l:!. such that w-I(l:!.) = Mb x l:!. (complex analytically). This means that we can choose coordinates (zj, t) such that, zj = fMzk' b) (independent of t). If .A is locally trivial, then each Mr is complex analytically homeomorphic to M 0; hence M r is independent of t. PROPOSITION 3.2.
If .It is locally trivial then (aMr/at V ) =
o.
Proo}: Trivial. We mention here a theorem of W. Fischer and H. Grauert (1965).
THEOREM. If each Mr is complex analytically homeomorphic to Mb , then .It is locally trivial. We now study some examples:
EXAMPLE I. Let R be a compact Riemann surface. Fix a point a E R. Let w be a coordinate in a neighborhood of a point bE R such that w(b) = O. We define a family {Mr) as follows: M r will be the branched two-sheeted
40
SHEAVES AND COHOMOLOGY
covering Rp of R with branch points at a and p; t = w(p). We have the question " is d: r = O?" Define the following neighborhoods on R: Wb = Wo = WI =
{wllwl < r}, {wllwl < r/2}, {wi r/4 < Iwl < r}.
We can write Mr = Uo U UI U U2 ••• Uj '" where Uo = n-I(Wo), UI = n-I(WI ), n(U j ) n Wo = 4> for j # 0, 1 and nis the map 1t: M, --+ R defined by the covering map n : Rp --+ R. We introduce local coordinates as follows on M,,(tE-!Wo): Zo
= J w - t on Uo ,
ZI
=Fw
on U I ,
and Zj on Uj can be an arbitrary coordinate which should be fixed and independent of t. Then we have Zj = jjiZk' t) for holomorphic!jk' In fact, Zo
= 101(zl' t) =
Jw - t = Jz~ - t,
and Zj
= jjk(Zk) (independent of t)
for (j, k) ¢ {CO, 1), (1,0)}. Then OCt) = {Ojk(t)} has only one nonzero component,
00l(t) = 0101 . (~) ot ozo
=_
1 2J zi
-
(~) = __ 1 (~)
t ozo
2zo ozo .
Let Vo = Uo , VI = UJ ~ I Uj • Then OCt) is a 1-cocycle on the covering "f/ = {Vo, Vd; OCt) E HI("f/, 0/) ~ HI(M" 0/). Suppose dMrldt = O. Then there are holomorphic vector fields Oy(t) on Vy such that
so
(6) We make the definition
3.
INFINITESIMAL DEFORMATIONS
41
Then y/(t) is a vector field on M, which is holomorphic on M, - {p} and has a simple pole at p. If the genus 9 of R is
LEMMA 3.2.
~
1, then such vector fields '1 do not exist.
COROLLARY. If 9 ~ 1, then dM,ldt =1= 0, that is, the conformal structure of the branched covering M, depends on t.
Proof
(of lemma) By the Riemann-Hurwitz formula, we have
X(M,) = [2 - 2g(M,}] = 2X(R) - 2 where X(M) is the Euler characteristic of M. Then the genus geM,) equals 2g. By the Riemann-Roch formula [see Hirzebruch (1962)], there is a holomorphicdifferential
o. Since Y/ = y(z)(dldz) has one (simple) pole, fez) = h(z)y(z) is a meromorphic function on M, with more zeros than poles [2(2g) - 2 ;::: 2]. This is impossible Q.E.D. (the number of zeros equals the number of poles).
EXAMPLE 2. Ruled Surfaces (See Chapter 1, Sections 3 and 4.) Recall that M, = U'l U U'2 where each V'v = C X ~I and (ZI, (1)-(Z2, (2)
if and only if '1
= Z2"2 + tz~,
and
ZI
= l/z 2
•
We are assuming m ;::: 2k, k ;::: 1. Then M, is independent of t
dM,ldt = 0 for t
=1=
=1=
0 for t =1= 0 so
o.
(For this, one could use the theorem of Fischer and Grauert.) What is dM Iidt 1'=0 ? Consider the covering of M 0, 6lf = {U 01, U02}; then
dM,ldtl,=o = 0(0) E Hl(6lf, 0 0)
£;
HI(Mo, 0 0 ).
Then
so that
012 (0) = Suppose dM ,Idt
(a/:2) at ,=0 (a) a'i = k(a)
= 0 at t
Z2 0'1
=
E
f(U ol n V 02 ' 0).
O. Then 012 (0) = O2
-
01
where each O. is a holomorphic vector field on V o•
= C x IP' 1.
42
SHEAVES AND COHOMOLOGY
LEMMA 3.3.
Any holomorphic vector field on C x IPI is of the form
0= g(Z)(:z)
+ [a(z)(2 + b(z)( + C(z)] (:,),
where g, a, b, care holomorphic functions on C. Assume Lemma 3.3. We have the following relations:
(a~J = z~(o;J, (o~J = mZI'I(a~J - ZT(o~J,
(7)
where (zv, C) are coordinates on UOv = C X IPI. Let us compare the coefficients of (0/0(1) in O2 - 01 and 0 12 (0). From Equations (7), we get z~
=
Z~C2(ZZ) - CI (ZI)
= Z2 c2 (zz) In
-
C I ( ;1 ;) ,
where the dz..) are entire functions. Expanding,
and 0
< k < m. This is impossible. Hence, dM t I/1;0 =1= O. dt
For the lemma we have:
Proof Let (z, 0 E C X IPI, where ( is a nonhomogeneous coordinate on At (= 00., the local coordinate on IPI is 1] = 1/(. Restrict the vector field to C x IPI - C X {oo} = C 2 • Here
IPI.
0= g(z, ()(:z) + h(z, ()(:,), where 9 and hare holomorphic on C 2 . At 00 we have
() = y(z, 1])(:z) + (J(z,
1])(:1])'
where' = 1/1] and y, {J are holomorphic. Then O(fOI] = -1/1]2 so (0/01]) - (z(%O. Hence at 00,
0= y(z,
IJ)(:J - (2{J(Z, 1])(:,),
=
3.
43
INFINITESIMAL DEFORMATIONS
since g(z, ') = y(z, 1'/), g(z, ~) is holomorphic on IC x [F»1. So g(z, as a function of , g(z,
0 is constant
0 = g(z).
Finally, h(z, 0 = _,2{3(Z, 1'/) implies that h(z, ') has a pole of order :::;;2 at So h(z, ') = a(zK2 + b(zK + c(z). Q.E.D.
00.
REMARK 1. The dime HO[M(m), 0] is the number of (complex) linearly independent holomorphic vector fields on M(m). We want to compute it. As usual, M(m) = VI U V 2 , Vv = IC X IPI, and
if and only if
We must count the number of parameters involved HO[M(m), 0]. By the lemma,
In
representing a
() E
on each Vy , and (}1 = (}2 on VI n V 2 • Changing coordinates,
(a~J = z~(a~J, Hence
+ (a2 zim'i + b 2ZT'1 + C2)ZI2(~) aZI
=
-Zfg2(a~J + [a 2zT'f + (b 2 + mz lg2)'1 + C2Z~m](a~J
= (}1
=
gl(a~J + (al'i + bl'l + CI)(a~J·
Equating coefficients,
g\(z\) = -Zigz(Z2)' al(zl) = ZTa2(Z2), b 1(z\)
= b 2(Z2) + mz t gz(z2), Ct(ZI) = Z~mC2(Z2)'
SHEA YES AND COHOMOLOGY
44
These functions are all entire functions of ZI. Let us investigate their behavior at ZI = 00. Since Z2 = lizi. we see that gl has a pole of order ~2 at 00, al has a pole of order ~m at 00 and CI has a zero at 00. Assume that m ~ 1. Then
= glo z~ + gl1 Z 1 + g12, a l = aloz'~ + ... + aim, C I = 0 (by Liouville's theorem).
gl
Consider the b terms: co
bl
1
co
= L blnZ~ = L b2n --;; n=O
()
mg ll
-
ZI
.=0
So bl(zl) = - mg lo zlb lO . blO). Hence,
mgloz i
-
1 mg I2 - · ZI
depends linearly on (glo, gil' g12,
alO, ... , al lll
,
(8)
We therefore have: THEOREM 3.1.
M(m) =1= M(n)
(complex analytically) if n =1= m.
REMARK 2.
M(2n) =1=
REMARK 3.
Let {M t It E C}, M t given by
as before. Then Mo
M(2n-l)
=
Mt lll ),
topologically.
M t
=
M(m-2k)
dMtldt = { 0, =1=0,
for t =1= O. And we have shown
for t =1= 0 for t = o.
Suppose we "reparametrize" and consider {Ms21 SEC}. Ms2 is defined by
1
ZI
Then Mo
= M(III),
Ms2
=-.
= M(III-2k), S =1= 0 as above.
Z2
But
dMs2 dM t ds 2 2s dM t -=_·-=--=0 ds dt ds dt for all sEC. We know that M t independent of t implies dM tldt = O. We have just seen that dM tldt = 0 does not imply that M t is independent of t. However, we have the following theorems:
3.
INFINITESIMAL DEFORMA nONS
45
THEOREM cx. If dim Hl(Mn 0 t) is independent of t and if aMt/at V = 0 for all v and t, then {M t t E B} is locally trivial and hence M t is independent of t. THEOREM p. The function t function of t. That is
-+
Hl(Mt, 0 t) is an upper semicontinuous
dim Hl(Mt, 0 t) ~ H 1(M., 0.), if t is in a sufficiently small neighborhood of s; that is, lim dim H 1(M" 0 r ) ~ Hl(M .. 0.).
t--.
THEOREM y. of s.
If H 1(M., 0.) = 0, then M t = M. for t in a small neighborhood
Theorem IX is proved in Kodaira and Spencer (l958a), Theorem p in Kodaira and Spencer (1960),and Theorem y is due to Frolicher and Nijenhuis (1951). Theorem P follows from some results which we will prove in a later chapter. Theorem IX will not be proved here. DEFINITION 3.3. We say that a compact complex manifold M is rigid if, for any complex analytic family {M t It E B} such that Mto = M, we can find a neighborhood N of to such that M t = Mto for tEN. (More precisely, if w:..It -+ B is the family {M t}, then w- 1(N) = N x Mto complex analytically.) The following theorem follows from Theorem y. THEOREM 3.2. If Hl(M, 0) = 0, then M is rigid. We will give a proof of this using elementary methods. We have the following:
PROBLEM. (Not easy?)
Find an example of an M which is rigid, but Hl(M, 0) =/; O.
REMARK. IP n is rigid. For n 2 2 the only known proof is to show H 1 (lP n , 0) = 0 [Bott (1957)]. Let us proceed to the proof. Proof (of Theorem 3.2) sists of two elementary ideas: (1) (2)
The proof will be elementary in that it con-
Construction of a formal power series, and proof of convergence.
The proof is actually long and computational, so please stay with us. It makes no difference for the proof and it makes the writing much easier if we assume
SHEA YES
46
AND
COHOMOLOGY
dim B = 1. The result is local so we may assume B = {tlltl < r} and to = O. We can cover W-l(~£), ~. = {tlltl < e} with coordinates flItj
= {(Zj' t)llzjl < ej , It I < e}.
Then
zj = fjk(Zk, t) on flit j n flit k' M is covered by u UJ = M where j
UJ = {zjllzjl < ej}X{O} s; flIt j .
Then M x B = u(UJ x B) where for (w., t) E U? x B, j
if and only if
that is,
wj = gjk( wk),
(9)
We can rephrase our result: THEOREM. If b is sufficiently small there is a biholomorphic map cp of W-l(~6) onto M x ~6 such that cp: maps w-l(t) onto M x t and cp: M = (i)-1(0) ~ M x 0 is the identity map. Suppose we choose b so that cp maps
u2
u2
into x B, cp(flItf) s; x B. Let (Zj' t) E flIt~. Then, cp(Zj, t) = (w j , t) = [cpiZj, t), t] so on each flIt~, cp is represented by holomorphic functions cp~(Zj' t) where Cpj(Zj' 0) = zj. On flIt~ n flItf,
zj = nk(Zk, t) ; so
implies
(10) Therefore we see that we can prove the theorem if we can construct holomorphic functions cpj(zJ' t) on flIt~ satisfying (10) and (11)
3.
INFINITESIMAL DEFORMATIONS
47
For simplicity we may as well assume that IlIi j is of the form IlIi j
= {(Zj' t)llzjl <
1, It I < e}, M
= U UJ,
UJ = {zjllzjl < 1 + v for some v > O} and U~ ~ M n IlIi j • If expand lPiZj, t) into a power series, we get lPiZj, t)
= z} + ({JJ/1(Zj)t + ({JJ/z(z})t Z + ... + ({JJ/m(z)t m + ... ,
(12)
where each ({J iI m(z) is a holomorphic vector valued function. If we expand both sides of (10) we get 00
00
I Fm«({Jm, "', ({Jjim) = m=O I Gm«({Jkll' "', ({Jklm)t m, m=O
(13)
where Fm and Gm are polynomials. We introduce some notation: If pet) = Pn t" and Q(t) = L Qn t n are two power series, PCt) == Q(t) means Qn = P n m
I
uij to n = m [that is, pet) == Q(t) mod (tm+l)J. Therefore, to solve (formally) Equation (13) we need only solve ({Jj[Jjk(Zk, t), t]
== gjk[({J~(Zk' t)], m
for each m, where ({Jj(Zj, t) =
Zj
+ ... + ({Jjlm(Zj)t m.
(14)
First consider m = 1. We have 00
Zj
= !jizk> t) = 9 jk(Zk) +
I
!jklm(Zk)t m •
m=1
Using (I 3)10 gjiZk)
+ !jkll(Zk)t + ({Jjll[gjk(Zk)]t == gjk[Zk + lPkll(Zk)t] 1
So
Now ejk =
~ (a~:k)t=o(a:j) = L !jk l l(a:j)
belongs to H 1 (M, 0). By assumption Hl(M, 0) = 0 so {Ojk} is cohomologous to zero. But Equation (13)1 says we must find {({Jk (1} so that ()jk = ({Jk 11 ({J j II' HI (M, 0) = 0 allows us to do this so the first step in an induction proof is completed.
48
SHEAVES AND COHOMOLOGY
Assume that ({>j(Zj' t) are determined so that Equation (13)m holds, that is, ({>m,,(!jk)-gjk«({>'k) == rjkt m+I , We must show that ({>jlm+I(Zj) can be m+1
' d so t h at d etermme
({>jm+l
= ({>jm + ({>j/m+lt m+I' satls fies (13) m+l> W h'IC h'IS
({>j+ 1 [Jjk(Zk , t), t] - gjk«({>'k+ I ) == O. m+l
This is equivalent to ({>j(fjk)
+ ({>jlm+l[Jji Zk, t)]t m+1 ==
gjk[
+ ((>klm+l(Zk) ' tm+l].
m+l
We use
+ ({>jlm+l ( Zj )tm+ 1
=
m+l
a. ( ({>km) gjk
OZa.
+ k.:;p({>klm+l ~ j (J ()tm+ 1 Zk ' vZk
Here we have used
and ({>'k(Zk, t) = Zk
+ ....
So if we can solve oza.
rjk(Zj) =
L vZk~ ({>klm+I(Zk) :l
({>jlm+l(Zj),
(15)
(J
for ({>Jlm+l we will have
(l3)m+l'
r jk
({>jllll+1
Let
= ~ rMZ)(o~j)' =
~ ({>jlm+l(Z)(O~j)'
Then we want to solve r jk = ({>k I m+ 1 -({>j Im+ I' We claim that {r jk } is a cocycle [belongs to Hl(M, 0)]. Then we would be done as before, since Hl(M, 0) = 0 and {rjd must be a coboundary, r jk = Ok - OJ, and set Ojlm+l = OJ, LEMMA 3.4, That is,
{r jk} is a cocycle, that is, r
ik
=
r ij + r jk on 0lI i n
IIIJ j n CIlI k n M,
3. Proof
49
INFINITESIMAL DEFORMAnONS
By definition
rik(Zj)t m+ 1 == q>~[fik(Zk' t), t] - gik[q>r(Zk, t)], m+ 1 so
gik(q>r) = giJgjk(q>r)] == giJq>j(fjk) - r jk t m+ 1J. m+1
Now rJk t m+ 1
== q>j(jjk) - gjk(q>r) and
m+1
So
By assumption
q>rEfij(zj, t), t] - YiJq>j(Zj, t)] == rij(z)t m+ 1, m+l
== rij[fjk(Zk, t)]tm+ 1 -
••••
m+l
Hence
Q.E.D. This finishes the construction of the formal power series
q>j(Zj, t) zjEiJ{tj
n M,
iJ{tj
=
Zj
+ q>j\1(z)t +"',
n M = {zjllz}' < I}
such that q> Jjjk(Zk, t)] = gjk[ q>k(Zk' t)] as formal power series. LEMMA 3.5.
The power series q>j(Zj, t) converges for
It I < ()
for some small
() > O. Proof We dominate q>j with a convergent series. We fix some notation. Let t/I(z, t) = L~=o t/lm(z)t m be a power series where t/lm(z) =[t/I~,(Z), "', t/I:,(z)] Z E U. Let a(t) = L~=o am t m , am :c: 0 be a series with real, positive coefficients. We write t/I(z, t) ~ aCt) and say that a(t) dominates t/I(z, t) if
50
SHEAVES AND COHOMOLOGY
It/I~nCz)1
::; am for all
max sup a
It/I~(z)l.
Z E U and all C( Consider the series
=
1, ''', n. The norm of
t/lm is It/lml =
zeU
b
(et)m
ro
L
A(t) = -16 -2 ' e m=1 m
where band e are constants to be determined later. Then b {
A(t)
converges for
em-Itm}
ro
= 16 t + ml;2 ~
It I < I/e. In Lemma 3.5 it suffices to prove t) -
Zj ~
(16)
A(t).
In fact, it suffices to prove for m = 1,2,3, .... First, we prove: LEMMA
Proof·
3.6.
[A(t)]2 ~ (b/e)A(t).
[A(t)]2 =
(~)2[ f (et~m] 16e
= (-
b
m
=I m
)2 L(et)" ro
16e
[f k
(ett] =1 k 1
L
22' m+k=n m k
n=2
Since 1
1
L 22::; 2k+m=n L 2 2' m+k=n m k m k mS;k
and m ::; k implies k
~
mk=n
n/2, we get
1 m2k 2
2 ::;
(n/2)2
ro
1
8
n;2
mY;l m2::; n2 6" <
16 n2 '
Hence,
)2 L (et)n 16 ~ -b - b L (et)n 16e " n e 16e n
[A(t)J2 ~ ( -
b
ro
ro
-2
=2
COROLLARY.
-2 .
n= I
A"(t) ~ (b/c)n-l A(t).
Let us prove (16) I : We want to show that
Q.E.D.
3.
INFINITESIMAL DEFORMAnONS
51
It is enough to prove I
L aaz~
We may (perhaps by shrinking Um to U! so that u V! = M) assume that the given functions fjk II are bounded. Also the
V:
and rjk(Zk)t m+I ==
t/I( t) = L t/I m t m, let
[t/I(t)]m+ 1 = t/lm+ It m +1. Then
Remember the definitions of U j and VJ, U j C VJ. Then gjk(Zk) is defined on UJ n U~ and gjk(Zk + Y), Y = (YI'''', Yn) is holomorphic for Zk E U j n Uk, lyl < v. So there is a K >0 so that gjk(Zk
+ y)
co
~
L Km(YI + ... + Yn)m.
(17)
m=O
For the moment, let t/liZk' t) =
since t/liZk' t) is a polynomial of degree we get
~
m in t. From Equation (17),
52
SHEAVES AND COHOMOLOGY
hence, [gjk(Zk
+ I/Ik)]m+i
co
~
L K"[nA(t)J'
r=2 ~ A(t)J2I(r nrGr-
i
n2 b 1 = A(t)-c- 1 - (Knb/c) , K2
because by induction I/Ik
ACt). lfwe choose c so large that (Knbjc) <
~
{gjk[CP~(Zk' t)]}m+l ~
2K 2 n2 b c
A(t).
1, then (18)
Now we want to estimate Ijk(Z)t m +1 == CPj[fjk(Zk, t), t] - gjk[cpr(Zk, t)]. m+1
Remember that
r is a co cycle so " -p OZ~ r ajk on U r ika = fa + L... OZj ij
i
n U j n Uk ,
where U i satisfies U i = {zillzil < I}. We choose Ui c Ui , Ui = {zillzil < 1 - P} such that u Ui = M. It suffices to estimate rik(Zi) for Z E Ui n U i , for take any Z E Uj n Uk, then Z E Ui for some i, and r1k(Z) =
La uZi ~z~ {rMzi) - ri/ Zi)}.
Then if rjk(Zj)t m+i ~ B(t) for Z E U1 n Uk, Z E n Uk, where
Ur
K 1 = 2n
rjiz)tm+l ~ KiB(t) for
n~a.x (sup IOZi oz~ \). 1,1
So consider CPj[fjk(Zk, t),
tJ
Ur n Uk' Expanding !jk ,
for Z E
!jk(Zk' t) = Zj
+ !jkl1(Zk)t + ... ,
we see we may assume that
for small t and some constants bo , cpj -
Zj
=
Co •
Our induction hypothesis is
m
co
,,=1
1
L cpn,,(z)t" ~ A(t) = L a"t".
3.
53
INFINITESIMAL DEFORMATIONS
Since each cP j III is holomorphic, we get a power series cpjliZj
+ y) -
cp.iliz)
L
= VI
where CV1
"' V
"=
+ ... + Vn 2: 1
cW
"
vny~1 ... Y;;",
CPjJll(Zj+Y) ( l)"f f y~'+l ... y~n+ldYl···dYn. 2rrj
...
ly~I=P
Since Icp n III < ap' for some constant all' on
Ic
VI
.•• Vn
ly,,1 = /3, we see that I ~ (a J.l //31'1' .. Vn).
We thus obtain
Summing we get CPJ~(zJ'
+ y, t) -
CPJ~(zJ" t) ~ L.. '" a.f . ~ Il~l
Vl+'"
Again, since cPj(Zj' t) only has terms of degree
~
00
Y;' ...
'" L..
+ /31',.
+v"~l
y~n
+ v" .
(19)
m in t m we obtain
[CPj[fjk(Zk, t), t]]m+ 1 = [cpj(Zj' (1jk, t) - cpj'(Zj' t)],n+ 1 •
Using (19) we get [cpj[jjiZk' t), t]],n+1
~
A(t)v,+
"~\'"~I
[
T
A (t)]l'l+
. + Vn
The corollary to Lemma 3.6 gives Ao(t)" ~ (bo/co)V-1 Ao(t), so
00 (A~ )V]" [L 1'=0 /3
1~ =
[00 1 (b~ )V-I Ao( t) ]n 1+ L ~ 1'=1/3 Co
1
[1 +~ 1 _(b~//3co)Ao(t)r-1.
Since we have chosen Co so that Ao(t) ~ fjk(Zk' t) - Zk' we may replace Co with a larger constant and assume that bo/2c o < t. Then
54
SHEAVES AND COHOMOLOGY
where K2 is a constant depending only on n, {l, bo , and co. Thus, [IPj[fjk(Zk' t), t]]m+ 1
~ ~2 A(t)Ao(t).
We may assume b > bo , c> Co and then A(t)
~
Ao(t), so
K2 K2 b 7i Ao(t)A(t) ~ p~ A(t). Finally,
b If - 211: 2n2) ~A(t),
K2 ~ ( for
ZE
Ur n Uk. Hence
(20)
8z
IJ
j IPklm+ p r"jk = "L- -;:P 1(Zk) uZk
" IPjlm+ 1(.. j )
and we want to estimate IPjlm+l(Z). As before, consider r jk = L rjk(Z)(O/ozj), IPjlm+l = L IPjlm+1(8/8zj). With these notations, r jk = IPklm+l - IPjlm+l. At this point we need a lemma which plays a crucial role in arguments of this type. Let r = {r jk } be a l-cocyc1e where r jk is a holomorphic vector field on Uj n Uk. Let 1/1 = {I/I J be an O-cochain where 1/1 j is a holomorphic vector field on U j. We define lin = max j,k 111/111 =
sup
max WMzj)l.
ZEVj"Vk
"
max sup max Il/Ij(zj)l· j
:E Vj
"
LEMMA 3.7. There exists a constant K such that if r is cohomologous to 0, then we can find 1/1 with bl/l = r satisfying 111/1 II ~ K II q where M is a compact complex manifold, Hl(M, 0) need not be zero, and K does not depend on r.
Proof Remember Vj = {zjllzjl < I} and Ur{zjllzjl < I - {l}. Suppose the lemma is not true. Let r(n = {lil/IlIlbl/l =
n < +00.
Ifr(r) ~ Kllq, then there is 1/1 such thatbl/l = rand 111/111 ~(K + l)lIni. Thus if the lemma is false, r(n/ltrll is unbounded. Hence there is a sequence {rev)}
3.
INFINITESIMAL DEFORMAnONS
55
such that r[r(V)]/lIr(v)lI_ 00 as v _ 00. Replace r(v) with r(v)/r[r(v)]. Then r[r(V)] = 1 and I r(v) I - O. Hence we can find {l/Ii V)} holomorphic on Uk so that III/I(V) I < 2 and rj~) = b{1/I1V)} = 1/I1V) - I/I)V). By Vitali's theorem we can select a subsequence {I/I)v).)} such that I/Ijv).)(Zj) converges uniformly on If Z E Uk then Z is in some U;. Hence for each k, I/Ilv).)(z) = I/I)v).)(z) + rj~).)(z) converges uniformly on Uk so CfJlv).)(z) -l/Ik(Z) uniformly on Uk. Since r)~).) - 0, 1/1 j(z) = I/Iiz) on U j n Uk· Let if;)v).) = 1/11 v).) - 1/1 j . Then r)~;') = if;~v).) - if;jv;.). But I if;(v;,) I < t for large v, and thus r[r(v;.)] < 1- This contradiction proves the lemma. Q.E.D.
U;.
and
By Lemma 3.7 we can find K and {CfJjjm+d so that r jk = CfJkjm+l - CfJjjm+l IICfJm+lll ~ K IlfII. Hence
CfJjjm+l (Zj) tm+l
(22 +73 K2)b ) ~A(t.
~KKI 2K n
We can choose c so large that KKl(2K 2 n 2 + K2/(3)(b/c) < l. Then CfJjjm+l t m + 1 ~ A(t) so CfJj+ l(Zj' t) - Zj ~ A(t). This completes the induction and proves that CfJ/Zj' t) - Zj converges for small t, thus finishing the proof of the theorem. Q.E.D. There is the following: THEOREM. [Kodaira, Nirenberg, and Spencer, (1958)] Assume H2(M, 0) = 0 (M is compact complex as always). Then for any element e E Hl(M, 0), there is a complex analytic family {M,lltl < r, r> O}, such that Mo = M and (dM,/dt),=o = e. PROBLEM. Find an elementary proof of this. (In the analogous idea of proof the convergence gives trouble.) We also have the completeness theorems.
DEFINITION 3.4. We say that the family (.$I, B, w) is complete at bE B if, for any family (JV, A, rr) such that rr-l(a) = w- 1 (b) = Mb there is a neighborhood U 3 a and holomorphic maps
: rr-l(U) -+.$1, h: U -+ B such that h(a} = b, rr-l(U)---+.H
"j U
, jm ---+
commutes,
B
maps rr-1(s) biholomorphically onto : rr- 1(a) = Mb-+ Mb is the identity map.
w- 1 [h(s)]
for each
S E
U, and
SHEAVES AND COHOMOLOGY
56
Roughly, (.It, B, w) is complete if it contains all sufficiently small deformations of M b. The (holomorphic) tangent space Tb at b is the set
map Pb: (a/at) -. (aMr/at)r=b Tb -. Hl(Mb' e b).
The
THEOREM.
E
Hl(Mb' e b) defines a linear map
[of completeness; see Kodaira and Spencer (l958b)] b ), then (.It, B, w) is complete at b.
If
Pb(Tb) = Hl(Mb' e
REMARKS. (1) Theorem 3.2 is a corollary of this theorem. (2) This theorem of completeness can be proved by the same elementary method used to prove Theorem 3.2.
4.
Exact Sequences As usual X is a paracompact Hausdorff space, [/ is a sheaf over X and
w: [/ -. X is its projection map. DEFINITION 4.1.
[/'
£; [ /
is a subsheaf if
(1) [/' is open, (2) w([/') = X, (3) W-l(X) n [/' = [/~ is an R-submodule of [/x' Let [/" be a sheaf over X with projection
w".
DEFINITION 4.2. A homomorphism h of [/ into [/" is a continuous map of [/ into [/" such that (1) (2)
w" 0 h = W, h: [/ x --+ [/~ is an R-homomorphism. h is a local homeomorphism. We define the kernel of h to be
REMARK.
ker h = {s I h (s) LEMMA 4.1.
=
O} where" t
ker h is a subsheaf of [/.
LEMMA 4.2. h([/) is a subsheaf of [/". The proofs are left to the reader.
= 0" means
tx
=
Ox E
[/x.
57
EXACT SEQUENCES
4.
Let [Il' E [Il be a subsheaf. Let Qx = [Ilx![Il~ which is an R-module. Define hx: [Ilx --+ Qx to be the natural homomorphism. Let Q = UXEX Qx' Define 11: on Q by 11:(Qx) = x; h is defined h : [Il --+ Q by h(s) = hx(s) for s E [Ilx' We give Q a topology by saying Illt is open if and only if h-1(°lt) is open in [Il (the quotient topology).
LEMMA 4.3.
Proof
Q is a sheaf and h : [Il
--+
Q is a surjective homomorphism.
Left to the reader.
Q is the quotient shea/of [Il by [Il' and we write Q = [Ilj[ll'.
DEFINITION 4.3.
A sequence ho
hll
III
[Ilo--[Ill--[Il2--"'--[Iln--[Iln+l
II,. +
I
-----+ ...
of sheaves is exact if hy([Il y) = ker(hv+ 1) for all v. Suppose given sheaves [Il and [Il" over X and a homomorphism h : [Il --+ [Il". Let U be an open subset of X and r(U, [Il) be the set of all sections of [Il over U, Then h 0 (J E r( U, [Il") if (J E r( u, [Il) so h induces a map h : r( U, [Il) --+ r( U, [Il"). Let Illt = {U j } be a locally finite covering and Cq(OZI, [Il) be the space of q-cochains cq = {(Jjo'" j) where (Jjo'" jq E r( Ujo n ... n Ujq , [Il). Then h induces a map h : Cq(lllt, [Il) --+ Cq(lllt, [Il") defined by h cq = {h(Jjo'" j.}' Then we have:
LEMMA 4.4.
Proof
h
0
{)
= {) h. 0
Obvious.
Hence h maps zq(lllt, [Il) into zq(lllt, [Il") and thus h induces a homomorphism h: Hq(lllt, [Il) --+ Hq(OZt, [Il"). Let "/f/ = {"/f/;J be a refinement of Illt, "/f/ > Illt. Then H'J(Uli, [Il)~Hq(OZt, [Ill!)
jn'l
1V
jn'll
iV
Hq("lf!, [Il)~ Hq( "/f/ [Ill!) commutes. Hence h induces a homomorphism h : Hq(X, [Il)
--+
Hq(X, [Il").
58 THEOREM 4.1.
SHEAVES AND COHOMOLOGY Assume that
o--+ fI"
~ fI' ~ fI'"
--+
0
is exact. Then there is a homomorphism b* such that
is exact.
Proof i is injective so fI" ~ i(fI") c fI', and i(fI") = ker h. Thus we consider fI" c fI' where fI" = ker hand i is the inclusion map. Recall that HO(X, fI') = Zo(X, fI') = reX, fI'). Since reX, fI") c fI'), we see that
rex,
0--+ HO(X, fI") ~ HO(X, fI') is exact. If a E reX, fI'), then ha = 0 if and only if a E reX, fI"); so HO(X, fI") ~ HO(X, fI') ~ HO(X, fI'") is exact. LEMMA 4.5. HO(X, fI') ~ HO(X, fI'")~ H1(X, fI") is exact (where we must define b*).
Proof Let a" E reX, fl'1I)' Since h is a local homomorphism there is a section Ty Ere Uy, fI') over a small neighborhood Uy of y such that h ty(x) = a"(x) for x E Uy . Now {Uy lYE X} covers X and we have a locally finite refinement Olt = {Uj } of {Uy}, that is, there is a map j --+ y(j) such that Uj £ Uy(j)' Set t j = rVj ty(j) E reUj, fI'). Then h t j = a" where defined. Let eO = {tJ E eO (Olt, fI'). Then: DEFINITION (4.4)1' b*a" = [beO] E H1(X, fI") where for any cq E zq(Olt, fI'), Ceq] denotes the cohomology class in Hq(X, fI') of cq. (One should check that b* is well defined.) Since beo = {Tk - tj} and hTk - htj = a" - a" = 0, we see that beo E ZI(01I, [/') so Definition (4.4)1 makes sense. Exactness means b*a" = 0 ifand only if a" = ha for some a E reX, [/). So suppose t5*all = [beO] = O. Then &0 = be o' where eO' = {tj} E CO(Olt, fI"). SO CO - co' = a E ZO(X, fI') = reX fI'), and ha = hco = {ht j } = a". Now suppose a" = ha. Then h(t j - a) = 0, so t j - a E reUj, [/'). Set c~ = {t j - a} = Co - (J E CO(Olt, fI"). Then bc~ = bC o since (J E ZO(X, fI') and hence b*a" = [bco] = [bc~] = O. Q.E.D. We now turn to check that
4.
59
EXACT SEQUENCES
is exact. Take e l ' EZI(OU, f/'). If [el'] = (j*(J" = [(jeO], then i[e l '] = 0, and if i[e l '] = 0, then e l ' = {)co; so 0 = he l ' = Meo = (jheo. Thus heo defines an element (J" E r(X, f/"). By definition, [el'] = (j*(J". We want to prove exactness
LEMMA 4.6. Given e4 " E Cq(OU,f/"), then we can find a locallyfinite refinement "fI/ and eq E C q("fI/, f/) so that eq" = heq.
n:,.
Proof We give proof for q = 2. Let OU = {U), eq" = {(J;:"d, where (J;:"k E r( U i n U j n Uk' f/"). Choose a covering "f/ = {Vj} such that Vj c UJ • Since OU is locally finite, a given Y E X belongs to only finitely many Uj • We choose a neighborhood Ny of y sufficiently small so that if Y E Uk n Uj n Uk there is T E r(Ny , f/) with (J;:"k(X) = hT(X) for Ny (remember h is a local homeomorphism), (2) for each y there is Vj such that Ny c Vj , (3) if Ny n Vj =F
xE
Then {Ny lYE X} covers X and we can choose "fI/ = {W)J a locally finite refinement of {Ny}. Hence there is a map A. -+ y). such that W). c Ny). . By (2) Ny). C ViA' so 1{/ > "f/ > OU. Define T = {T).l'v} E C 2 ("fI/, f/) as follows: we have W). c Vi' WI' C Vj , Wy C Ck where i =j.. , j =j.. , k =j•. By (3) if Ny). n Vj =F
Q.E.D. Let us prove that Hq(X, f/)
i
-+
h
Hq(X, f/) - + Hq(f/")
is exact. hi = 0 is clear. Suppose 11 E Hq(X, f/) and hl1 = O. Then 11 = Ceq], eq EZq(OU, f/) for some OU and n~heq = ()cq-I" for some "f/ and eq- I " E Cq-l("f/, f/"). By the lemma n~eq-I" = h (q-I for some "fI/ and ,q-I E Cq-I("fI/, f/). Thus hn:, eq - h (jt q-
1
= 0,
so
n:' eq -
(jt q- I = eq'
where eq' E zq("fI/, f/') !;;; cq("fI/, f/'). Finally we get 11 " = il1' where ,,' = [e4'] E Hq(X, f/').
= Ceq] = [e q,]
so
60
SHEAVES AND COHOMOLOGY Next we prove that
is exact. We must define 15*. Take r( E Hq(X, .9"). Then r( = [eq"J, eq" E Zq(OlI, .9"). By the lemma there is t q E Hq(1f'", .9) such that h t q = IT~ eq". DEFINITION 4.4. 0*'1" = [otqJ E Hq+ leX, .9'). Again we should check that 15* is well defined. For the moment denote n~ by n. Suppose 0*'1 = O. Then ITot q = fJb q ' for some bq ' E Cq (1Y, .9'). Thus o(IT t q - bq ,) = 0 and IT t q - bq = eq E zq(1f'", .9). So let '1 = [eqJ E Hq(X ,.9). Then h'1 = [heqJ = [IT htqJ = [IT eq"J = '1". Suppose conversely that '1" = h'1. Let '1 = Ceq]. Then n" = [heqJ and ht q = IT heq by definition of (q. So t q - neq = aq' E cq(1f'", .9') and fJ*'1" = [fJtqJ = [oaq,]. Thus 0*'1" = O. Finally we prove that
is exact. Certainly i 0*'1" = O. Since by definition 0*'1 = [otqJ where t q E Cq ('"If/",.9) so io*'1 = O. Suppose i'1' = 0, '1' = [e q + 1'J E Hq+I(X, .9'). Then IIe q+l' = ot q for t q E C Q(1Y, .9). Then 0 = 15 ht q E zq('"If/", .9") and '1H = [htqJ E Hq(X, .9"). Since 0*'1" = [fJtqJ = '1' we are finished. Q.E.D. [If the reader wishes more details for these elementary properties of sheaves he may consult Hirzebruch (1962).J Next we prove functoriality. THEOREM 4.2.
If i
h
0--+.9'--+.9--+.9"-0
l~'
r
1~
k
l~"
O--+ff'-ff--ff"--+O
is exact and commutative, then
is exact and commutative.
4.
Proof
61
EXACT SEQUENCES
We need only prove commutativity. We check that 6·
Hq(X, Y")_Hq+l(X, Y')
\_.
\.
'"
Hq(X, !Til) _
Hq+ l(X, !T')
commutes. The rest is easy. Let r( E Hq(X, Y"). Then there is eq" E zq(1fI, Y") and t q E cq(,,#r, Y) such that 1]" = [e q"] and eq" = ht q. Thus 0*1]" = [ot q]. However, cp' 0*1]" = [cp' ot q] = [ocptq] , and CP"I]" = [cp"ht q] = [cp"ht q] = [h cpt q]. Thus o*cp",," = [0 cpt q] = cp'15*1]". Q.E.D. We give a brief discussion of fine sheaves.
DEFINITION 4.5. Y is afine sheaJif for any locally finite covering {U j } of X there exists a set {hj} of homomorphisms hj: Y -+ Y such that (1) (2)
= 0 for x¢ L hi = id. hjY x
Wi' where Wj
S;;
U j is a closed subset of U j
,
j
EXAMPLE. Let f!) be the sheaf of germs of differentiable functons on a differentiable manifold X. We have a partition oj unity subordinate to UJ ; that is, a set {Pi} of differentiable functions Pj = pix) on X such that (1) (2)
pix) = 0 for x ¢ Wj Pj = 1.
,
L
For any local differentiable function J = J(x) on X, define hJ = pix) J(x). Then h j induces a homomorphism h j : f!) -+ f!). Using these {hj} we see that f!) is fine.
THEOREM 4.3.
If Y is a fine, then Hq(X, Y)
= 0 for q ~ I.
We give the proof for the case q = 2. Let e 2 be a cocycle, = {O"jjk} E Z2("II, Y), with O"jjk E qUj n Vj n V k , Y). By fine-ness we have the {hj} in Definition 4.5. Since &2 = 0, if V j n Vj n Vk n Vt :f. ¢ then O"jkt - O"jM + O"jjt - O"jjk = O. Since hi O"jjix) = 0 for x ¢ Wj , hi O"ijk can be extended to 'ijk E qU j n Uk' Y) by setting 'ijix) = 0 for x E Vj n Uk - Wi. For fixed (j, k) we have only a finite number of Vi with Vi n Vj n V k :f.
Proof
e2
= hiO"jkt - hjO"jjt + hjO"jjk· 'jt + 'jk so e 2 = oe 1 where c 1 = {'jd. hiO"jkt
'kt -
Thus O"jkt = proof for any q
~
1.
Q.E.D.
It is easy to give this
62
5.
SHEA YES AND COHOMOLOGY
Vector Bundles
We give a brief review of vector bundles. Again, a good reference for this section is Hirzebruch (1962). Let M be a complex (differentiable) manifold. 5.1. By a complex analytic (differentiable) vector bundle (C" or IR" bundle) we mean a complex (differentiable) manifold F together with a holomorphic (differentiable) map n: F -+ M onto M such that, for a sufficiently fine locally finite covering OU = {V J of M: DEFINITION
(1) There is an analytic (differentiable) equivalencefj between n-1(U j ) and Vj x C" (or U j x IR") such that
n-\V)~ Vj x C"
.j .' j" Vj-Vj
commutes, where nj(zj, () = Zj' (2) If(z, (j, ... , ()) E U j X C" (or Vj x IR") and (z, (L"', (or V J' x IR"), then
fj
0
f k-I( z,
WE Uk
X
C"
p = " fajkP ( Z )V~k'
vi VII)" ~k' ••• '\.k 1...
/l=l
wherefj'kp(z) are holomorphic (differentiable) functions on Vj n Uk' In vector notation fjk is the matrix (!j'kp) and (j = «(j , .. " ()), and then
We call n-I(z) the fibre of F over. By (I) and (2) we can give it a vector space structure n-I(z) = z x e" [or n-1(z) = z x IR"]. DEFINITION 5.2. We say that F and F' are holomorphically (or differentiably) equivalent if there is a biholomorphic (bidifferentiable) map cp : F -+ F' such that
(I)
F~F'
\/ M
commutes.
5.
63
VECTOR BUNDLES
(2) On each fibre qJ is a linear transformation, that is, if Uj is chosen as in Definition 5.l, then there is a holomorphic (differentiable) matrix-valued hjp(z) (~. function h j on Uj such that Ij 0 qJ °/j-I = hj' that is, G/. = In particular, if F is a I(: 1 bundle and F is trivial over Uj (that is, 7t -I (U) = U j x IC), then z E Uj n Uk implies that (z, () is identified with (z, (k) if and only if (j = Ijk(zKk whereljk(z) is a nonvanishing holomorphic (differentiable function) of Z E U j n Uk' Let 0* (or g,*) be the sheaf over M of nonvanishing holomorphic (or differentiable) functions in which the module operation on each stalk is multiplication and the ring R = lL so here instead of rt./(z) + {3 g(z) we have [f(z)]"[g(z)]p, rt., {3 E lL. Consider Ijk as an element of
Lp
Hence {fjk} E Zl(OZt, 0* (or g,*» and two bundles F and F' are equivalent if there are nonvanishing functions hiz) on Uj such that Ij -I 0 h j O/j = I: -I 0 hk o/k on U j n Uk' This is equivalent to
or {fjk} = {fjd . c5{h;I}, that is, {Ijk} is cohomologous to {fik}' Thus an equivalence class of bundles defines an element [{fjd] E HI(OZt, O*(or .@*» £ HI(M, O*(or .@*». Conversely it is easy to construct a bundle from an element of HI(M, l!1*(or .@*». Thus we have a I-I correspondence between equivalence classes of 1[1 bundles and classes in HI(M, l!1*(or .@*». We shall always identify equivalent 1(:1 bundles and we caU Hl(M, O*(or g,*» the group 01 1(:1 bundles over M. It has a natural group structure if we define F· G = {fjkgjk} where F = {fjk} and G = {gjk}' We construct an important invariant of 1(:1 bundles. For any germ of a holomorphic function!, e 2nij E l!1*. Thus we get an exact sequence 0--+ lL
--+
0
--+
l!1*
--+
0,
where lL is the sheaf of germs of 10caUy constant integer valued functions on M. We also have the following commuting, exact diagram:
o~:r~[~~~~O O--lL--l!1--l!1 * - - 0 ,
64
SHEA YES AND COHOMOLOGY
since (9 c f!) and (9* c f!)*. This yields the exact commutative sequences ···-H'(M,Z)-H'(M,(9)~H'(M,(9*)
_ H 2 (M, Z)----.H 2 (M, (9)----. ... ... _
H'(M, Z) -
H'(M, ~)- H'(M, !0*) d*
---+
Now
~
H 2 (M, Z) - - - + H 2 (M, ~) - - - + ....
is a fine sheaf, so Hq(M, f!) = 0 forq --+ H 2 (M, Z).
~
1. Thus (i* is an isomorphism
(i* : H 1(M, ~*)
DEFINITION
5.3.
c(F) = o*(F) is the (first) Chern class of F.
We remark that c(F) = c(G) if and only if cp(F) and cp(G) are equal in H 1 (M, E&*) where cp : H 1 (M, (9*) --+ H1(M, ~*) is induced by (9* c f!)*, where F, G E H'(M, (9*). Hence F and G are differentiably equivalent; that is, there are nonvanishing differentiable functions hj such that {fjk} = {h j g jk hi: 1 } where F = {fjk}, G = {g jk}. Thus: 5.1. The Chern class c(F) of a complex analytic 1[' bundle represents the differentiable equivalence class of F.
PROPOSITION
Let us give an explicit description of c(F). If F
= {fjk}, lij . jj" . Iki
= 1
and log/ij + log/jk + loglki
= 2lti Cijk '
J=t.
where i = Then c(F) = {C ijk } E H 2 (M, Z). Next consider en bundles. Let n ~ Z and F be a en bundle defined by {jjk}. Let ij) be the sheaf over M of germs of matrix-valued holomorphic functions/(z) = I;(z) with det/;(z) '" 0 where the module operation is matrix multiplication. Note that the operation is not commutative. We cannot define the higher cohomology groups of ij) but we can define the following objects: Let I1lt = {V j} be a locally finite open covering of X. A O-cochain CO = {jj}, Ij E nVj' ij) is a set of sections of (fj over V j ; a l-cochain c' = {jjd where jjk E V ik, ij); a l-cocycle c 1 is a l-cochain such that jjk(Z) = lij(z) . Ijk(Z ) for Z E Vi n Vj n V k . Let ZI(I1lt, ij) be the set of all l-cocycles. We note that ZI(I1lt, ij) is not a group.
n
5.4. We say that {fik} and {g id E Zl(l1lt, ij) are equivalent if there exists {hJ such that gik = hJikh;;l. Let H 1(11lt, (fj) be the set of equivalence classes of l-cocycles. We define H'(M, (fj) = lim Hl(l1lt, (fj). DEFINITION
"IJ
5.
65
VECTOR BUNDLES
5.2. Each element of Hl(M, OJ) represents an equivalence class of complex analytic en bundles.
PROPOSITION
Proof
Left to the reader.
We now list some methods of forming new bundles from old bundles. Let F be a en bundle, G be a em bundle, OIJ = {U j} a trivializing covering and F = {fjk}, G = {gjd. We define the following new objects: (I) Whitney Sum FrJ;> G. the cocycle {h jk} where
This is a
hjk(z) =
en+ m
bundle which is defined by
(~k g~J
(2) Tensor Product F® G. This is a h jiz) = Jjk(Z) ® g jk(Z). Recall that •..
enm bundle defined by hll nl
••.
{h jk } where
hll) nm
, h~:
and here h'j;PIl = /jkP g7kll. A point in F ® G has coordinates (z") I, •.• , m , ••• , 'T), where (z, '1) and (z, 'k) are identified for Z E Vj n V k if and only if
'1
,j). = L fjkp(z)gfkll(zK~Il. Dual bundle F* of F. This is the bundle defined by {fjt} where (z, '1 ..) is identified with (z, if and only if
(3)
fjt = (fjk I)' = (fkj)' which is the transposed inverse of Ijk· Then
,:P)
'r
Sometimes we write
=
L fJMzK: P = L ffj..(zK: p•
'T. for n". Then we have '1.. = P=I L" nk. (zK:p •
(4) {fjk}.
Complex Conjugate F of F. This bundle is defined by the co cycle
Let us now define subbundles and quotient bundles. Suppose that, by a suitable choice of Olt = {U j} and of fibre coordinates ,;, the matrices Jjk in the l-cocycle {fjd defining F can be written as follows:
66
°°
SHEA VES AND COHOMOLOGY
Hencefj~p(z) = for I ~ f3 ~ m, m + I ~ ex ~ n. Thus(j = Lp=m+t!jk/l(f for ex> m, and if (f = for f3 > m, then (j = 0 for ex > m. Let F' = u U j X em where em = {«(J ' .. " (j, 0, .. " O)} !;; and we identify (z, () and (z, (k) if (j = A jk(Z)(k' Then F' is a subbundle of F. The quotient bundle F" = F/F' m bundle defined by the l-cocycle {C }. is a jk
en,
en-
DEFINITION 5.5. A holomorphic (or differentiable) section of F over V!;; M is a holomorphic (differentiable) map cp: z -+ cp(z) of V -+ F such that ncp(z) = z where F is a holomorphic (or differentiable) en bundle. We see that locally cp is a set of n-functions. Since local sections and germs of sections are defined, we get a sheaf of germs of sections of F. We denote by (9(F) (or !,)(F» for sheaf over M of germs of hoi om orphic (or differentiable) sections of F. Then locally, (9(F) = (91 U j EEl" . EEl (91 V j (sum n-times), where {!} 1Vj means (9 restricted to Vj and (!}z(F) = {!}z EEl'" EEl {!}z (n-times). We now review tangent bundles and tensor bundles. Let M be a complex manifold and {VJ an open covering of M with coordinate patches with coordinates (zj, .. " zj) on V j • A (holomorphic) tangent vector at z is an element of the form v = L:= 1 (j(%zj). It is easy to see that the set Tz(M) of all complex tangent vectors at z is a complex vector space TiM) ~ en. If z E V k another chart at z, then we identify Lp= 1 W%zf) with L (j(%zj) if n
(j =
L fjk/l(zK~,
P=l
This is a linear identification so the vector space structure of Tz(M) is well defined. The set T(M) = UzeM T.(M) is a complex analytic vector bundle defined by the I-cocycle {fj~p(z)}. T(M) is the holomorphic tangent bundle of M. T(M) is the conjugate (holomorphic) tangent bundle of M. And !Y(M) = T(M) EEl T(M) is the (complexified) tangent bundle of M. Then 0, the sheaf of germs of holomorphic vector fields, is (!}(T(M». If M is a differentiable manifold with local coordinates (xj, .. " xj), then the (complex) tangent bundle !Y(M) = UxeM!Y iM), where
The real tangent bundle !Y R(M)
=
UXEM !Y xR(M) , where
The relation is !Y(M) = !YDiM) ®R e, where bundle over M considered as a real bundle.
e is
the trivial (complex) line
5.
67
VECTOR BUNDLES
Let T*(M) be the dual bundle of T(M). If Tz*(M) the transition relations are
= {(C;t> "', C;n)}, then
We use the following notation: An element U E Tz*(M) shall be written = L~ C;~ dzj, where dz'j(%z1) = op as an element of Tz*(M). Briefly let T = T(M), T* = T*(M). A tensor bundle is a bundle of the form
U
T ® ... ® T ® T* ® . " ® T ® ... ® T*.
We denote T ® ... ® T = (® T)P for the p-fold tensor product of T. We remark that T is not a holomorphic bundle so (® T)P ®(® T*)q is a holomorphic bundle but (® T)P ®(® T*)q ®(® T)r ®(® T*Y is only differentiable. A holomorphic (differentiable) tensor field is a holomorphic (differentiable) section of a tensor bundle. We now give a brief treatment of differential forms. 5.6. A differential form of type (p, q) (or a (p, q)-form) over an open set W_£ M is a differentiable section cp: z -+ [z, cp j~, ... ~pPI ... P.(z)] of (® T*)P ®(T*)q over W such that the fibre coordinates cP jll, ... ~pPI ... P. are skew-symmetric with respect to 0(1 ••• O(p PI ... Pq • If p = 1, q = 0,
L:=1
cp(z) =
+,
L
p.q.~, .... ~p
fI" "', fl.
where dz P, = dz l1 ' and " 1\ " is the wedge product of skew-symmetric forms and satisfies for example, dz~ 1\ dz P = -dz P 1\ dzlJ. [Note: We write zlJ = z~
= zit.]
If M is only differentiable we still have Y(M) and Y*(M). Then if W is open in M, we make the following definition: DEFINITION
5.6'.
A differential form of degree p over W 1
cP = p!
L
is a section over W of yP which is skew-symmetric in the indices 0(1 ••• O(p • If cP is a p-form and x is a q-form, we define the wedge product of forms
1
1\
t/l = p! q! ~
p
t/ljPI "'P. dxj'
1\ ... 1\
dx?
1\
dxf'
1\ •.. 1\
dxf·.
68
SHEA VES AND COHOMOLOGY
F or example, if cP =
-1 L CPap dxa 1\ cP
1\
tJt
=
dx li and tJt = L 1/1 Y dxY, then
1
2" L CPap I/Iy dx a /\ dx P 1\ dxY
where
Hence
DEFINITION
5.7.
If
then the exterior derivative dcp is
d 1\ ... 1\ . 't',p. ex.ex,.·L... ex pocp ja,a ex... a d d x·a
dm = -1
p
Xj
X~ /\ 1
1
x·p
1
1
(1)
where
tJtao"'exp = (o:.o)CPexl ... ap - (o:exl)CPex ex '''ex + ... O 2
+ ( -lY(a:exp )cpexo
P
"'''p- 1 •
5.3. dcp is well defined (that is, the definition is independent of the choice of local coordinates Xj)'
PROPOSITION
Proof We write out the proof for p = 2 and leave the general case to the reader. So consider cP =
t L cP jap dxj 1\ dx~ = -1 L CPH/l dx; 1\ dx~
on Uj n Uk' We want to see that
ocpa '"p t "L.. 1y dxj Xj
1\
aCPkJ./l dxj /\ dXjp =!" L.. --;-;- dx~ vXk
1\
dX kJ. 1\ dx~.
(2)
5.
69
VECTOR BUNDLES
The following rules of transformation are given to us. OX~ ox~ CPM = I ;a ;-p CPkAI' , A,I' uX j uX j
dx; =
I'"
ax;" ;'l
:
dx} .
UXj
The sum of the first two terms is zero, so
~ =t"
L...
OCPUI' OX~ OXl'k ox~)
dx~
oxt dx": ax":) )
1\
)
1\
ox~ dx~ ox~)
)
Q.E.D.
and (2) is proved.
REMARK. For other types of tensor fields such a "nice" operator does not exist. To correct for the difficulty one introduces the idea of a connection.
PROPOSITION
Proof.
ddcp
5.4.
= O.
dcP = -1 "oCP"""''''Pd''' L... x p! ax'" ddcp
1 "02cp,,, .. '"
= -
p!
L...
' p dx oxfJ ax'"
1\'"
{J
1\
1\
dx'"
d'" X p.
1\ ... 1\
dx"'p
=0, since 02cp"" ...",)ox pox'" is symmetric in PROPOSITION
Proof
5.5.
Easy.
IX,
fJ and dxP 1\
If cP is a p-form, then
dx'" is skew-symmetric.
70
SHEAVES AND COHOMOLOGY
THEOREM 5.1. (Poincare's Lemma) Suppose that a p-form qJ, p ~ 1, satisfies dqJ = 0 on a neighborhood U of 0 = (0, .. ',0) s;;; /R n • Then there is a (p - I)-form t/I on a neighborhood W, 0 E W s;;; U, such that qJ = dt/l (locally dqJ = 0 if and only if qJ = dt/l for some t/I).
Proof
Let qJ = qJ(x)
I
= -, LqJ.. ,..... (x) dx'" p.
1\ ••• 1\
dxaF •
I'
We fix p and prove the theorem by induction on the dimension n. The first step is the case n = p, and qJ = qJ12 ... p(X) dx 1 1\ ••• 1\ dx p• Define ljI(x) = g(x) dx l
1\ ••• 1\
dx p -
l
by
f
xl'
g(x 1,···,xP)=(_1)p-l
qJl ... P(X 1,···,XP-l,t)dt,
o
where W is a star-shaped neighborhood of 0 and if x E W so is the line joining 0 and x). Then og(x) dljl = , , - - dx" i.J ox ..
1\
dx 1
1\ .•. 1\
og(x)
= - - dx P 1\ dx 1 ox P
1\ ... 1\
(Xl, ... ,
dxP
dxP
x P)
E
W (that is,
_ 1
_ 1
= qJ. Now assume the result for n = m - 1, m > p and consider n = m. Then
1 p!
q>(x) = -
+
L
qJ .. , .....1' dx'"
l~d~m-l
"In
1 ,
(p -1).
l..J
l~").sm-l
1\ ..• 1\
't'mClt:J ... Clp-l
dx"p
dxm
1\ ... 1\
dx"p-'.
Let
and
1
t/I = (p -1)!
L
l~fl).sm-l
9 dXfl' .. , .....1'-,
1\ ... 1\
dx"p-'
.
5.
71
VECTOR BUNDLES
Then dljJ
=
1
m-1
L
L
(P-1)!1:s;,,;.:s;m-1 ,,=1
+
1
,L
(p -1).
og
-dx"
ax"
A
dX"1
qJm"I"'''P_I dxm
A
A'"
dX'"
A
dX"p-'
A •. , A
dX"P-'.
1:s;"... :s;1II-1
Hence qJ(x) - dljJ(x)
=, L 1
m-1
p.
,,;.=1
X"I"'''/X) dX'"
A ••. A
dx"p
=X and dX = dcp - ddljl =dqJ =0.
But 1
dX = -
p!
L aX" ...m "
m- 1
I
OX
P
dx lll
A
dX'"
A ••• A
dx"p
+ terms not involving dxm. So 0X"I' .,,)a~ X(x 1 , " ' , xm -
= 0 and this implies that X is independent of xm, X =
1 ).
By induction X = du and thus qJ
= d(1jJ + u).
Q.E.D.
We denote the sheaf over M of germs of p-forms (COO p-forms) by AP. Then we have: COROLLARY. (to Poincare's Lemma) and only if cp = dljl, IjJ E AP-1.
Let qJ E AP, P ~ 1. Then dqJ = 0, if
Proof We need only remark that d can be defined on the germ level and then use Theorem 5.1.
We notice that A O = ~, the sheaf of germs of COO differentiable functions; and qJ E A 0, dqJ = 0 if and only if qJ is locally constant. Thus: THEOREM 5.2.
The following sequence (of sheaves over M) is exact d
d
d
d
0----+ C - - + A 0 --+A 1 ----+ ... ----+ An ----+ 0,
where n = dim M. THEOREM 5.3.
The sheaf AP is a fine sheaf.
72
SHEAYES AND COHOMOLOGY
Proof Given any locally finite covering {Uj} of M, we have a partition of unity {Pj}, LPj= 1 and
{xIPj(x)#O} £: U j ,
where Pj(x) is a Coo function on M with pix) h j : AP ..... AP by
Then L hj
= id and hiA~) = 0 if Pj(x)
=
o.
~
O. Define a homomorphism
Q.E.D.
THEOREM 5.4. (de Rham's Theorem) Let dAP be the image of AP under d. Then dr(M, Aq-l) £: r(M, dAq-l) and
HO(M dAq-l) Hq(M, C) ~ dHO(~, Aq-l)· Proof We note that dAp-l cAP is the subsheaf of AP which consists of germs of p-forms cp such that dcp = O. A form cp is called d-closed if dcp = O. A form cp is exact if cp = dtfJ. Thus Theorem 5.5 says that the closed p-forms modulo the exact p-forms is isomorphic to the cohomology with complex coefficients. For the proof we use Theorems 5.4, 5.3, 4.3, and 4.1. Since 0 ..... dAp-l ..... AP ..... dAP ..... 0
is exact for p ~ 1,
0 ..... HO(dAP-') ..... HO(AP) ..... HO(dAP) is exact. But Hq(AP)
=0
for q
~
-+
Hl(dAp-l) -+
.••
1. Thus,
0-+ HO(dAp-l) -+ HO(AP) -+ HO(dAP) -+ Hl(dAP-l) -+ 0
is exact, and Hq-l(dAP) ~ Hq(dAP-l) for q ~ 2. Equation (3) gives
H'(dAP-')
~
HO(dAP)/dHO(AP).
For p = 0 we have d
0---1(: - - - AO - - + dAD - - - O. Thus 0-+ HO(IC) -+ HO(AO) -+ HO(dAO) -+ H'(IC) -+ H'(Ao) -+
Since
H1(dAo) -+
... -+
Hq-l(dAo) -+ Hq(1C)
-+
O.
(3)
6.
73
A THEOREM OF DOLBEAULT
for q ;::: 2. Thus,
Hq(C)
~
HQ-1(dAo)
~
HO(dAQ-1)
~
HQ-2(dA 1) ~ ... H1(dAq-2) for q ;::: 2.
= dHO(Aq 1)
Q.E.D.
These two statements prove the theorem.
REMARK. We have actually proved more. Let Y be a sheaf over X. By a fine resolution of Y we mean an exact sequence (al), h
h
h
(f!A) 0 ---. Y ---. .910 ---. .91 1---. .91 2---. ...
such that each .91 P is fine. *THEOREM 5.4.
If (al) is a fine resolution of Y, then Q ~HO(X,h.91q-1) H (X, Y) = hHo(X,.91 q 1)
Proof
6.
for q ;::: 1.
Same method.
A Theorem of Dolbeault (A fine resolution of (9)
Let M be a complex manifold with covering {Vj} by coordinate patches where (z~, ... , zj) are local complex coordinates on V j ' Let
1
p.q!
L <Pa., ... p, '" P (z) dza., 1\ ••• 1\ dz Pq • q
DEFINITION 6.1.
where
and
Om "f'
1
a
= -" '\' -:Ol P "f'a., m ... {J q dz P 1\ dza. L... p. q. «., "·P.{J, "·P. uZ
1\ ••. 1\
dz Pq ,
SHEA YES AND COHOMOLOGY
74
where
REMARK.
I t is easy to verify that
aa = 0,
00 = O.
(1)
From the complex coordinates we get real coordinates x Y defined by
z~
= X2~-1 + i x2a. Then
dz a = dx 2a dz~
= dx 2a -
1
+ i dx 2a
1 _
i dx 2a •
So rn = -r
LEMMA 6.1. Proof
d=
1 (p +q)!
*
~ rn 't -r)."
dxJ., /\ ... /\ dxJ. p + q •
... ,).p+q
a+ o.
We easily check that
a + dz -a = dx -a + dy -a. az az ox oy
dz -
In fact
a+ () = a=1 L n
COROLLARY.
((}
dz a ;-:. uZ
+ dz~
a) = L 2n
ii
:l
uz
0 dxJ. -;, = d.
),=1
Q.E.D.
ax
00= -0 a.
THEOREM 6.1. [Dol beau It's lemma (an analogue of Poincare's lemma)] If a coo (0, q)-form cp satisfies ocp = on a neighborhood U f; en of zo, then there is a Coo (0, q - 1)-form IjJ on W with Zo E W f; U (W open) such that cp = oljJ on W. First, we turn to:
°
LEMMA 6.2.
LetJ(z) be a bounded Coo function on U ') =-lfJf(Od~d'7 glz ,
(-z
1t U
where (
= ~ + i'7. Then 9 is Coo
on U and
og/oz =f
<;;
C.
Suppose
6. Proof
Pick Zo ~I
E
75
A THEOREM OF DOLBEA ULT
U and let
= {zllz - zol < 6},
~2 =
{zllz - zol < 26}.
Suppose z E ~I' Let I I (0 be a COO function which is identically equal to I in and is zero outside of ~2 • Then
~1
where
h(O = 0 for (1: ~2' 12 «) = 0 for' E ~1' Then we have g(z)
=
gl(Z)
+ g2(Z),
g.(z)
where
= -=-!. Iff'(Z) d~ dlJ. n
u
(-z
For Z E ~1> g2(Z) is holomorphic, so (a/a'2) g2(Z) = O. Since h = 0 outside of we can consider
~1
Let
( - z=
W
= S + i t = re ilJ •
Then 1 Iff1(W gl(Z) = -n:
+ z) dsdt=
W
1 fffl(re ilJ -n:
IC
IC
= -
~ If II (re ilJ + z)e- i8 dr de. I[:
Thus gl(Z) is COO in z. Then
a I, (w + z) ds dt = --1 If -:. n
and
az
w
+ z)r dr de °IJ re'
76
SHEAVES AND COHOMOLOGY
We use Stokes' theorem. If W is a domain in C with boundary C a Jordan curve, then
t
u dt - v ds =
If (~: + ~~) ds dt. w
= udt - vds, then d({)
If (()
=
(-auas + -au) at ds
1\
dt so
We notice that
dw
1\
dw = (ds
+ idt)
1\
(ds - idt) = - 2ids
1\
dt.
Hence,
a 1 oz gl(z) = 2ni
If owa fl(w + z) dw W dw = 2ni - 1 If [ dW] d fl(w + z) -; , 1\
c since
ofl dw 1\ dw =----ow w Thus, if we set
r=
{zllz - zol = 2e},
y = {zllz - zol = e}, then
a
--:: gl(z) = - -1. oz 2nl
(Ir fl -dw - fy fl -dw) - 2nl -1. If d (f ldW) W
W
W
41
1 = -. 2m
f fl(w + z)-dw - -2m1. If d(fl -dW) . W
y
W
Taking the limit as e -+ 0 we see that
og oz
=
ogl oz
= fl(z) = fez).
Q.E.D.
6.
A THEOREM OF DOLBEAULT
77
Instead of as~uming that 1 is bounded we could have proved, using the same proof: LEMMA
is
COO
6.2. If I(z) is
COO
for Izl < R, then
for Izl < R - e and
o~~) = I(z) where e > Proof
for Izl < R - e,
°and R - e > 0. (of Theorem 6.1 ) We may as well assume U
= UR = {zllzl < R, where
lal
= maxlz"l}.
Then we want to prove that if oq> = 0,
1 q.
q> = -,
L q>P! ... p
dzP!
1\ •.. 1\
dzP,
q
is a Coo(O, q)-form on U R , then for any e > 0, R - e > 0, there is a COO(q - 1)form l/J on UR - t such that
ol/J
=
q> on U R-t
(q ~ 1).
Suppose
By this we mean that form q> does not involve differentials of coordinates for i > m. The proof will be by induction on m with fixed q,n. First we consider m = q. Then Zi
and
Hence, (ojoz")q>, ... q =
°for
(X
~
q + 1. Define
If
g ( z ' , ... , z") = --1
q>,oo.i('Z2'···'Z")dJ'd y , .. ",. \,-z
TC
~2 + ~2
< R - (tl")
78
SHEA YES AND COHOMOLOGY
ThengisC'" on UR-(£/R) and og/OZI = (P!. .. iz).SoletljJ(z)= g(z)dz 2 dz q • Since O(p/oz« = 0 for IX 2 q + 1, og/oza = 0 for IX 2 q + 1. Thus,
±
aljJ(z) =
a= 1
0:« dz a OZ
/\
dz 2
/\ ••• /\
/\ ••• /\
dz q
= <po Now assume the lemma is proved for m
= -1, p.
"En L.
"1'(11 '" /J.
~
k - 1 and consider m = k. So,
dZ(11 /\ ... /\ dz(1.
/J1';'k-l
and
Thus, 0 = (%z«)
= 0 for
If
9P2 ... /l.(z) -- -;-1
~2~2 <:
IJ.
2 k
+ 1. Define
•••
,
Zk-1 (
"
Zk+ 1 ...
"
R -(I
for
+ 1.
!X
2 k
/\
dz-/J· •
Let W
=
1 ,
(q - 1).
2 " L. g (12'''(1. d Z/l
/\ •••
P,$.k-I
Then
'Ow =
1 (q - I)!
ZR)
k- I
og
L L -
P,$.k-l a=1 oza
dz a
1\
dZ/l 2
/\ • • •
/\
dz fJ •
d~ dlJ ,
6.
79
A THEOREM OF DOLBEAULT
Then
". = U
1
(f) ."
"u L... 1l""fJ
0'" = - I \.U
p.
fJisk-1
q
dill,
1\ ••• 1\
dill.
is Coo on U R-£,. By induction there is a 1/1 which is Coo on U R-£'-£2 such that u = 01/1. So cp = o(w + 1/1) on UR-£'-£2' Q.E.D. We can make an improvement. THEOREM 6.2. (also called Dolbeault's Lemma) If cp is COO on U Rand ocp = 0, then there is a COO 1/1 on U R such that cp = 01/1 on U R. We first consider the case of a q-form cp with q ~ 2. Let U. = R. On each U d1 there is a I/Iv such that bl/l. = cpo We construct Ifr I' Ifr 2 , ••• by induction on v such that olfr v = cp on U v+ 1 and Ifr v = Ifr v+ 1 on U v' First set Ifrl =1/11 on U 2 = U R-(rI2)' Then 0(1/12-1/11)=0 on U 2 · Hence there is a 0 on UR-(r/2)-£ such that = 1/12 - 1/11' Take p(z) ~ 0, Coo and such that Proof
U R-
rlv,
°< r <
oe
p() z
I, on ={
U1
outside U R-(r/2)-2£'
0,
Thus p, 0 are defined on U R, and p(}
Define Ifr 2 = 1/1 2
-
=
{O, 0,
outside U R- (rI2) - 2£ onU1=U R_r ·
o(p . 0). Then olfr 2
=
01/1 2 on U 3
and Ifr 2
= 1/1 2 - o(} = 1/1 1 = Ifr 1 on
U1 •
Suppose we have Ifrl' "', Ifrv such that olfr). = cp on U H1 for A. ~ v and Ifr). = IfrHI on U). for A. ~ v - l. Then 0(1/1 v+1 -Ifrv) = on UV + 1 = U R-(rjv+l) and 1/1 v+1 -Ifrv = D(} for some () on UR-(r/v+I)-,' Define IfrV+I = I/Iv+l - o(p(}) where
°
p=
{I,
on ~v 0, outside UR-(r/v+I)-2£'
Then al/l v+! = cp on U v+2 and Ifr.+l = Ifrv on U y • Then 1/1 is well defined, Coo on U R, and 01/1 =olfrv = cpo Next consider the case q = I. We assume 1/1 v on U + 1 constructed so that O(I/IV+I - I/Iy) = on U d1 , cp = ol/lv on U v+1, and I/Iv is a Coo function where cp is a given (0, I)-form such that ocp = 0. Then 0(1/1 v+1 - I/Iv) = implies that
°
y
°
SHEA YES AND COHOMOLOGY
80
IjJv+1 -ljJv = Iv is a holomorphic function on VV+I . Let ~I = 1jJ1' Since IjJ 2 - ~ I = It is hoi om orphic on U 2 let PI be a polynomial such that Ipi - Itl < -1 on VI' Set ~2 = 1jJ2 - Pion V 3 • In general, given ~v such that oljJv = o~v on Vv+1 and OIjJV+1 = cp o~v on U v+1, we can find a polynomial P. such that Ipv - /.1 < 1/2v on Uv since 1jJ.+I -~. is holomorphic on VV+I · Then I~.+l - ~vl < 1/2v on V. so the limit exists, lim ~. = IjJ and
eo
1jJ=1jJ.+L(j~-P,,)
,,=v
= 1jJ.
where
on
Uv
+ h.,
oh. = O. So oljJ = oljJ. = cp for all v.
Q.E.D.
COROLLARY. Let AO,q denote the sheaf of germs (over M) of Ceo (O,q)forms. Then AO = AO,o = ~ and D
D
Ii
Ii
D
o_m_~_AO,l_Ao,2_"'_AO,n_o
is exact.
Proof
COROLLARY. Proof
AO,q is a fine sheaf. Hq(V R ,m) = 0 for q
Q.E.D. ~
1.
Use Theorems 6.2 and 6.3 (Ho(V R, oAO,q-l) = oHo(U R, AO,q-l)).
Q.E.D. We can generalize these results. Let cp be a Ceo(p,q)-form. We have the following sequence of statements whose proofs are similar to the previous proofs [see Gunning and Rossi (1965)]. LEMMA 6.2'. On U R, a (p,q) form cp,q cp = oljJ, where IjJ is a (p,q - I) form. COROLLARY'.
~
I satisfies ocp = 0 if and only if
Let OP be the sheaf of germs of holomorphic p-forms over M.
6.
81
A THEOREM OF DOLBEAULT
Then
c
c
c
O-OP-Ap,O-Ap,I_"'-AP,"-O is a fine resolution of OP,
REMARK. Let cp(p,q) be a (p,q)-form. Then ocp(p,q) = 0 if and only if cp(P.q) = arjl(p.q 1). So,
ocp(p,q) = 0 if and only if cp(P.q) = orjl(p,q
1)
Hence (on U R) ocp(p,q) = 0 if and only if cp(p,q) = orjl(p-1. q), P ~ 1. IfJ(z) is holomorphic on /z/ < R, then dgjdz = Jwhere g(z) = J~ g(O d(. LEMMA 6.3. Let cp be a holomorphic p-form on U R , P ~ 1. Then there exists a holomorphic (p - I)-form rjI such that drjl = cp if and only if dcp = O.
Proof d = 0 + 0 so dcp = ocp if q> is holomorphic and if dq> = oq> = O. The rest of the proof uses the observation made just before the statement of the lemma. The details are essentially the same as in the previous proofs and are left to the reader. d
THEOREM 6.4. is exact.
O-IC-
REMARK.
is not a fine sheaf.
QP
(!)-
d
0'- 0 2 -
••• -+
0"-+ 0
Now we consider holomorphic vector bundles F over M. Let F be defined by the l-cocycle {fjk(Z)} wherejjk(z) = [fjkP(zHz,p= ' ..... m' On each coordinate patch (!)(F) = (!) E9 ... E9 (!) (m times). Let q> be a section of (!)(F) over an open set W ~ M. On W n UJ '
(p(z) = [cpJ(z), "', cpj(z)], where CPJ is a holomorphic function on W n U j' and
cp7(z) = L JJkl'(Z)CP~(z)
for z E W n U j n Uk'
I'
By a (p, q)-form cp with coefficients in F over W we mean cp(z) = [cpj(z), .. " cpj(z)] where each q>1(z) is a (p, q)-form over W n Uj such that m
cp;(z) =
L fMz)q>~(z)
1'=1
SHEA YES AND COHOMOLOGY
82
as differential forms for Z E W n U j n Uk' We define Ap,q(F) to be the sheaf over M of germs of (p, q)-forms with coefficients in F. At each point x E M, the stalk A~,q(F) = Af,q EB ... EB A~,q (m times). We have: iJ
THEOREM 6.5. 0 ~ (!)(F) ~ AO(F) ----- Ao,I(F) ~ '" ~ oAO,n ~ 0 resolution of (!)(F).
IS
a fine
Proof We remark on the definition of 0 and leave the proof to the reader. The functions !jk are holomorphic so (O/OZ)Jjk(Z) = O. Hence,
So 0 is well defined and if cp is a (p, q) form with coefficients in F, ocp is a (p, q + l)-form with coefficients in F. We note that Ocp~(Z) =
2: f;k,,(Z) ocpHz) + 2: af;k"(Z)CP~(z), "
11
and ocp is not well defined on AO,p. We also notice that Ap,q(F) = Ap,q®(9(!)(F). Remember that cp(p,O) is a differentiable section of T* ® ... ® T* which is skew-symmetric. We define T* /\ ... /\ T* to be the subbundle of T* ® ... ® T* consisting of those (z, (<1\ ...
[3] Geometry of Complex Manifolds I.
Hermitian Metrics; Kahler Structures
Let M be a complex manifold. We want to introduce in M something analogous to a Riemannian metric which is "compatible" with the complex structure of M. Remember that for a Riemannian manifold the element of arc length is given by ds 2 = L gaP dxadx p. 1.1. An Hermitian metric on a complex manifold M with local coordinates (zj) is given by DEFINITION
n
ds 2
=
L a,p=
gjallCz) dz'j dz~, 1
where gj«lJ(z) is a COO section of T* ® f* such that (1)
gjalJ(z)
(2)
L:, fl =
= gjPii.(z) 1
g jall(Z) (a
(fl
(Hermitian symmetric) ~ 0, and equality if and only if ( = 0 (positive
definite). 1.1. Given any complex manifold M, we can introduce an Hermitian metric on M.
THEOREM
Proof Let cJlf = {Uj} be a locally finite covering of M wIth coordinate patches U j and let (zJ ' ... , zj) be coordinates on U j ' On each Vj we have a metric n
n
L dz; dz; = ll,fl=1 L ~afl dza dz fJ , '\=1 the usual Euclidean metric. Let {p j} be a partition of unity subordinate to cJlf; that is,
Define
83
84
GEOMETRY OF COMPLEX MANIFOLDS
We claim this is an Hermitian metric. On Uk ds 2=
n
L gkaP(Z) dz'k dzf . a.{J= I
Since
Then
t
"./l.A=1
if (
=1=
t ozj ("1 a.A=lOZ'k
(ozj)(ozj)(,,(~ = 1 oz'k
oZ~
2
>0
O. Thus ds 2 is positive definite and the rest is easy to check.
Suppose given an Hermitian metric ds 2 = A. , fA'lJe{1 " 2 ... ' " n I "'ii} by
( )_(0 gPIl -
gN./l
L gjap dzj dz~.
Q.E.D.
Define gjAp. for
g jap) 0 .
Then 2 L gjap dzj dz~ = LA.Il E {l.2 .....ii} gjAp. dz; dzj, where, as has been our custom, = zj. We associate to ds 2 a differential form of type (I, I),w = iL gjap dzj 1\ dz~, where i =
z1
J=l.
Or if one
REMARK.
prefers W«(, '1) = tigjap('j'1~ - '1j'~)
= -!igj{Jii("'1~ - '1"~) = w«(, 17). So
W
is a real form.
DEFINITION 1.2. 2 L g"ll dz" dz~ is called a Kahler metric if dw = O. Then we call w a Kahler form. M is called a Kahler manifold if we can define a Kahler metric on M.
REMARKS
(I) zero.
An Hermitian metric can always be defined, but generally dw is not
1.
HERMITIAN METRICS; KAHLER STRUCTURES
85
(2) Compact Kahler manifQlds have many prQperties similar to' (prQjective) algebraic manifQlds. (3) Every algebraic manifQld is a Kahler manifQld. (Recall that algebraic manifQlds are by definitiQn cQmpact submanifQlds Qf IPN fQr SQme N.) (4) There are many nQnalgebraic Kahler manifolds. CONJECTURE. A cQmpact cQmplex manifQld Qf cQmplex dimensiQn 2 is Kahler if and Qnly if the first Betti number is even. We knQw that M Kahler implies that the first Betti number is even (TheQrem 5.4, CQrollary 2). We have the fQIIQwing facts: (1) If hI (M2) is even, then M2 is a defQrmatiQn Qf an algebraic manifQld where by M2 we mean a cQmpact cQmplex manifQld Qf cQmplex dimensiQn 2. Thus there is a cQmplex family {Mtlltl < 2} such that M5 is algebraic and Mi = M2 [KQdaira (1964)]. (2) Any small defQrmatiQn Qf a cQmpact Kahler manifQld Mo is Kahler; that is, if Mo is Kahler, then M~ is Kahler fQr small enQugh t. (3) It was cQnjectured that any defQrmatiQn Qf a (cQmpact) Kahler manifQld is Kahler. This turned Qut to' be false [H. Hironaka (1962)] fQr dimensiQn ~ 3. SO' we make the cQnjecture fQr M2. NQW we cQllect SQme facts abQut Kahler manifQlds. Let U = {zllzl < I}
c:
en.
PROPOSITION 1.1. A fQrm
A
dz ll Qn U satisfies drp
= 0 if and
Proof
(I)
If
ao f) = (0 + fJ)( 00 f) = ooof = -008f
=0.
(2) SupPQse d
o= orp = 001/1(1,0) = - oal/l( 1.0)
86
GEOMETRY OF COMPLEX MANIFOLDS
and
So
a (al/l p
az). az~ -
a1/1~)
azfJ = 0
and a1/1 (1 ,0) is a hoI om orphic 2-form. According to Chapter 2, Lemma 6.3, there is a holomorphic I-form '1 on U such that al/l(I,O) = d'1 = a'1. Thus 0[1/1(1,0) - '1] = 0 and there is a Ceo function f such that 1/1(1,0) - '1 = af (see the remark after Chapter 2, Theorem 6.3'). Hence, q>
= 01/1(1.0) = 0('1 + of) = oof = oo( -f).
Q.E.D.
L
L
Let 2 gMI dzj dz~ be an Hermitian metric and w = i gMi dzj " dz~ be the associated form on Uj where the manifold M = u U j' Assume U j = {zjllzjl < I}. THEOREM 1.2. w is a Kahler form (that is, dw = 0) if and only if there is a differentiable function K j on each U j such that w = 00 K j on U j .
Proof
Use Proposition 1.1.
PROPOSITION 1.2.
jf)n is a Kahler manifold.
Proof jf)n = Uj=o(U), where U j = g I C= (Co, ... , Cn), Cj =1= OJ. On U j ' 0 '- 1 ) then we have (a ffi) ne coord mates Zj = (Zj, .•. , Z) , Z}'+ 1 , ... , z'j, where zj = C~/Cj' We set K j = 10g(1
+ L,IZjI2) "*J
= IOg(JoICaI 2) - 10gICY on U j , Then
Kj
-
Kk
=
10gICklCl
= loglz~12
= log z~ + log z~ on Uj
n Uk'
Therefore aOK j = ooK k on Vj n Uk' and we can define a global form w = i aOK j on each Vj • Clearly dw = O. If w = i L gj(f.jj dzj " dz~, we must show that gM is positive definite; Hermitian symmetric will be left to the
reader. Take j
87
HERMITIAN METRICS; KAHLER STRUCTURES
1.
=0
and let za
= z~ . Then
10g(1+ J,lzaI2),
Ko = so
and
Then
where n
(C z) = The Schwarz inequality implies that I(C is a positive definite form. Q.E.D. THEOREM 1.3.
a;I ,,aza. zW ~ «(, 02(Z, zW hence Equation (1)
Any submanifold of a Kahler manifold is a Kahler manifold.
Proof Suppose that N eM and w is a Kahler form on M. We claim that the restricted form wiN' which we shall shortly describe, is a Kahler form on N. To define the restriction we proceed as follows: On each coordinate patch Vj with Vj n N", ct> we choose local coordinates (z} , ... , z}, ... , zj') = z~ 11.,1 ••• w'.) with r = m - n m = dim M n = dim N so that (z'J ' ... ' J ' J' 'J ' ,
N n U j = {(zj,· .. , zj, 0,···, O)}. Suppose
= ~"
..•
ap
1\ •.• 1\
dz~pJ
(z.J' w.) J
dZ~1 1\ . . . 1\
(z. w.) dz al J'
J
+ "L,
...
IIp
J
dw Jaq
1\ •.• 1\ dw~p
(2)
dzrl:J p
(3)
J '
then
1
= - I "L,
...
IIp
(z.J' 0) dzrl:J 1
1\ .•• 1\
88
GEOMETRY OF COMPLEX MANIFOLDS
is a form on N since
zj =
fj~(Zk' wk),
o=
g;k(Zk, 0),
and on N,
Similar remarks apply to forms of type (p,q). We see that eplN is a form of type (p,q) and that
d(ep I N) = dep I N. Thus we see easily that wiN is positive definite and d(w I N) = O. COROLLARY.
Proof
Q.E.D.
Any algebraic manifold M is a Kahler manifold.
M
$;:;;
Ill" for some n.
THEOREM 1.4. If M is compact and Kahler, then the Betti numbers b2k(M);;::: 1 for k = 1, "', n, n = dime M. Before giving the proof we shall embark on a brief discussion of integrals of forms, specifically m-forms, on a compact differentiable manifold M of dimension m. Suppose xi = (x] , .. " xj) is a local coordinate on Vi c M. Then an m-form
=
A ••. A
A ••. A
A •.. A dx'~
To be able to define
J
f
dxj on Vj
dx,!: on Uk' and on U j n Uk,
axa.) = det (j dx I ;) P k
A ... A
UX k
dxmk'
(4)
ep we make the assumption that M is oriented. This
M
means that there is a covering of M with local coordinate patches (U j ,Xj) such that det (axj/ax~) > 0 on Vj n Uk' Any local coordinate (Xv, ~v> is positively oriented if det (a~~/ ax~) > 0 for all j such that Uj n Xv "1= cJ>. Let {p j} be a partition of unity subordinate to {U j}'
J
f pix)
M
= Lj
Uj
M
1.
89
HERMITIAN METRICS; KAHLER STRUCTURES
dent of the choice of covering by local (positively oriented) coordinate patches and partition of unity subordinate to it. We leave this to the reader.
PROPOSITION
1.3. If qJ = dt/J, then f
qJ
= O.
M
t/J = Lj P/x) t/J j(x), so
Proof
f
qJ= M
f~d(pjt/J)=Lf d(pjt/J), J
Uj
J
where
Uj = {XjJJxjJ < We will show that each
f
rJ and
pix)
0 for x
=
rt Wj ~ Uj.
d(pd) = O. To simplify notation let us drop the
Uj
subscripts. Then m
Pjt/J/x) = L h,.{x) dx l
,,= 1
and h,.{x)
A •.•
= 0 for x ¢ Wj' We calculate
dX,,-l
A
dxa+!
A'"
A
dxm
J d(pjt/J): Uj
f
f =f
d(pjt/J) =
Uj
d(L heX dx 1
A ... A
dx,,-I
A
dX,,+1
A ... A
dx m )
Uj"
L ( -1)"+ 1 oh" dx 1
ox"
Uj "
•••
dxm
=0, since
oh" J~ dx" r
-ruX
= h..(r) - h,,( -r)
Q.E.D.
= O.
A complex manifold M is naturally oriented. For if (zJ ' .. " zj) are local coordinates and zj = xJ" - I + i xJ'x, then the coordinates (x ~, ... , x;") give a covering of M and the determinant of a change of charts is detJozj/ozW > 0 hence M is oriented and (x] , ... , xJn) is a positively oriented chart.
Proof
(of Theorem 1.4)
Let w = iL9"1l dz"
A
dzlJ. Then
90
GEOMETRY OF COMPLEX MANIFOLDS
We claim
f
w" > O. For
M
w" = (i)"
11) sgn (If3 L sgn (a1I ... ... all I
a. fJ
... f3n) g~ PI ... 1
..."
and
1 ...
n)
= sgn ( f3 ... f3 I
=
where 9
II
gIlT ... g"ii
n)
1 ... sgn ( f31 ... f3" g,
= det(gaP). Therefore, w" =(i)"n!g dz l
1\ ••• 1\
dz ii •
Since
dz l
1\
dz T = (dx l
+ i dx 2 )
= -2 i dx 1 w"
=
1\
2" n! 9 dx 1
1\
(dx l
-
i dx 2 )
dx 2 , 1\ ••• 1\
dx 2 ".
We have assumed (gap) is positive definite so 9 > O. Thus SM of > O. If w" = dlji then M w" = 0 by Proposition 1.3. Thus w" =f. drj;. In fact, we claim w 1\ ..• " w = wk =I drj; for any Iji. Since dw = 0, d(w k ) = 0 and if w k = drj; then w" = wk 1\ W"- k = drj; 1\ W,,-k = d(rj; 1\ W"- k ). Now recall that
J
.
2k
•
HO(M, dA zk -
b u = dime H (M, IC) = dime dHo(M, A 2k -
1)
1)"
The facts just proved show that w k E HO(M, dA 2k -
wk
Thus, bZk
I ),
rt dHo(M. A lk- I).
z I.
THEOREM 1.5. If M is compact Kahler and if N eM is a compact complex submanifold, then N is not homologous to zero in M. Proof
(sketch)
We first prove:
I.
HERMITIAN METRICS; KAHLER STRUCTURES
91
PROPOSITION 1.4. (Stokes' theorem) If W,,+I is a compact differentiable manifold with boundary aw,,+1 = M", then Jw"+! dlj; = JMn Ij; for any n-form Ij; on W= W,,+I.
Proof We cover W with a locally finite family of coordinate patches {U j } such that U j = {xjllxjl < r j } if Uj ~ int(W) and if Uj n M =I- cp, then Uj={xjllxjl I, -rj<x)::;rj }, and MnUj={xjlxJ=,). We choose a partition of unity {Pj} with the following properties: (I) supp p) = {x I pix) > O} c Uj if Uj n M = cp. (2) supp Pj £:; {xjl-rj < x) ::; r j , Ixjl < r j , ex ~2}. (3) Pj is Co, Pj ~ 0 and L Pj = I.
Then Jwdlj; = LjJwd(pjlj;) = LjJuJd(Pjlj;). From Proposition 1.3 JVjd(Pk Ij;) = 0 if Uk n M = cp. Suppose Uj n M =I- cp and Ij; = L::: Ij;j dx), /\ ... dxj-l /\ dx a + 1 /\ ••• /\ dxj+l on U j • Then
and
For ex
= I,
Hence, f
VJ
d(p"")=f p )'1') .. I,I(r.) 'x~ ... ' x~+I)dx~···dx"+1 )'1' )' ) ) ) Vj
Thus,
where M =aw. For the proof of the theorem, suppose M" is compact Kahler and N m is a complex submanifold N m eM". Suppose N = aWe M for Wan embedded suhmanifold. Then if w is the Kahler form on M, 0 < IN w m = Jw dw'" = O.
92
GEOMETRY OF COMPLEX MANIFOLDS
This contradiction proves the theorem in this case. Generally if N is homologous to zero, we do not have such a convenient situation. One must change the proof, and we supply no details here.
2.
Norms and Dual Forms
Let QP be the sheaf over M (a compact complex manifold) of hoI om orphic p-forms. Let AM be the sheaf of Coo (p, q)-forms on M. We want to introduce an Hermitian scalar product (cp, 1jJ) for cp, IjJ E r(AM), which makes r(AM) into an (incomplete) inner product space. We introduce an Hermitian metric 2 L gja.fJ dzj dzq = 2 L gap dz a dz P on M. Associated to this metric we have the form w = i L gap dz a 1\ dz P and w" = 2" n! g dx l 1\ ••• 1\ dx 2n as before where x 2<1- I + i x 2a = za and g = det(gaP)' We denote the inverse (g
LgapgPY=c5~.
P
P
The length 1" of a tangent vector ( is given by 11) = La.1I g
«(.
1'2, = La.1I gap (af!
al cp(z) = _1_ p! q! L cpal ... Pq dz
1\ '"
1\
dz Pq
tjJ(z) = _1_ L tjJ P dz
1\ •.• 1\
dz Pq
and the
and •
Then at each point z E M we define 1 = __ «(tl't" "')(z) 'I' , ,
---"L- gA.a •... gPIIl • ... gPqllq(tl _ 't'a • ... Pq."'1'.1. •.•. Ilq'
p. q. a.p . .1..11
DEFINITION
2.1. (lp, 1jJ)
The inner product of two forms cp, IjJ is
=
f (lp, tjJ)(z) w:n. = f (lp, 1jJ)(z)2"g dx M
M
(,) satisfies the following properties: (cp, tjJ) = (tjJ, lp), (2) (aljJ + b x, cp) = a(ljJ, lp) + b(X, cp), (3) (cp, cp) ~ 0, (4) (cp, cp) = 0 if and only if cp = o. (1)
We define
(1)
I!cpl! =J(CP, cp) as usual.
l ...
dx 2".
(2)
2. THEOREM 2.1.
93
NORMS AND DUAL FORMS
There is a linear map *: r(Ap.q)
of =
r(An-p,n- q) such that:
*i/I(z) ,
(1)
(
(2) (3)
*1/1 = *i/I (that is, * is a real operator), **I/I(p,q) = ( -l)p+q I/I(p,q).
A
~
Proof Before giving the proof let us fix some notation. Let n = dim M. We denote as follows:
and «(Xl' " ' , a p ' (Xp+1> " ' , (Xn) is a permutation of (1, "', n). For example, n=5 and A 2 =(2,4), An-2=(1,3,5). Similarly for Bq=(p[,· .. ,pq), Bn- q = (P q +l, "', Pn). Then with this notation we write a (p, q)-form (3) where dz Ap =dz'"
A •• , A
dzl1. p, and so on. We denote
q _ "gillAI .1.ApB 'I' L..
.••
gjiqP ••'I'I. Al
...
Apjil ... jiq'
).,1'
Then
L L (-lyql/l ApjJq dz Bq A dz Ap = L L (i/lhqA q dz Bq A dz A•.
=
Thus,
We can now write Equation (1) as
(
L
Ap, Bq
B p
q
I/IApB.
= (-l)pq
L
Ap, Bq
B
p.
i/lBqA P •
Remember
Then we define
*.1. 'I'
= (i)n( _l)t
A
dz B.. - p.
(4)
94
GEOMETRY OF COMPLEX MANIFOLDS
First we prove Theorem 2.1 (I); that is,
If tf; is as in Equation (3),
*.T. = (i)"( _ 1)tn(n-l J+qn 'I'
,
,
L. L.
A ,,8,
9
.T.8 qAp dz A"- p ApA"-pli,8"-, 'I'
/\
dz 8,,- q.
(5)
Let
where Ap ({) /\ *J, ~ 0/
= V'l' "', ,l.p), Nq = (VI' "', vq) with
= (+ iii), 9 ~
-
the usual conventions. Then
;r.B,Ap(n dz"p /\ dz Nq /\ dzA,,-p /\ dz B,,-, rApRq
ApA,,-pBqBn-qo/
=(±in)(-l)q(n-p)I("')elz"p /\ dz A.. - p
/\
dz/Vq /\ dz Bn - o,
where (±n = in( _l)t(n-l)n+qn from (4). Now, dz"p /\ dzA,,-p #- 0 if and only if Ap = Ap; similarly we need only consider N q = Bq. Thus (() /\ *.T; = in(_l)tn(n-lJ-qp, ~
~
in
_
J,B,A pg
~ ~ApBqo/
ApAn-pBqBn-q
dzApAn-p /\ elz808n-q.
Recall as in the proof of Theorem 1.4
9 AI1An-pBq Bn-q -- sgn
dzBqBn-o where sgn
Cl
= sgn( 1 ...
11 )
Cl l .,. Cl n
Cl •
sgn
= sgn f3 dz I
f3 . 9 , /\ ... /\
dz ii ,
and so on, and
dz l /\ ... /\ elz n /\ dz T /\ ... /\ dz ii
= (_1)n(n-l)/2 dz l /\ dz T /\ ... /\ dz n /\ dz ii . Thus,
cp /\
*if/ =
inc -l)pq
I
CPApBo if/BoApg dz 1 /\ dz T /\ ... /\ dz ii
= 2"(cp, Ij;)(z)g dx 1
/\ ••• /\
dx 2"[by Equation (4)]
wn = (cp, Ij;)(z) . - [by Section 1, Equation (5)]. n!
Thus Theorem 2.1(1) is proved. For Theorem 2.1(2), *,1, 'I'
= (_i)n(_I)"(1I-1)/2+ PII'g _ ,1,8pAodzA"-Q/\dzBn-p L. BpB'l-pAqAn-q 'Y
2.
95
NORMS AND DUAL FORMS
SInce 111(n - 1)
+ pn + 11 + (n
- q)(n - p)
+ pq
= 111(/1 - I)
+ 112 + /1
== 111(11 - 1)
+ nq (mod 2).
-
nq
Last we must check Theorem 2.1 (3). Before doing this we make an assumption which we could have made for 2.1 (I) and 2.1 (2), which simplifies the calculations. We must check 2.1(3) pointwise and at any point Zo. We may assume by a change of coordinates that giljj(ZO) = biljJ . This will not be true in a neighborhood of Zo; we only assume it at zoo But since we only check 2.1(3) pointwise, each time we verify it at a point we may assume that gajJ = bill! . Then
=
_ ( *'/') 'I' A,1- q 8 n -
p
sgn(p),
= (i)"(_1)1"(II-ll+IIP sgn(ct.) ,I, _ f3 'f'BpAq
at zo and *(*,/,)
_
'P BpA.
= (i)n(_1)1 n(n-l)+n(lI-q)
sgn(BII-pB p)(*,/,) A A 'P A .. -qB .. - p n-q q
= (_I)"+II(n- q)+q("-q)+p(II- P)+"P,/,'I' BpAq_ = (-I)p+q,/,
-.
'I' B.A.
Q.E.D.
* has one more property: PROPOSITION
Proof
But also, qJ
2.1.
ip
1\
* t/!
=
t/!
1\
* ip for
qJ,
t/I
E f(AP,q).
* t/I(z) = (qJ, t/!)(z)wn/n! = (t/!, qJ)(z)w"/n! = t/! 1\ * ip(z). (Remember that w n is real.)
qJ
1\
1\
* t/I
= ip 1\
* t/I.
Q.E.D.
We next define the adjoints of 0, G, and d.
96
GEOMETRY OF COMPLEX MANIFOLDS
It is easy to see that
9: r(A(p,q»
--+
r(A(p,q-l»,
r(A(p,q»
--+
r(A(P-l,q»,
~:
15: reAP) - - . r(AP-l), PROPOSITION
2.2.
Assume M is compact. Then
(ocp, 1/1)
=
(cp, 91/1) for cp
(ocp, 1/1) = (cp,
~I/I),cp E
E
r(A(P,q-l), 1/1
r(A(p,q», 1/1
E
E
r(A(p,q»,
r(A(P+1.Q»,
(dcp, 1/1) = (cp, 151/1),cp E r(Ar), 1/1 E r(Ar+ 1). Proof (Integration by parts; we only prove the first equation.) X= cp A * l{i is of type (n, n - 1). Hence, OX = 0 and dX = OX. Thus,
6
o = fM ox = fM ox = fM o( cp 1\ *l{i)
So
= -
f cp
1\
*(*0(*1/1»
M
=
t
cp
1\
*(:)1/1) = (cp, 91/1).
Q.E.D.
Thus the coefficient (ocp);:P is similar to a .. rotation." We claim that 9 acts like a divergence,
2. PROPOSITION
2.3.
For
t/I E r(A(p,q+l»),
(9t/1)A p P, ... P. = ( -l)P+ 1
t P= 1
REMARK.
97
NORMS AND DUAL FORMS
(~+ a log g)t/lAPPP, ... fl •• 8z fl
8z li
In Euclidean space div 1/1 = L (al/lp/az P) for a I-form, 1/1 =
L t/lP dx P. Proof
(of the proposition) By definition (ocp, 1/1) = (cp, 91/1).
For convenience we omit the index Ap and use the notation
_
a
ap=p' az Then
Since
where
Pi means" omit" Pi' Then . (ocp, 1/1) =
~ (-1)P fLOp CPp, ... fl. t/lPfl, ... Pqg 2n dx l q.
•••
dx 2n
M
= -( -1)P ~ f L q.
CPfl, ... pq ap(I/IPP, ... Pqg)2n dx 1
., •
dx 2n
M
(integrating by parts). Thus,
Q.E.D. Next we make: 2.3. 0 = 09 + 90. 0 maps r(A(p,q») into r(A(p,q») and is called the (comple9 Laplacian. 0 is a partial differential operator and we want to compute its principal part. DEFINITION
GEOMETRY OF COMPLEX MANIFOLDS
98
PROPOSITION 2.4.
(0
,1,) __ 'I' AB -
Let A = (lXI' "', IXp),
~ L.
gVA
2
a8Z A
Ii
A.v=1
P= (PI' "', Pq ). Then
+ "L.
+ hMNX 8
(hMN)' 8
8Z A
AB
A,M.N
AZ
+ M,fiI L k':1:
82
n
L. g iiA
"
A,v=1
If giiA =
REMARK.
8).;)
Z
vZ
ii'
then
(iVA,
- tv (i "
82 VA
OZA
82
n
oz· =
oz· oz·
"
- v";-I
1 =--L -a )2 4 2n (
k =1
Proof
(of the proposition)
(9
8Xk
.
From Proposition 2.3,
= -( -I Y L 8p
Pq-,
+ order zero terms.
We note that
a ma.,·· pp, ... Pq-, = "ga.,A, ... g[JfJ . g[J,P, .•. 8 L.
m
-
P'f'A'··~llt .. ·
P'f'
+ terms of order zero. Thus" lowering indices,"
+ order zero terms. Then (o9
q = _ I Lg [JP { L( _ 1 i+I )
P,Il
i= I
+ terms of order::; I. On the other hand,
Op,
8p
2.
99
NORMS AND DUAL FORMS
Thus,
1"
(90CP)A ... {i. p{i, " {iq=(-1)P + L.,.g/l-P OP(OCP)Apji{i,
+ terms
of order
~
L giiP{op 0/lCPApPdJ, ... -
= -
(7)
1
op op, ({JA p/lP2 ..
+ ... }.
P,I'
Many terms cancel when Equations (6) and (7) are added yielding
(Dcp)Ap{i,{i,···P. = -
L g iiP op O/lCPApP, '"
P,ji
+ lower order terms.
0 = 09 + 90, 6. = dfJ + fJd and
Similarly we define PROPOSITION 2.5.
{i.
Q.E.D.
prove:
L giiP 02jozPozii + lower order terms - L 2g iiP 02jozP ozii + lower order terms.
[J = -
!l =
(.6 is the real Laplacian.)
Proof
Left to the reader.
The operators D, D, and .6 are second-order partial differential operators. Since d = 0 + 0, fJ = 9 + 9, we get
+ 8d = (0 + 0)(9 + 8) + (9 + 9)(0 + a) D + D + 09 + 90 + 09 + 90.
.6 = d8 =
LEMMA 2.1.
Proof.
09
+ 90 is
a first-order operator:
(9cp)~, ... ~pli2 ... = (-lY+ 1
L giiP Op CPa, ... ap/lP' ...
/l,P
+ order zero terms. (09cp)"o ... "plll ... = (-lY+ 1 L giiP{Oao Op CP"II'2 ... -0"1 Op CP"oa2 ... /l.P
+ lower
+ ... } (8)
order terms.
(9)
(09cp)"o ... aplll ... = ( -l)p L giiP op{ oao CPa I ." -;PI ... - 0"1({J"O"2 ... /lPI'''+'..) iiP
+ lower order terms. When we sum Equations (8) and (9) the second-order terms cancel.
Q.E.D.
100
GEOMETRY OF COMPLEX MANIFOLDS
6 =D
COROLLARY.
+ D + first-order terms.
For our purposes the following theorem emphasizes the most important fact about D, D, and 6. For the definition and standard facts about elliptic operators we refer the reader to Palais (1965) or Hormander (1963). THEOREM 2.2.
Proof
3.
D, D, and ~ are strongly elliptic partial differential operators.
Q.E.D.
galJ is positive definite.
Norms for Holomorphic Vector Bundles
The main purpose of this section is to extend the results of Section 2 to vector bundles. Let F be a holomorphic vector bundle over the complex manifold M and r(A(P,q)(F» the space of COO (p, q)-forms with coefficients in F. Let {fJh} be a l-cocycle defining F on the coordinate covering d/i = {U j} of M. Then locally (fJ E r(A(p,q)(F» is given by (fJ
= «fJj(z), ... , (fJj(z» on V j ,
where
_1_" I I
rnJ.(z) -'t'j
p.q.
dzall\ ...
rn
~ 't'jal"'PI'"
1\
dz P1
1\ •.•
,
m
(fJJ(z) = Lfji.(z)(fJ~(z) . •=1
By definition O(fJ = (O(fJJ(z») which is well defined since o/Jdz) = O. Let 2 L g).. dz l dz' be a given Hermitian metric on M. An Hermitian/orm on the fibres of F is defined by specifying on each Vj a positive definite form
aiC ') = L aj)..(zKJfj .I.,v
such that ajJ.v(z) is COO and
ai', 0 = a ('1, '1) where '1 =h j . ,. k
REMARK. Such forms always exist. Let {piz)} be a partition of unity subordinate to the locally finite covering d/i and seta(C O(z) = LjPj(z) Lr=l 1'112. Let
(fJ,
t/J
E
r(A(P,q)(F», (fJ(z) = «fJJ(z)), t/J(z) = (t/JJ(z».
Then we define a Coo function «fJ, t/J)(z) by m
«fJ, t/J)( z) =
L ).,1'= 1
am((fJj , t/J'])( z),
3.
NORMS FOR HOLOMORPHIC VECTOR BUNDLES
101
where (
=
J(
(dime M = n)
M
We want to define the adjoint 3a of 0 (with respect to the metric a). We want to solve (o
for 9a. Let
E
r(A(p,q)(F)), 1/1
E
= (
r(A(p,q+1)(F)). Then m
1:
=
_
L aj).ji
1\
.1/1'1
defines a differential form of type (n, n - I) and hence d1: is a 2n-form. Then
o = JM d1: = fM 01: = fM L a
j).Y
0
1\ *ljIj
But
Thus,
Let m
. "L.j ia)j .a jAy t hat IS,
-
-
1:1''
U
v
).=1
Then
(*9 a 1jI)1' = - ( -l)p+q ~ a;l'( a~ ajvx(*ljIj») , (9 a ljl)1' = Hence:
~ a;I'*(o ~ ajVx(*ljIj»).
GEOMETRY OF COMPLEX MANIFOLDS
102
If we expand this out we get (,9 a t/J)'j = -(*a*t/J)'j -
L a;~*(aajvX 1\ *t/Jj)
A,v
= (9t/J)'j
+ terms
of order zero.
The reader can then easily verify the following: Let
Da = 09 a + 9a o, then
Da = 0 + terms of order
~
I.
Hence: THEOREM 3.1. Da = - L gfJl1. a2 /iJ zl1.az P + lower order terms and hence, D. is a strongly elliptic second-order operator.
4.
Applications of Results on Elliptic Operators
For the results about elliptic operators on manifolds that we need Palais (1965) is a good source. First we fix some notation: Let !£,q = r(A(P, q)(F» where F is some holomorphic vector bundle. We drop the subscript a and let 9 = 9., 0 = Da' PROPOSITION 4.1.
0 is self-adjoint, that is, (Dcp, t/J) = (cp,Dt/J).
Proof
(0 cp, t/J) = «0,9 + 90)cp, t/J) = (9cp, 9t/J) + (ccp, ot/J) = (cp, (09 + 90)t/J) = (cp, 0 t/J). Q.E.D.
The following is the fundamental result about elliptic operators on compact manifolds: Let
THEOREM. (a) dim,YEq < + 00. orthogonal direct sum; so every cp Cp=1}+('
E
(b) !£,q = ,YEq EEl D!£'P. where EEl means !£,q has a unique representation,
(=Dt/J.
4.
APPLICATIONS OF RESULTS ON ELLIPTIC OPERATORS
PROPOSITION 4.2.
103
Dcp = 0 if and only if ocp = 9cp = O. Q.E.D.
(I], ocp)
= (91'/, cp) = 0,
(I], 91/1) = (01].
1/1) = 0,
(ocp, 91/1) = (aacp.
1/1) =
o.
Thus, ytq, 8::e q-1, and 9::e q+ 1 are orthogonal and .!l'q = .}'fq EEl D::e q = .}'fq EEl 09.!l'q EEl /)o::e q £;
.}'fq EEl o.!l'q EEl /).!l'q
£;
.!l'q.
Q.E.D.
We next have the important theorem relating cohomology groups and harmonic forms. THEOREM 4.1. [Kodaira (1953)] Let F be a holomorphic vector bundle on a compact complex manifold. If o.P(F) is the sheaf of germs of holomorphic p-forms with values in F, then
Proof
Recall HP(M, o.P(F»
~
r(oA (P·q-l)(F»/or(A(P.q-l)(F» and
r(oA(P,q-\F» = {cp I cp
Let z.tJ (!e q) = {cp I cp
E
::e q, ocp
E
r(A(p,q)(F», ocp = O},
= O}. Then Z-(!e q )
Hq(M , o.P(F» ~ - -~-o::eq- I •
We claim (1)
The inclusion
GEOMETRY OF COMPLEX MANIFOLDS
104 is obvious. Let cP
E
ZDUl'q), cP
= I] + 01/1 + 9a. Then ocp = 0 so 09a = 0 (09a, a) = 0 (9a, a) = 0
and
9a = O. Thus Equation (1) is true and implies the theorem. COROLLARY.
dim Hq(M, OP(F») <
Q.E.D.
+ 00.
EXAMPLE. Let M n = en/G be a complex torus. If z = (Zl, "', zn) are coordinates on en, then ill = i L:= 1 dz" 1\ dza. defines a 2-form on M n associated to the metric on M n lifted from the Euclidean metric of(.". If z" = X 2,,-1 + i X2« then 0 = -! L;~I (iJjiJxk) 2. Let
1
m - -" m p!q!~'f'",
'f' -
... "pp, ... pq dz'"
1\ ... 1\
dz Pq .
OCP
Then = 0 if and only if L;~l (OjOX k )2 CPAB = O. Such solutions CPAH are necessarily constant so
THEOREM 4.2. [Serre duality; see Serre (1955)] Let F, M be as in Theorem 4.1. Let F* be the dual bundle of F. Then Hq(Mn,OP(F» ~ Hn-q(Mn,on-p(F».
Proof Let ill = i L g"P dz" 1\ dz P be the form associated to a metric on M. If F is represented by UJkv} on r1I1 = {U j } (that is, (J = L~=lfJkv on Ujn Uk) then F* is represented by UZ j,\} (that is, I]j'\=L~"=I.njl]k,\On U j n Uk) and L)J~j,\ ·rjkp = bX· If aiC 0 = (jaj~j (matrix notation; At = transpose of A) is an Hermitian form on F it is easy to see that a transforms as follows:
(z
(2)
We define a conjugate linear map p: F -+ F* by pee) = L aj),ij, that is, p«(j) E F* acts on I]j by p«()(I]) = L aj},v I];(~. From Equation (2) we see
4.
APPLICATIONS OF RESULTS ON ELLIPTIC OPERATORS
105
this is well defined. Using p we introduce a metric on the fibres of F* by b('7, '7)
= a(p-l'7, p-l'7).
It is easy to check that the matrix of b('7, '7) is
bj = aj-1 = (VA) aj on Uj
•
Let .1fp,q(F) = {q> I q>
E
f(Ap,q(F», Dq> = O}
Jlt'n-p,n-q(F*) = {q> I q>
E
f(An-p,n-q(F*», Dq> = O}.
If F were trivial we could define a map, r(Ap,q) - --+ f(AM) - -*+ f(An-p,n- q) q>--+ip--+*ip= #q>. In general we define # q> by (#q»j). = L ajJ.v(*q>j) = p LEMMA
4.1.
#: r(Ap,q(F)
0
*.
-+
r(An-p,n-q(F*» satisfies
#
0
# = (-ly+qid,
where id = identity.
Q.E.D. Recall
»),-
11 j - - "ji)..o( (t:fq> L... a j a j'ji * q> V). j ,
and since aj is nonsingular (8q»j=O Conjugate this to get
that is
Thus
ifandonlyif
o(~ajvjJ*q>j)=O.
106
GEOMETRY OF COMPLEX MANIFOLDS
and
So
and
# : .ifn-p,n-q(F) - - + .ifp,q(F). Lemma 4.1 implies that # is a (conjugate linear) isomorphism. EXAMPLE.
Q.E.D.
Let M be a compact Riemann surface. Then HI(M, @) = HI(M, QO) ~ HO(M, QI)
and HO(M, Q I) is the space of holomorphic differentials
on M. Thus,
Further HI(M, 0)
= Hl(M,
@(T» ~ HO(M, QI(T*»
= HO(M, @(T* ® T*» which is the space of holomorphic quadratic differentials on M. Thus, dim HI(M, 0) = (
0, 1, 3g - 3,
°
if genus (M) = if genus (M) = 1 if genus (M) :2: 2.
[For example, see Teichmliller (1940).]
s.
Covariant Differentiation on Kahler Manifolds
In this section we want to exhibit some of the special facts that a Kahlerian structure imposes on the Hermitian geometry of M, a complex manifold. For instance, 0 = [] = 16 holds on a Kahler manifold. First we must review the idea of a covariant derivative. Suppose ~j(z)(a/azj) = ~k(z)(a/azn is a given COO section of T. If Zj E Vi " V j " V k =1= cp, then in general,
La
La
5.
COVARIANT DIFFERENTIATION ON KAHLER MANIFOLDS
107
because
and thus, 01'~
_"'_J
= '\
OZ~
_"'_k + '\ __J_
L... OZ~ OZ;
OZ;
0 2 Z~
01'/!
_J
L... OZ~ OZ~
~e
.
We would like to define a "correction" term rj;.p(z) such that
where V
a1'~
t.
;. t.,j
"'i = ;-;: + '\ L... uZ,
r.j;'fJ "'j1'P •
(J
We temporarily fix the following notation: !2(T), !2(T*) will be written
eX!
sections in !2(T), !2(T*)
L ~j(a~j). L CPj~ dz}, L I]~(a~~)' L I/Ij. dz~, respectively. We suppose we have fixed an Hermitian metric ds 2 = L 9Jali dz} /\ dz~ on M. Later we will assume that this metric is Kahler. Let V be a coordinate patch with coordinates (ZI, ... ,z"). Let A;. = %z\ 0;. = D/ai;'. The '1~ transform as follows:
'1~(z) =
t ( oz~ '1~(z). OZ~)
Since
o;.'1~(z) = t
(!;D
o;.lJf(z).
Similarly for 1/1 j~' We define
V;. '1~ =
a;. 1]1' V;. 1/1 ja. = 0;.1/1 jii. •
Let (g~~) = (g j~p) -I. Then p(~) =
L" gjy/i~j dz P E f2(f*) y= I
(1)
108
GEOMETRY OF COMPLEX MANIFOLDS
and we define
that is,
Thus,
and
where G = (9«p). Similarly,
n
L q", cP jy ,
= a;. cP j« -
y= I
since "g L- ",p gPY
=
fJ
{)Y
""
and 0= L a;. g«pgllY fJ
+ Lg«j1 a;.gllY. II
We also notice that if (Zl, ... , zn) are different coordinates on
so
Suppose now that
Zj
= z. Then we see that V;.
and
e" dz;' ® (~) E ~(T* ® T) az'"
5.
COVARIANT DIFFERENTIATION ON KAHLER MANIFOLDS
109
that is, ~ _ " oz~ ozj
Vjl~j -
f..., ;:) ;.;:) p /I,P uZ j uZk
P
Vk/l~k'
This remark is used in differential geometry as motivation for defining V : ~(E) - ~(T* ® E), where E is a Coo vector bundle over M a differentiable manifold. We also define V, Vl~~
=
o),~~, V),qJ~
=
o),qJ~,
V),r,jj = o),r,jj + L r~pr,P, p
V), t/fjj = 0), t/fjj -
Lp r~", t/fp.
In fact, we could define V;. qJ analogously for any tensor field qJ
E ~(T
® ... ® T ® ... T* ® ... T*),
by raising or lowering indices until p(qJ) E ~(T ® .,. T*) taking 0), and then P -1. We will not write out the result in local coordinates here.
Proof
V;.g",P=Lg"'tio;.(gtiYgyp)
= L g"'ti 0i<5~) Q.E.D
=0. THEOREM
5.1.
w = i L g~P dz'" /\ dz P is Kahler if and only if r~p
Proof
= r py .
w is Kahler if and only if dw = 0 and
dw = i
L iza: dz)'
/\ dz'" /\ dz/J
Now r~p = Lag tiCl 8ig pti) implies OlgPti = L",g"'ti r~p, and so on. Therefore,
dw =
i
2L g./J(r!), -
na.) dz),
1\
dz Cl /\ dz/J
GEOMETRY OF COMPLEX MANIFOLDS
110
Thus, dw = 0 if and only if r~p = r py . PROPOSITION 5.2. Let Kahler manifold, then
Proof
cp =
l/p!q!
Q.E.D.
L Cp(l.""(l.pP,
... /1q dz(l.~ ···"dz/1·. If M is a
We prove the case q = O.
Since r:«) = r:j(l. and dz«" ... " dz«· is skew-symmetric all terms sum to , zero except the first which sums to acp. Q.E.D. Similarly: PROPOSITION 5.3.
In the Kahler case,
Om {() ..,.. -- "V L. ",..,.."" THEOREM 5.2.
Proof
... ",./1, ... /1.
dz« " dz(l., " ... " dz/1· .
In the Kahler case
By Proposition 2.3 -( -l)P(.9cp)'i' ... • pp, ... (J.
=
t
(O(J
+ ~ a(J g )cpli' .. «p(J(J, ... P••
By definition
L V(J cpAp(JP' ... P. = L op cpAp(J(J, ... P. + L rffy cpAp(J, ... (Jp (J
fl
y. fl
+ L rg~cpA,pYP2 ... P. + ... y.fl
=":l L...,uflCP Ap(JfJ,···P. (J
+ "rfJ A,P,···fl p L..., pyCP . y.fJ
5.
COY ARIANT DIFFERENTIATION ON KAHLER MANIFOLDS
This is true because q~ (/3/31 ... y ... /3q)' We claim
=
111
r~p and cpiipPP''''Y'''P q is skew-symmetric in
where 9
= det(gaP)'
(2)
If we know (2), then
L Vp cpii pPP ,'"
fJq =
L (o(J + ~ o(J g)cpii pP'" flq
(3)
9 = _( _ty+ 1(.9cp)ApP ... Pq
P
P
by Proposition 2.3. We remark that
V A ~a(J = (V A ~)iiP, since
and
thus
L giiYg3flVA~yJ'
V).~ilfl=
Similarly we may prove that (3) implies
-( -1)P L VfJCPtBq = (9cp)A pBq , fJ
so -( -l)P
I
11. Y
g/lY VyCPAp/lP''''/lq = (9cp)A pBq'
Thus we prove (2). Recall r~p
=
L gaa OA(gpa-). (I
Let AAv be the cofactor of 9 AV' Then 9
VA
AA'i
=-
9 But
og ::)- =AAV' ug AV
so
og
-A
-;-- = gg" . ug).v
(4)
GEOMETRY OF COMPLEX MANIFOLDS
112
Using (4) and the fact that M is Kahler
Q.E.D.
We now introduce an operator A mapping HAP,q) into HAp-l,q-l). Let tn '1'
1 p.q.
= - ,-, " L...
tn dZ"" /\ .. , /\ dz P•. '1''''' "''''pll, "·P.
Then Atn '1' -
1
" igP"'tn dZ"'2 /\ ... /\ dz ll • (p -l)!(q _ I)! L... np"'2"·"'pP2"·P. '
J=l; that is,
where i =
(Alp )"'2 ... "'p1l2 ... 11.
=
L ig P"lp"P"2 ... "plh ... P.
",p
-- ( - 1)P - 1 L... " . 11", Ig lp"'''''''' "'pPP2 ... P•• ""p
If p - 1 < 0 or q - 1 < 0 then we set HAP-l,q-l) =
o.
LEMMA 5.1. A is a real operator; that is, Alp = Aip.
Obvious.
Proof
PROPOSITION 5.4.
iJA - AiJ = if), aA - Ao = - ;9.
The first implies the second. We shall prove the first, by Propo-
Proof sition 5.2,
(iJAlp)""", "'pB = V",,(Alp)"'2'" "'pB - V"'2(Alp)"""'J'" "pB
where fj =
P2 ... p
q•
+ "',
Thus,
(iJAlp )"" ... "'pB = i( - I)P- 1
L {V""gll"'lp"'«2 ... B -
«,p
V«2 gP"'lp"'
(II "'B-_ --1'(_I)P-l"gP"'{t7 L V!l1 't"CJCE2 "',p
t7
tn
V Cl:2't"ClClIClJ' • •
B-+"'} •
5.
COVARIANT DIFFERENTIATION ON KAHLER MANIFOLDS
113
Again by Proposition 5.2
and
Thus,
Q.E.D.
by Theorem 5.2.
THEOREM 5.3. On a Kahler manifold, 1:::. = 20 = 20. (1)
Proof
We prove 0 = D.
0= 08 + 80 = -i{o(oA - Ao) + (oA - Ao)o} = -i{ooA - oAo
o = 09 -
90 =
=
i{ ooA - oAo -i{ooA
+ oAo -
Aoo}
+ oAo -
Aoo}
+ oAo -
oAo - Aoo}.
Thus D= O· (2) Proof of b. = 20 = 20. b.
= db + bd = (a + 0)(8 + 9) + (8 + 9)(0 + 0) =~+M+M+M+~+~+M+M
= D + 0 + 08 + 80 + M + 90. But i(08
+ 80) = o(oA -
Ao)
+ (oA -
Ao)o
=0
so 08
+ 80 = 0, 09 + 98 = o.
Thus, b. = 20 =20.
Q.E.D.
Now we wish to derive the important relations between Hr(M, C) and Hq(M, QP).
GEOMETRY OF COMPLEX MANIFOLDS
114 THEOREM 5.4. M we have
(1) (2)
(Hodge, Kodaira, deRham) On a compact Kahler manifold
Hq(M, QP) ~ HP(M, Qq), W(M, C) ~ EB HP(M, Qq). p+q=r
Proof
(I)
Hq(M, QP)
~
:Yfp,q = {IP I IP
HP(M, Qq)
~
:Yfq,p = {IP lIP E r(Aq,P), DIP = O},
E
r(Ap,q), DIP
=
O}
The map IP --+ iP is an antilinear isomorphism from r(Ap,q) to [(AM). If DIP = 0, then DiP = 0, But M is Kahler so D = D, and conjugation thus gives an isomorphism :Yfp,q ~ :Yfq,p, (2) By the de Rham theorem,
But
Hr(M, C)
~:Yfr
= {IP lIP E [(A r ,b..IP = O}.
(5)
This is the Hodge theorem, We have not proved it, but its proof is similar to that of Theorem 4.1, using de Rham's theorem and the decomposition !l'q = :Yf q + d!l'q-l + ()!l'q-l analogous to that of Proposition 4,2, We leave it as an exercise to the reader. We have the following decomposition: r(Ar) =
EB [(AM).
p+q=r
We claim
:Yf r = EB :Yfp,q, p+q=r
(6)
For if b..IP = 0, then DIP = tb..IP = O. D maps (p, q)-forms into (p, q)-forms and thus DIP = 0 implies DIP(p,q) = 0, where IP E r(Ar) and IP = Lp+q=r IP(p,q), IP(P,q) E [(AP,q). Thus b..IP = 0 implies rp(p,q) E :Yfp,q. So
:Yfrs; EB:Yfp,q. p+q The reverse inclusion is also easy and (6) is proved. This proves Theorem 5.4. (2). Q.E.D. Let hp,q = hp,q(M) = dim Hq(M, QP) and br = hr(M) = dim W(M, C); hr is the rth Betti number of M. COROLLARY I.
On a compact Kahler manifold
h2p
~
hPoP
~ I.
5. Proof
COVARIANT DIFFERENTIATION ON KAHLER MANIFOLDS b 2p ~
hP'P
is clear. Let w = i L ga/Idza 1\ dz li w P = w
115
1\ ••• 1\
w
(p-times). Then
Owp=o, because
and V,ga/I = 0, V,gall =0.
So, aw p = 0, Sw P = 0, by Proposition 5.3 and Theorem 5.2. Thus, Ow P = O. From the calculations in Theorem lA, we know w P =1= O. Thus dim .Yt'p,p > O. Q.E.D. From Theorem 1.4 we already know b2p
REMARK.
COROLLARY 2.
~
1.
On a compact Kahler manifold b2 k+l == 0 (mod 2)
Q.E.D.
Proof
PROPOSITION 5.5.
On a compact Kahler manifold, every holomorphic = O.
p-form cp satisfies dcp
Proof
dcp = ocp
+ ocp and
ocp = 0 so dcp = ocp.
Thus,
IIocpl12 = (ocp, ocp) = (cp, !Jocp). From Proposition 5.3, oA - Ao
=-
i !}. so
II dcp 112 = - i (cp, oAocp - Aoocp). ocp is holomorphic so oocp = 0 and A maps r(AP,O) into O. Thus IIdcpl12 = 0 and dcp = O. Q.E.D. EXAM?LE. We show that Kahler is needed in this proposition. Let be the subgroup of GL(3, C) defined by
C'~{(Z"Z"Z'))~(ZIZ~G
1~m"GL(3'c)
(:3
116
GEOMETRY OF COMPLEX MANIFOLDS
Let
!' f:)lg,~m'+in,;m"n'EZ)
G+~ (~
One easily checks that G is a subgroup of 1[;3. The quotient space 1[;3/G is a compact complex manifold. For
+ 9 1 Z 2 + 9 2 + 9 3 Z 1) , 1 Z3 + g3 = Z, 001 1
Zg
Z1
= (0
Z;)
I and each point of M is represented by some Z' = ( 0
z~ 1
z;
o
1
1
i y~ where 0
~ x~ ~
1, 0
~ y~ ~
with Z). = x).
+
1. The terms of 2' satisfy
Thus
= dZ 2 + (z; - Z3) dz ,. Hence, cp = dz; - z;dz~ = dZ 2 - Z3dzi is a holomorphic I-form on 1[;3 invariant under G; if n : 1[;3 --+ 1[;3/G = M is the canonical (holomorphic) map, there is a well-defined holomorphic I-form t/I on M such that n*(t/I) = cpo Similarly there are nonzero forms~, 17 on M such that n*(~) = dz 1 , n*(I7) = dZ 2 Since dZ 3 1\ dz , =1= 0, ~ 1\
But dcp = dZ 3
1\
17
=1=
o.
dz , and hence
dt/l = ~
1\
17
=1=
O.
Thus the theorem is not true in general
6.
Curvatures on Kahler Manifolds
We assume throughout this section that M is compact and Kahler. We wish to find some expressions for the Laplacian in terms of curvature tensors which will be used in Section 7 in the proof of the vanishing theorems. Let w = i L gClP dZCl 1\ dz P be the Kahler form on the Kahler manifold M. We define the bracket of V). and
Vv •
6. PROPOSITION
6.1.
117
CURVATURES ON KAHLER MANIFOLDS
[V).,
Vv]~a =
Ip= 1 Rpv). ell, where
-
a
R pv). = vr~/I' DEFINITION
The tensor field R pv). is called the curvature of the metric w.
6.1.
Proof
(of the proposition).
Vve a = ave a, so
Similarly,
vv
l\(o).e a
+ ~ r~/I)ell
= Ov o).ea + I r~/I Ove ll + I ovr~lIe/l. P /I Thus
- -
v). Vv - Vv v). = -
,,3u vr~/I ~ /I . L...
/I
Q.E.D.
L:=
We wish to investigate the symmetries of Rp'i)" Let R~/lv). = 1 gIJ~ R~v.t· By Theorem 1.2 we can find real Coo Ku on each small coordinate patch U eM such that gaP = 0,,0/1 Ku on U. We claim: PROPOSITION
6.2. n
R~II'i.t
L
= oao/lovi\K u IJ,
t=
gjjt(oIJopo.tKu){i3taaovKu} 1
on any small open set where Ku is a Coo real function.
Then
L« g"tRP'i). = #.« L gat ovg jjao ), g/ljj + ovo,gllt' Since
GEOMETRY OF COMPLEX MANIFOLDS
118
Thus
=
Oyo).cJ.opKu -
PROPOSITION 6.3.
(I)
L giilZ(OaOtOy Ku)(op.opo). Ku)· a,p.
R apv ).
=
R vpa ).
= Rv).ap =
Q.E.D.
Ra).vp'
Proof Proposition 6.3 (1) is clear from Proposition 6.2. For part (2), conjugate 6.2 and remember that Ku is real. Q.E.D.
DEFINITION 6.2.
The tensor field
is called the Ricci curvature. PROPOSITION 6.4. Proof
R v).
Rv;. = Oy
= 0). Oy log g, where g = det(ga/l)'
Lp r1p
= Oy 0). log 9 by (2), Section 5 of this chapter.
We have a few more simple computational results.
Proof
Vy((>a
=
Oy ((>a, so V). Vy((> a = 0). 0y((> a -
Lp r1a '0 y((> p •
Since we also have n
V).((>a
= O).((>a -
L r~a((>p,
P=I
we get,
V V).((>IZ = rMV).((>a) y
= OYO).((>IZ -
Lp ;\rt((>p - L r~aOy((>p. p
Thus Q.E.D.
Q.E.D.
6.
PROPOSITION
Proof
6.6.
CURVATURES ON KAHLER MANIFOLDS
119
[V A' V.]CPii = - III Rl VA CPp .
Conjugate Proposition 6.5 to get [VA' V.]qJa =
I
R~VAqJp
Q.E.D. We could similarly prove [VA' V.J~a/l1
= -
I
R~J..CIl'i
+I
t
Rp,\.~at'i
t
+I
Rh.~alli·
t
THEOREM
6.1.
For any (p, q)-form cP = ljp!q!
(OCP)a, .··P.
= -
I
a.1I p
I
CPa,"'p. dz a, A .• , A dz Pq .
gPaVaVp CPa, ,,·P. q
+ i=lk=lt,a L I IRta'Pkiicpa,."a,_,ta,+,·"Pk_,iiPk+'''·P.
Proof
As usual let A denote (OCP)A/lo" P.
and
0(1 •••
a p ' Then
= (-1)P{Vpo CPAP'P2'"
-
VII, CPAPOP2'" + ... }
120
GEOMETRY OF COMPLEX MANIFOLDS
Also
so
Thus
= -
L"
a., /1= 1
gPa.Va. V/I ep AP
j
.. ,
P.
(1)
Let us calculate the second term on the right-hand side of (1), For a form ep of type (l, 0)
[V,t, Vv]epa. =
L R'a.v;,ept t
and
,t For a form ep of type (0, 1) [V;" VJepp=
-'I Rp'v,tept t
so
We also have ~
L. /I
R Pt p, P -=
~
L.
a.,/I,y
gta.gPy R pa.pjY
L g'a.Rp,a. = Rp,t. II
Similarly, we see
L gfJa.[Va.' Vp...]ep APPj ,-, PA --- P.
a.,/I
(2)
6.
121
CURVATURES ON KAHLER MANIFOLDS
Since Rpktp/ is symmetric in is zero. Thus, using (2)
i/3 and
qJ is antisymmetric in
i/3, the last term (3)
p
=
L LfJ Rta,p}qJa, ...
(t), ...
app ... 71,. ···P.
i= I t,
- L Rp,:q>a, ... aptP,···iA·'·P., t
where (r)i means that 1: occurs in the ith place. Multiplying (2) by (-1)). and plugging into (I) yields the theorem. Q.E.D. We want to derive a similar theorem for Do acting on r(AM(F)) where F is a complex line bundle defined by the I-cocycle C(;k}' (As usual in this section we are assuming that M is Kahler.) A form q> E r(AM(F)) is given locally by a family of (p, q)-forms {q>j} on {Uj} where {Uj} is a covering of M with coordinate patches over which F is trivial such that q>j =fjkqJk on U j n (Tk'
(4)
Let w = i L gap dz a 1\ dz P be the Kahler form of M and suppose we have chosen an Hermitian form (,) on the fibres, so
(C 0 = aj "jI2, where Cis a fibre coordinate of, and aiz) is a real positive Coo function on U j Then aj "j12 = ak "k1 2 implies
•
(5)
For two forms qJ, IjJ E r(AM(F)) their inner product (q>, 1jJ) is then (q>, 1jJ) =
f ajq>j
1\
* i/ij.
M
The integrand is well defined since by (4) and (5), ajqJj
1\
* i/ij =
akqJk
1\
* i/ik on
Uj n Uk'
Recall that we defined
and by Equation (l), Section 3
* (a i loa j 1\ * q» = 9q> j - * (I a j lOa a j d za 1\ * qJ j ) = - * (0 * qJ) - * (L aj10aa j dz a 1\ * q>j)'
(9 a qJ) j = 9qJ j
-
122
GEOMETRY OF COMPLEX MANIFOLDS
Recall also by Proposition 5.2 and Theorem 5.2
9cpj
* (0 * cp) =
= -
-
*
Ct,Va dz
a /\
* cPj)
and
Thus we have: PROPOSITION
6.7.
(9 a CP)a, ... p, ... P._, = -( -l)p Proof
L gl1a(Va + oa log a)CPja, ... a I1P, ... P._,' p
a,p
We need only show
To do this we assume gpa(zo) = lJPa and cP j = dz Ap
/\
dz l1•. Then
* cP j = sgn(Bq Bn- q ) inc -1 )tn(n-I )+np dz Bn ApAn_p
q /\
dz An - p.
Next
* (dz a
/\
dz Bn -.
/\
dz An - p ) = IJdz Ap
/\
dz x,
where 17 and X are as follows: X is the increasing set of numbers (x, ... x q _ J) complementary to the set rxBn - q £;; (1 .. ·n). If we order rxBn _ q in increasing order Y let
s = sgn
(BY)' rx n-q
Then 17 = sine _l)tn(n- 1)+n(n-q+ 1) sgn
_ 'n(_I)tn(n-')+n(n-q+ I) I
Thus,
Hence,
sgn
(~-p}) (An_pAp) rx B n-q X'
6.
123
CURVATURES ON KAHLER MANIFOLDS
where X = (Xl'" Xq-l) is an increasing set of numbers from (I ... n) so that X is the complement of a in B q • But we also have
L
(-1)P IX
ca({JjApax, ... xq_,dzAp/\dzx
v X=Bq
= (-1)P
L Ca sgn (BaXq )
dz Ap /\ dz x
ae Bq
corresponding to the right-hand side of (6). Letting Ca = aj laa. aj we see (6) is verified at Zo in the case gPa(zo) = o/1 a, ({J j = dz Ap /\ dz p•. The general case Q.E.D. follows easily from this. We want to define covariant differentiation of sections ({J E nAP·q(F)). Let ({J = {({J E qA(p,q)(F)). Then ({J jAP = !jk ({JkA/J' One can easily check the following fact: If ({J is a form and! is a COO function,
J
and
But in our case
Ojjk = 0 so
Thus we define (7) However,
Va ({J JAB = !jk Va ({JkAB SO
+ aa!jk ({JkAB
we must make a different definition of Va. which depends on
Hence,
aj
1
({J JAB
=
-J ak ({JkAB' jk
Now
so that
aj'
We know
124
GEOMETRY OF COMPLEX MANIFOLDS
thus proving
We define
v~a)
THEOREM
{;j
v~(aj
{(V"
(8)
+ a" log aj)(
6.2. in - " gP"v(a)v D a't'jApB. - - ~ ,,{J't'jApB. tf)
q
+ L L (X Tpk k= 1
where
- V{J et and
xl =
Proof
RpkT)
f
e = La gt"oalog a
j •
We omit the subscriptj from our calculations. From Proposition
6.7, (8a
= (8
t -
(
-l)p L ~P
We denote the last term in this expression by (e
= (D
We also have
7.
VANISHING THEOREMS
125
So (9)
We now use Theorem 6.1. First note that
- L gPa.V« V{J -
L gl1«("1« + a«log aj)V{J - L gl1a."1~D)V{J .
e11V{J = -
=
Now use (9), (10), and Theorem 6.1 to finish the proof. REMARK.
L gA.X'jj = = -oA, log aj
X Ajj =
(10)
Q.E.D.
vjja A log aJ
(11)
is called the .curvature of the metric a.
7.
Vanishing Theorems
We wish to use the computations of Section 6 to show that the cohomology groups Hq(M, eJ(F)) must vanish under certain circumstances. The first result to this effect is the following theorem. The technique is due to Bochner. THEOREM 7.1. If the Hermitian matrix XTa - RaT is positive definite at each point of the Kahler manifold M, then for q
Proof
~
I.
We first prove:
LEMMA 7.1. Let w = i L 9«11 dza. any I-form on M. Then
f
A
M
dz 11 be the Kahler form on M. Let ell be
bell
w~ = o. n.
Proof ({J A *Vi = «({J, IjI)(z)(w"/n!) from Theorem 2.1. Let ({J = 1, Vi =/ be a differentiable function. Then we see that */ = /(z)(w"/n!). Recall that JM d'P = 0 for any (2n - I)-form 'P. Thus, 0=
f
M
d(*ell)
=±
f **d(*ell) = ± f *bell = ±fbell . (w"/n!). M
M
Q.E.D.
GEOMETRY OF COMPLEX MANIFOLDS
126
Now for the proof of the theorem. Recall that
Hq(M, 19(F»
~ ~o,q(F)
Thus we want to show that any sarily zero. Let
= {qJ I qJ E qJ E
f(Ao,q(F», DaqJ
= O}.
r(AO, q(F» satisfying Da qJ
=
0 is neces-
Then is a (0, I )-form so
o=
f
M
ron
<5<1>. - = n!
f .9<1>.-n! ron
M
since
is the zero map. Thus,
The term in the braces in the last term is always nonnegative since gaP and gPa. are positive definite. Thus the second integral is nonnegative and the first integral is nonpositive. It is clear from the derivation of Equation (3) in Section 6 that for a (0, q)-form qJ using Theorem 6.2
Thus,
since Da qJ j
= O.
Recalling the definition of qJ~q we see __ ron
O>fa.q J
"L, Aq _
1.
Bq -
'(X mrAq-l.m"Bq-l_ L, ra -R)g ar alPI "'g aq_IPq_I'f' 'f' ,. 1 a t n . I
By assumption (Xra - RaT) is positive definite. Hence, (p'a" .. aq _ I = 0 and = 0 for q ~ I. Q.E.D.
~(O,q)(F)
REMARK.
For q
=0
the theorem says nothing.
7.
127
VANISHING THEOREMS
We now discuss the meaning of the curvature X).ji' Recall the sequence
o------. 7L ------. (I) ------. (!) * ------. 0 in Section 5, Chapter 2. We get the exact cohomology sequence b*
'"
-----+
HI(M, (!) ------. HI(M, (!)*) -----+ H\M, 7L) -----+ ...
and we defined c(F) = ch(F). Since 7L c C we map H 2(M, 7L) send c(F) --t c(F)c. De Rham's theorem then says
-+
H2(M, C) and
r(dA 1) dr(A I)
H2(M C)~--. ,
THEOREM 7.2. The de Rham cohomology class of c(F)c is represented by (l/2ni) Xlv dz;' /\ dz v.
L
Proof This is an exercise in tracing the de Rham isomorphism map. Let F = {fjd. Then c(F) = [{cudJ, where
1
Cijk
= 2-' {log!u + log!jk + 10g!ki}' nl
We wish to find Y E rcdAI) representing c(F)c. We can find Coo I-forms a j on U j such that
1
= a k - a·. 2m. d 10gJ.k J J Then y Thus
= da j = dak . Remember that log !jk
!!jk!
+ log !jk
2
= adaj and X.<ji =
-
-o;.oJl log aj'
= log a k - log a j
and
Let
1
ak
= -. alog ak' 2m
Then
1 "L.. X;'ji dz ). /\ = -. 2ITI
dzJl._
Q.E.D.
128
GEOMETRY OF COMPLEX MANIFOLDS The line bundle over M defined by the l-cocycle 8(zf ... zZ) K={Jjk},J jk = 8t ZjI ••. Zj")'
with respect to a coordinate covering {V) of M, is called the canonical bundle K of M. Then it is easy to see that gj - -I) jk 12 .
gk
The first Chern class cl(M) of M is then (this may be taken as a definition) c1(M) = - c(K). THEOREM 7.3.
The de Rham cohomology class of CI (M)c is
L: Rii .. dz). [~ 2m Proof
1\
dz ii ].
Left to the reader.
THEOREM 7.4. Let M be a Kahler manifold. Let F = {ljk} be a line bundle. Ifa(l, I)-form ')I
1 L: ')1 .. - dz ). = -.
2nl
"
1\
dz"-
is real (that is, y = y), d')l = 0, and if [y] = c(F), then there exists {a j }, aj Coo functions on V j , aj > 0 satisfying aj Ifjkl 2 = ak such that ')I
Proof
and
a
j l.fjkl 2
= -i 8 0'" log aj' 2n
Choose any metric = ilk' Then define
a = {a j} on
~ =_1. L X;'ii dz" 2m
F. That is,
1\
where that is, ~
i
=-8 ologtl .. 271:
J
dz ii ,
aJ E C'''( V j), aj > 0
7. Then as in Theorem 7.2 dcp is a (1, I)-form and
129
VANISHING THEOREMS
['J = c(F)c so ,
- y = dcp, where cp is a I-form. Thus,
dcp = , - y = '1
+ Ot/!,
where '1 and t/! are (I, I)-forms and 0'1 = O. But then
A'1 = 2 0'1 = 0
d'1 = (j'1
so
= o.
Also ddcp = 0 so
dcp = '1
+ t(d(j + (jd) t/!
implies
0= d(jdt/!
(1)
and hence
«(jdt/!, t5dt/!) = (dt/!, dt5dt/!) =
o.
Thus,
(jdt/! = 0 so
dcp = '1
+ t d(jt/!.
Then
('1, dcp) = «(j'1, cp) = 0
= ('1, '1) + t ('1, d(jt/!) = ('1, '1). Hence '1 = O. Using Equation (I) 0= «(jdt/!, t/!)
=
(dt/!, dt/!)
so
dt/! = O.
(2)
From (2)
o = dt/! = iN + at/!. Thus, at/! = ot/! = 0 since t/! is of type (1, 1). So
dcp =
e-
y = 09t/! = - i( aAat/! -
= i 0 aAt/!. Thus,
0 aAt/!)
130
GEOMETRY OF COMPLEX MANIFOLDS
where I is a Coo function on M. But
~ - y= ~ -
y= -
2~ 0 aJ
i 0= 2n: a If
since ~ - y is a real form. Hence, ~ - y = Uj2n:)a 0(1)(1 + j) and thus we may assume that ~ - y = (ij2n:)a 0 I where I is real valued. Finally
Y=
i
_
aoj = 2n:
~ - -
i_ a(log 2n:
-
a
a. - f). J
i
y=-aologa.
(1)
2n:
J
Q.E.D.
REMARK.
Perhaps we should explain this proof a little more clearly. We
claim: PROPOSITION
7. I.
If
and if [1/1] = 0 (that is, IjJ = d({J), then there is a Coo function 1/1 = 0I when M is a Kahler manifold.
a
Proof
Let 'II
I
such that
= {I/I 11/1 = dcp, 1/1 of type (l,l)}. Then a 0{0 s; 'II, where Y EB a0 {0, where
{0 is the space of differentiable functions, and'll =
Y = {I]I I]
E
'II, (I], aof) = 0, for all IE 22}.
We note that (I], a 0I) = 0 if and only if 8!}. I] = O. We claim that if M is Kahler, then Y = {O}. For Kahler implies -!-d = 0 = 0 and a8 + 8a = 0 = 2 1'/ = DOl] = (08 + 8o)(a[}. + 9a)I] for 1'/ E Y. Since I] is of 08 + 8e. Thus type (I, I) and I] = dcp, 0 = dl] = al] = 01]. Thus,
±d
±d 2 1]
= (o8a[}. + 9a(9)I] = -
009!}. I]
+ 9080 I]
=0. Thus ,121] = 0 and (,121].1]) = (,11], ,11]) = O. So ,11] = 0 and hence (jl] Finally, ('1, 1]) = (dcp, 1]) = (cp, (jl]) = 0 and I] = O. Q.E.D.
=0
7.
131
VANISHING THEOREMS
DEFINITION 7.1. A complex line bundle F over any compact complex manifold is said to be positive if there is a y = (l/2rri') X"il dz" 1\ dz il , dy = 0, y = y, and [y] = c(F)c such that X;'il(Z) is positive definite at every point z of M.
L
L
REMARK. If F over M is positive, then ()) = i X"il dz" 1\ dz il is a Kahler form. Hence M is a Kahler manifold. Rewording Theorem 7.1 gives: THEOREM 7.5.
If F - K is positive, then Hq(M, (D(F)) = 0 for q
THEOREM 7.6.
If -F is positive, then Hq(M, (D(F»
~
I.
= 0 for q::; n -
l.
Proof Serre duality gives Hq(M, OP(F)) ~ Hn-q(M, on-p(-F)) where dim M = n. Notice that on ~ (D(K) and let p = O. Then
Hq(M, OO(F» ~ H"-q(M, 0"( -F» ~
H"-q(M, (D(K - F»
= 0 for n - q ~ I if K - F - K
= -F is positive.
Q.E.D.
We also have: THEOREM 7.7. If F is "sufficiently" positive, then Hq(M, OP(F» = 0 (where F is a line bundle) for q ~ I.
Proof
Again we use
Hq(M, OP(F»
~
.Yfp·q(F)
= {cp I OaCP
For cP
E
= 0, cP of type
(p, q)}.
.yt(P·q)(F) we let the reader check the following inequality:
Thus if X'ij is sufficiently positive definite, then the integrand is positive for cP # 0 we see .ytp.q = O. Q.E.D. We now proceed to a generalization of Theorem 7.6 due to Nakano (1955). As usual M is a compact Kahler manifold and F= {fjk} is a complex
GEOMETRY OF COMPLEX MANIFOLDS
132
line bundle with metric {aj}. Remember that (9 acp)j = {(I/a)9(aj cp)}, and so forth. LEMMA 7.2.
(ooa
r (AP,q(F». Proof
+ oao)cp
X /\ cP, where X = -00 log aj and
=
We have (oa cp)j =
cP
E
{~ o(aj' cPj)}
= {ocpj
+ alog aj
/\ CPj}.
Thus, O(OaCP)j={Oocpj+oologaj /\ cp-ologaj /\ oCPj}.
Add o/Jcp j = {o()cp j
+ 0 log aj
/\
()cp j} to get (ooa
+ oa o)cp = X
/\ cpo
Q.E.D. THEOREM 7.8. [Nakano (1955); Calabi and Vesentini (1960)] r (AP,q(F» be such that ocp = 8acP = O. Then
Let cP
E
Os )-=1(X /\ Acp - A(X /\ cp), cp).
s Fl (oaCP, OaCP) = (9o acp, cp) (ot/!, tIt) = (cp, 8at/!) and (9cp, tIt) = (cp, oat/!).
Proof
0
since - J-=19 = oA - Ao. Hence
By Proposition 5.4
O:s; (Boacp, cp) = J-=1 (oAoacp - Aooacp, cp)
(3)
=)-=1 (Aoacp, 8acp) - )-=1(A(ooa
=
-)-=1 (A(X
+ oao)cp, cp)
/\ cp), cp).
But we also have Os (Bcp, Bcp)
= (oaBcp, cp) = )-'=-1 (oaoAcp =
- oaAocp, cp)
)-=-1 (oaoAcp, cp)
= )-=-1 (oaOI\'I'+ oOaACP, cp) since ocp = 8acp = O. Thus,
J-=-1 (X /\ Acp, cp) ~ O. Now as Equations (3) and (4) to get the theorem. THEOREM 7.9.
[Nakano (1955)J
Q.E.D.
If F is negative, then
Hq(M, QP(F» = 0 when n = dim M.
(4)
for p
+q s
n- 1
7. Proof
133
VANISHING THEOREMS
By the harmonic theory
Hq(M, OP(F» where P·q(F)
=
~
{
E
P,q(F), O
r(AP,q(F». By Theorem 7.8 if
1\
i\
E
,1fp,q(F)
1\
We let the reader verify the following equations:
P
+
L L(-lYgji).XlXtii,
1',). i= 1
q
+
L
L (_l)k g ji ).X).p.
1',). k= I
p,q
+
~
L.
~
L.
( - 1)
i+k
~
X aiP. g
1'.). i= I.k= I
Thus,
~
- P L. g
ji).X vji
/I •
jiAX ).v <Pa, ... IXpjiP2 ··P.
Since F is negative, - X).ji is positive definite at each point. (We should use Theorem 7.4 here; that is, we choose aj so that -X).ji is positive definite.) Now
134
GEOMETRY OF COMPLEX MANIFOLDS
and OJ satisfies
dOJ = d(I - X Aii dz l Thus we may use Then
Xlii
1\
dz ii ) = O.
as a Kahler metric on M. Hence, assume g"P
= - X"p.
and
Finally
o ~ fM ,n. -,-, {( - n) L + p L + q L}, p.q. OJ"
O~
So 0
~
(-n
1
OJ" I "A ---(-n+p+q)L.,CPAliCPpq. fMn!p!q! pq B
+ p + q)(cp, cp).
But p
+ q < n and
we see that cP must be
o. Q.E.D.
8.
Hodge Manifolds
Recall that by de Rham's theorem a Kahler form on a manifold M determines an element of H2(M, 1[:). We also have the image of the canonical map
H2(M, l) - - + H2(M, I[:) which we denote
C
-4
L
Co.:: •
DEFINITION 8.1. ga11 dz" dz P is a Hodge metric on M if [OJ] = Ca:; for some C E H2(M, l) where OJ = i gajJ dz a 1\ dz P• If M has a Hodge metric, then we say that M is a Hodge manifold.
L
THEOREM 8.1. M is a Hodge manifold if and only if there exists a positive line bundle FE HI(M, (l)*).
Proof Suppose FE HI(M, (l)*) is pOSitive. Then, by Theorem 7.4, c(F) E H2(M, l) is cohomologous to (l/2ni) X where X = -a"O log aj =
LX Aii dz!'
dz ii and (X Ail) is positive definite. Thus a Hodge metric on M. 1\
gAil =
(1/2n)X lji defines
I
Next we assume M has a Hodge metric, that is, OJ = i gap dz a 1\ dz P with OJ cohomologous to Ca:; for C E H2(M, /f), and (gaP) positive definite. It
8.
135
HODGE MANIFOLDS
suffices to show that there is a line bundle F such that c(F)
wi th X ",11
= c; because then
= 2n g"'11' and hence - F is positive. Recall the exact sequence
... _ _ HI(M, (9*)~ H2(M, 71.)~H2(M, (9)-- ... F--c(F).
Thus, it suffices to show J.lC = O. Let c be defined by the 2-cocyc1e c = {Cijk}' The proof involves chasing through thede Rham and Dolbeault isomorphisms. Consider the following diagram: /H2(M, C) ~ J'(dAI)/dr(AI) CE
H2(M, 71.) ~ jJ.
Cc - - -
H2(M, (I))
~
t/I
(1)
J'(oAO.1)/of(Ao,l)
J.lC _ _ _ q>(0,2).
As in the argument of Theorem 7.2, we can find differentiable functions Aij such that Cijk = c5(A};jk = Ajk + Aki + Ajj' Then we can find differentiable I-forms t/lj such that dAjk = t/lk - t/lj. Then t/I in Diagram (I) is obtained by t/I = dt/lk = dt/lj. For the Dolbeault isomorphism, OA jk = q>k - q>j' where the q>j are (0, I)-forms. Then q> = Oq>k' We can split up t/lj = t/I/I,O) + t/lP'O) into forms of type (1,0) and of type (0, I). We know that d = a+ 0 so we compute ;) 1 _ UIL jk -
OA jk
./,(1,0) 'I' k -
./,(1,0) 'I' j ,
= t/llo,t) - t/ljo.I).
Thus we may assume that q>k = t/lk(O,I). Then q> = ot/l/O,I) = t/I(O, 2) [the (0, 2) part of t/I]. Thus, if Cc - t/I, J.lC - t/I(O, 2). Now we have assumed Cc '" w which is of type (1,1). Thus t/I = W(I.I) + dl], with I] = I]li. O) + 1](0.1). Thus t/I(0,2) = 01](0,1) which means J.lC = O. Q.E.D. With the obvious definition of elements of type (I, I) in H2(M, 71.) we have: Let M be a compact complex manifold. Then the il1'age of the map HI (M, (9*) ~ H2 (M, 7L) is the set of elements of type (I, 1). COROLLARY.
We now give the proof of the main theorem of this chapter which can be considered as a generalization of the fact that every compact Riemann surface is algebraic.
136
GEOMETRY OF COMPLEX MANIFOLDS
THEOREM 8.2. [Kodaira (1954)] Every Hodge manifold is algebraic (that is it is a submanifold of some IP N ). We first outline the idea. We know there is a positive line bundle
E E H 1(M, (!)*). Let F = mE where m is a large positive integer. Let dim HO(M, (!)(F»
=N +1
choose a basis {f3o, ... , f3N} for HO(M, (!)(F», and let F be defined by the I-cocycle {fjk} with respect to some covering {Uj} of M (remembering that the !jk are never zero). By definition
f3v = {f3v/z)}, f3vi z) = fik(Z) . f3viz), where the f3vj(z) are holomorphic on U j . Consider the candidate for a map
= (f3o j(z), ... , f3N J(z»
for
Z E
U j'
It is easy to see this is well defined as a point of IPN if for every Z E M there is an index v such that f3v(z) :f:. O. We want
(l) q> =
Given Z
E
M, at least one f3v(z) :f:. 0, that is, there is a q> E HO(M, (!)(F», [Then (I) implies that
L cv f3v such that cp(z) :f:. O.
holomorphic on M.J (2)
(acp~~~z j»)
:f:. 0
.
We first prove (2). Let Y' = (!)(F - p - q) be the subsheaf of (!)(F) consisting of germs of holomorphic sections of F which are zero at p and q. Let us investigate the stalks of Y'. Clearly, Y'.
= (!)(F). ,
if z :f:. p, z :f:. q
Y'p= {cpE(!)(F)plcpip) =0, ;fpE U j } and similarly for q. We have the exact sequence
0 - - - Y' - - - (!)(F) - - - Y''' - - - 0,
(2)
where Y''' = (!)(F)/Y' is the quotient sheaf. Then Y'~ = 0 except at p or q. Clearly Y'; ~ C, Y'; ~ C and the isomorphism depends on the choice of local
8.
HODGE MANIFOLDS
137
coordinates around p and q. This shows that HO(M, f/") = C Ei3 C. The exact cohomology sequence of (2) is
0----. HO(M, f/) ----. HO(M, m(F») ----. C EB C ----. Hl(M,f/) ----. ... (CfJj(p), CfJk(q))·
CfJ -
We sometimes use the suggestive notation f/ = m(F - p - q). To prove (2) it is sufficient to prove: PROPOSITION 8.1. If M is compact and Kahler and FE Hl(M, m*) is "sufficiently positive," then Hl(M, m(F - p - q)) = O.
The proof makes use of the quadric transformations Qp, Qq. Let M = Qp QiM) and let P be the holomorphic map P : M ~ M of M onto M such that C = p-l(p) and D = p-l(q) are isomorphic to IP n - 1 with dim M = n, and P is a biholomorphic map on M - C - D, P : M - C - D -+ M - p - q. Let f/ = m(F - p - q), 9 = m(F - C - D), where F is the holomorphic line bundle on M induced by P and 9 is the sheaf of germs of holomorphic sections of F which vanish on C and D. Let 0Ii = {Uj} be a covering of M. Then dii = {OJ}, OJ = p-l(U) is a covering of M. We recall that Proof
Hq(M, f/) = lim Hq(o/i, f/) u
and for q = I, the map Hl(OIi, f/) LEMMA 8.1.
-+
Hl(M, f/) is injective. We prove:
If
then Hl(M, m(F - p - q»
= O.
Proof It suffices to show Hl(o/i, f/) = 0 for all coverings 0Ii. Take a l-cocycle CfJ = {CfJij} E Hl(OIi, f/), CfJij E reU i n U j ' f/), where CfJij is a holomorphic section of F over U i n U j such that CfJij(p) = 0, CfJij(q) = 0 if p, q E U i n U j • P induces
- = P *CfJij = CfJij 0 P E reUi - n Uj - ' m(F» , CfJij where ipij vanishes on C and D if C £; OJ n OJ, D £; Oi n OJ. Thus {ipij} represents an element of Hl(dii, 9) £; Hl(M, iJ) = O. Hence ipij ..... 0, that is, ipij = tjJj- tjJi where each tjJi E reUi' m(F» and vanishes on C and D. If U i £; M - p - q, then P: Oi -+ U i is biholomorphic. In this case there is CfJi E reU i ' m(F)) such that tjJ i = P*(CfJJ If, for instance, P E U i, then
138
GEOMETRY OF COMPLEX MANIFOLDS
P : 0 i - C ~ Vi - P is biholomorphic. Hence there is cP i E f( V i - p, (!J(F» such that P*(CPi) = I/J i on 0 - C. We can always assume F is trivial over Vi' so r(vj, (!J(F» ~ i , (!J). Thus we consider CPi as a holomorphic function on Vi - p. By Hartog's theorem CPi can be extended to all of U i ' Then P*CPi is defined on all of 0 i and must equal I/J i (by continuity, or the identity theorem). Thus CPi(P) = O. Hence we have found CPi E nVi> (!J(F - p» such that I/J i = P*cp i ' We have proved that there is a o-cochain {cp J, cP i E Vi' Y) such that I/J j = P*cp;, and thus P*CPij = P*cpj - P*cp j ' But P is surjective, so C(ljj = C(lj - CPj· Thus {cpij} ,.., 0, and Hl(ott, Y) = O. Q.E.D.
nv
n
REMARK.
Relations between Hq(M, Y) and Hq(M,
!J)
are not easy to see.
To prove Proposition 8.1 it now suffices to prove H1CM, (!J(F - C - D)
e
= O. Let [C] and [D] be the corresponding bundles of the divisors and D. Then we must show that HI(M, (!J(F - [C] - [D]» = O. To prove this it suffices to show that P - [e] - [D] - K(M) is positive, and then quote the
vanishing theorem. We want to show that
P - [e] -
[D] - K(M) > 0
if m is sufficiently large, where F = mE, and K(M) is the canonical bundle of M. Therefore we would like to compute e([e]) and e([D]). First we find a I -cocycle on M representing [C]. Let z be a coordinate chart map centered atp E M, and let V = {zllzl < 2/:}. Let P: M -+ M. Let us describe the normal bundle W of e in M. Let
VA = {u where
U I , •• "
Un
E
!pn-II U
= (u l , "', Un), UA i= O},
are homogeneous coordinates for !P" - I. Then
!pn- 1 =
" VA'
U
A= 1
and n
w=
U
(VA
X
C),
A=I
where we identify (u, w).) and (v, wI') if and only if
u = v,
(3)
We could define P: W ~ U by W.1.
P(u, w).) = - (u l , "', un)
(4)
U).
n = (WAUl , ... , W ... ,.\. , , W).u ) .
U).
U).
8.
139
HODGE MANIFOLDS
Then IPn-l = u). (V). x {O}) s; Wand we can identify C = Qp(p) with \fDn-l in W. Thus we consider a small neighborhood p-l( V) = 0 of IPn-l in Was a small neighborhood of C in M. On each V). x IC, C is defined by IV). = 0. Let 0 A = 0 II (VA X IC) for A = I, "', n, and let 0 0 s M - C be such that
Qp QiM) = We set
Wo
= I on
00 ,
M = 00
U
Then the line bundle [C] is given by the I-cocycle on
Then (5) implies g;.o
=
0 1 U··· U On. 0).
0 •.
II
(5)
w)..
Recall that, in general, if F is defined by {Fjd and if aj positive Coo functions {at}, then c(F)
~
i
-
2n
Ifj kl 2
= ak for
_
f)
alog a
j .
(6)
We want to find such COO functions for C. We make use of a cro function ex. on M with the properties (1) (2)
ex.(z) = Izl2 for z E V, Izl < ex.(z) = 1 for z EM - V.
e
We define
Ao(w) = ex.(P(w»,
WE
ex.(z)
Aiw) =
-I- 2 ' WE
w;.1
00
_ V)..
Notice, on CliO). local coordinates are (A "f 0)
( ~ , •.• , U).-l , U).+ 1 " .•. Un) . U). U). U). U). Thus,
so the definition has meaning, and
A). > 0 on
0).
A. = 0, "', n.
140
GEOMETRY OF COMPLEX MANIFOLDS
The A;. satisfies so
c([C]) "" -
i
2n
a0 log A)..
(7)
We notice that
a0 log A). = a0 log(l + v,.;. L 1 v12) U U).
and this is just the standard Kahler metric on IP n - 1 0 a(z) is a Coo 2-form on M of type (1, 1). Since
a
= C. We also remark that
a0 a = d(oa) a0 a
is cohomologous to zero on M, and the induced form Q cohomologous to zero on M. We then define
= P*(ooa) is
1
(1c = - . (00 log A). + Q). 2m
Then (1 c ""
(8)
c( - [ C])
by (7), and in a neighborhood of C,
2ni(1c
=
00 log(l
+
L 1U 12) + oo(lzI v
2 ).
v,.A UA
Recall that Z
= (z l' ... ' n z)
= W;. -u
= (. .. ,
W A , ••• ) •
U;.
Then
oO(L ZvZv) = L dz v = dWA
1\
1\
dz v
dw). + ....
Hence (1c is positive definite in a neighborhood of C. We get similar results for D. Next we want to find a relation between K(M) = K and K(M). We prove: 8.2. K(M) = K + (n - 1) [C] + (n - l)[D], where bundle over M induced from the canonical bundle K of M. PROPOSITION
Proof
Suppose
M = QiM). It is K(M) =
K is
the
sufficient to prove
K + (n -
1) [Cl
(9)
8.
141
HODGE MANIFOLDS
We choose V3P as in the previous proof. Then we choose {Vi' Zj} coordinate systems on M such that {V} u {V j} covers M. Let (Zl, ... , zn) be a coordinate system on M. Then the canonical bundle K of M is defined by the l-cocycle {J jk } where
dz]
1\ ..• 1\
dz'j = Jjk I
dz~ 1\ ••. 1\ dz~
dz l
1\ .•. 1\
dz n = J;l
dz~ 1\ •.• 1\ dz~
on V n Vk'
On Sf we may use {O)., OJ} as a coordinate covering where OJ = P-l(V) = Vj and 0= P- 1(V) = U1=1 0)., using the notation of the previous theorem. On 0). we have the local coordinate system
and Z
'"
W;.U",
=--,
if 0: '" A.
U;.
z)'
= wA,
if 0: = A..
Computing, we get
since
dz'" = Let {Ijk' Then
f;.v, f)'k}
d(W~:"') = w). d(~:) + (::) dw)..
be the Iacobians on Sf [which are used to define K(M)]. -I I jk = J-I jk
1
-I
~_I
I )'k = -;;-=t J ok W).
so
This proves (9) since
W;.
= 0 defines c.
GEOMETRY OF COMPLEX MANIFOLDS
142
We now return to the proof of Proposition 8.1. Recall that E is positive, that is, c(E) '" y = - i
L Ya.P dza. 1\ dz P,
where (Ya.p) is positive definite. For simplicity we write Y > 0 or E> O. We want to show
F-
[C] - [D] - K(Nt) =
for large m. Let c(K) '"
K.
F- K -
n[C] - nED] > 0
(10)
Then
c(F - K - n[C] - nED]) -my -
K + Me
+ mlD,
where y = P*y and K = P*K. We choose m so large that my - K is positive definite on M. Then my - K is positive semidefinite on .Nt and is positive definite on.Nt - C - D. But (1e > 0 near C and (1D > 0 near D. Then Equation (10) follows. This proves Proposition 8.1, and thus part (l) and (2). REMARK. It is an easy compactness argument to see that one can find an integer m such that HO(M, mE) separates points for all p, q EM, P =1= q. The proof of C is almost the same as the proof of B. We want to show that is biholomorphic at each p EM. Consider Y = (!J(F - 2p) which is the sheaf of germs of holomorphic sections of F which vanish at p up to order 2. Again we compute the stalks Y z and write down the exact sequence
O-Y-Q(F)-Y"-O, Y
z
= (!J(F)..
z
=1= p
Yp= {CfJICfJ = CfJj,CfJj(z) = kl
L
+"'+k"~2
akl"'k"Z;l"'Z~"'CfJE(!J(F)p}.
Then if z =1= p if z = p. Thus HO(M, Y") ;;;; C+ I. We write down the exact cohomology sequence
6
1
-H(M,Y)-···. It is easily seen that to prove is biholomorphic at p we need only show HI(M, Y) = O. To prove this we once again use Nt = QiM), C = Qip).
8. LEMMA 8.2.
HODGE MANIFOLDS
143
If HI(M, (!J(t - 2[C]»
= 0,
then
Hl(M, (!J(F - 2p» = O. Proof The proof is the same as that of Lemma 8.1. One only has to notice that if cp has a zero of order 2 at p, P*cp has a zero of order 2 (at least) on C and vice versa.
LEMMA 8.3. Proof
HI(M, (!J(l' - 2[C]» = 0 if m is large enough where F
= mE.
Using Proposition 8.1. we find
F - 2[C] - K(M) = ml!. -
= mE -
K-
(n - 1)[C] - 2[C]
K - (n
+ 1)[C].
Hence
e(l - 2[C] - K(M»if m is large enough.
my -
K + (n + l)uc > 0
Q.E.D.
REMARK. We again use compactness to see that there is an m which will work for all P E M. This completes the proof of Theorem 8.2. We now derive some consequences: THEOREM_ 8.3. [Kodaira (1960)] If M is compact Kahler and H2(M, (!J) = 0, then M is projective algebraic. Proof
The exact cohomology sequence of 0----+ 7L ----+ (!J ----+ (!J* ----+ 0
yields
... ----., HI(M, (!J*)~ H 2 (M, 7L)----" O. Thus everything in H 2 (M,7L) is the Chern class of some bundle. Let {bl' ... , bm } be a basis for the free part of H 2 (M, 7L) so that
H2(M, IC) = Cb l
+ ... + Cb m •
144
GEOMETRY OF COMPLEX MANIFOLDS
Each b;. = c(F;.) and hence is cohomologous to a real 2-form of type (I, 1). Let
w=
iL gall dza 1\ dzll
be a Kahler form on M. We wish to modify w to get a Hodge metric on M. Since w E H 2 (M, C)
w'" L p;.b;. where P;. E ~ (w is real and the b;. are real). Given e, we can always find integers k;., r E 11 such that
A=l, .. ·,m. But then for a small enough e
,
W =
(P). - k).) w - '" L... --r- Y;.
defines a Kahler form on M where y;. '" b;. is a real (1, 1) form. Hence w= rw' is also a Kahler form. But
Thus
en defines a Hodge metric on M, and M
is algebraic.
Q.E.D.
Theorem 8.4. [Kodaira (1954)J Let M be a compact complex manifold. If the universal covering manifold Sf is complex analytically homeomorphic to a bounded domain rJI c:;;; cn, then M is algebraic. Proof We make use of the Bergmann metric on rJI [see Helgason (1962)]. We have M = rJIjG where G, the set of covering transformations of rJI, is a collection of biholomorphic maps from rJI to rJI. Let ds 2 = L gall dza dz(J be the Bergmann metric on f!4. We claim
L
(I) ds 2 is invariant under G and hence induces a metric gall dzlJ dz(J on M = rJIjG. (2) If w = (ij2n) gall dzlJ 1\ dz(J, w'" c( - K); so we have a Hodge metric on M.
L
This gives the theorem, thus we need only prove (I) and (2). Let :It' be the Hilbert space of all holomorphic functions f on rJI which have bounded norm
IIfl12 =
f If(zW dX, 1M
8.
145
HODGE MANIFOLDS
where dX = dX 1
dX 2n and
•••
Za = X2a-l
+ iX2a'
Let {I.} be any orthonormal base of .Yr. Then the Bergmann kernel K(z, z) is given by 00
K(z, z) =
_
L f.(z)f.(z)
.=
[= K(z)].
1
The kernel K(z) is actually independent of the choice of orthonormal basis {I.} [see HeJgason (1962)]. Then L ha/1 dza. dz P is a positive definite Hermitian metric where 2 h ( ) = 8 log K(z)
8Zex 8-zp
a./1 z
Let y : f!A LEMMA
-+
.
f!A be a biholomorphic map, y(z) = z'.
8.4.
I
K(z) = det
o(z~, ... , z~) 12 K(z'). 8(ZI' "', zn)
Proof 00
_
K(z) = L f.(z)f.(z) .=1 and
f f.(z')jiz') dX' = f f.(z)f;'(z) dX = D.).. ~
~
Let F.(z) = f.(z')
det(~~~;)
and notice
Thus,
and {F} gives a new base. Hence, K(z) =
L Fv(z)Fv(z) = v= 00
1
I
0(Z,)\2 det - 8) Lfv(z')Jv(z') (z
I
o(z') 12
= det 8(z) K(z').
Q.E.D.
146
GEOMETRY OF COMPLEX MANIFOLDS
Since G is a group of biholomorphic maps this proves (I). Now let K be the canonical bundle of M. Let TC : B -+ BIG = M. Let Vj be an open set in M on which a local inverse of TC is defined, and choose one Jl j = TC -1 to use as a coordinate chart for V j (Jl j(p) E en if P E V). Suppose P E V j n V k' Then there is Yjk E G such that Jlip) = Yjk(Jl j(p». The canonical bundle K on M is defined by the l-cocycle
and we have K(z)
= Ifj kl 2 K(Zk)
by the lemma. Recall that if we have positive Coo functions aj on V j such that aj Ifj kl 2 = ak, then c( - K)
i ~ =
-
2TC 0
0 log a j
.
Therefore, let aj = K- 1(zj)' Then j
c( - K)
i
= -2TC aa log K(zj) = -2TC L gall dzj
1\
dz~.
This proves the theorem. There is much interest in nonalgebraic Kahler manifolds. Kahler manifolds give examples of the minimal surfaces of differential geometry.
REMARK.
[4] Applications of Elliptic Partial Differential Equations to Deformations I.
Infinitesimal Deformations
We want to study analytic families of compact, complex manifolds. Informally, we are only interested in small deformations. We may as well assume our base space Br = {til tl < r, tEem} is an open disk around the origin of em. We want a manifold .JI and a holomorphic map w: .JI -+ Br with maximal rank so that w is proper and each fibre Mr = w-1(t) has the structure of a complex manifold which varies analytically with t. We want a covering {Oft i} of vft so that 1,
It I < r}
e = (C,"', ej), w(e
t) = t,
Oft i = {(e i
,
t)IIU <
j
j ,
and
'j = fik(e j , t) on Oft j
n Oftk>
wherejjk is holomorphic in (j and t. We notice that under these circumstances Mr is diffeomorphic to M 0' and in fact, vft is diffeomorphic to X x B r , where X is the underlying differentiable manifold of Mo. Thus Oft j = U j x Br where U i = {(i I(i < I), and M
= U U j x Br • j
If x is a point of X, t E Br we notice that
(j = (i(x, t) is a differentiable function of (x, t) and we have (j(x, t)
= fM(k(X, t), t).
(1)
Let M = M 0 = X and use the complex coordinates z of M as differentiable coordinates so that (j(x, t) = ej(z, t), where (j(z, t) is a differentiable function of z and t. Because holomorphic in z (otherwise it is only differentiable). 147
t
= 0, (j(z,O) is
148
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
DEFINITION 1.1.
Let
-at (-ata) =I,c (a) m
y=1
y
y
belong to the tangent space to Dr at the origin. We define (aMI/ot),=o to be the cohomology class in HI(M, 0) given by the l-cocycIe (Jik =
f
<1=1
aj't".(Ck' t) at
I (-;). 1=0
aCi
We want to represent {(Jik} with a I-form by using Dolbeault's theorem. Let T be the holomorphic tangent bundle on M and 0 = (!)(T) be the sheaf of sections. Then we have the resolution D
3
0 - 0 - A O ( T ) - A o . 1( T ) - · · · ,
where AO·q(T) is the sheaf of COO vector (0, q)-forms. Locally, such a thing has the representation
where
and
Oq> =
±oq>p(-;).
p= 1
OZ
Let us trace through the Dolbeault isomorphism 1 r(oAo(T)) H (M, 0) ~ or(Ao(T)) .
Let (aM,/at),=o -11. Then 11 is defined as follows: Pick ~i E r(Ui' AO(T)) such that Then
PROPOSITION 1.1.
"= _
f o(amz,at t) I )(-;). aCi
«= 1
1=0
1.
Proof
INFINITESIMAL DEFORMATIONS
149
Let
Then Equation (I) yields
Thus,
If we set ek = - L Wf)/f)(~), we get - L oC~(f)/f)(~). Q.E.D.
(Jik
= ek-
ei'
Therefore, '1 =
Oei =
We want to define a vector (0, I)-form ((J(t), t E B which describes the complex structure of M,. With respect to the local complex coordinate z on a neighborhood W of M we have the differential operators
and
Recall that
where Irk is a holomorphic function of (k' Thus
and
150
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
At t = 0, (Zl, ... , z") are local complex coordinates on M and «(j(z, 0), ... , 'J(z,O» are also local coordinates. Hence
So, for small enough
Itl,
Let
Consider the local I-form,
L AJ(I a(j(z, t) = cp1(z, t). (I
We claim the form CP1(a, t) is well defined independent of (j. For !jk«(k' t) holomorphic in (k implies
which yields A
cp j(z, t) =
'\'
a(j
L.,
()rfJ
fJ
'ok
(I,
=L fJ
A Aj(l
fJ
a(k(Z, t)
A: a,f(z, t). fJ
We therefore define cpA = cpj(z, t). Then cp\z, t) is a (0, I)-form independent of C, but it still depends on the local coordinate z. Let z = Zw. Suppose V is another open set in M with local coordinate z". Let cp:" = cpA(ZW' t) and cp~ = cp4(Z" , t). If we set
then
cp(t) =
~ cpfy(zw, t)(a:fy)
= ~ cpe(z" , t)(a~~)
1.
INFINITESIMAL DEFORMATIONS
is a well-defined, global vector (0, I )-form on M. Now by the definition of "ira(
u'>j Z, t
151
A;.,
) = "L. cp .l. oz;"· oCj
Therefore
(0 - ~ cp.l.o;,.)(j(Z, t) = O.
(2)
PROPOSITION 1.2. The complex structure on M, is determined by cp(t). More specifically a differentiable function f defined on any open subset of M is holomorphic with respect to the complex structure of M, if and only if
(0 Proof
L cpfl(t)op)j(z) = O.
(3)
We represent cpp(t) by
Cp'1(t) =
I
cp(t)f dz~
a
If f satisfies (2), then for all (/..
(4)
We use
and Equation (4) to get
Equation (2) implies lhat the first term is zero so (5)
Since (j(z, 0) is holomorphic in z, cp(O) small for small t. Thus,
=
O. Continuity tells us that cp(t) is
(u,,- ~CP(t)gDp)~j is invertible for small t. Hence Equation (5) yields
o! = o.
a~jvy
152
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
Thus f is holomorphic in (j(z, I). We can read the argument backwards, so Q.E.D. the proof is concluded. We want to introduce a bracket operation in the algebra of vector 0, q)-forms. Let us recall the Lie bracket of vector fields. If u = ulI. all., W = waoll.' then
L
L
[u, w] = L(UIJOaw p - wIJoauP)a p '
a,p
For the generalization, let
DEFINITION
1.2. n
[ep, I/J] =
L
(epa
A
oaI/JP - (-I)pQI/Ja
A
OaepP)Op
a, fJ= I
where :'l
.I,P
Va'f'
PROPOSITION
1" Va :'l mP 't' Xl'"
= -, p. L..
Xp
dz-.l.l
A
"
•••
A
"
dz-.l. p •
1.3.
[qJ, I/J] is bilinear [I/J, qJ] = -( - I)Pq[qJ, I/JJ (3) o[ep, I/J] = [oep, I/J] + (-!)P[ep, oI/J] (4) (-l)PTep, [I/J, T]] + (-I)QP[I/J, [T, ep]] (I) (2)
where ep is a (0, p)-form, I/J is (0, q) and,
T
+ (-l)rQ[T, [ep, I/J]]
= 0,
is (0, r). (This is the Jacobi identity.)
Proof Uninteresting; we leave it to the reader. We collect our facts into the following theorem: THEOREM 1.1. If OJ: vIt --+ Br is a complex analytic family of compact complex manifolds, then the complex structure on M t = OJ-I(I) is represented by a vector (0, I)-form ep(t) on Mo such that (1)
(2)
(3)
oqJ(t) - HqJ(t), ep(t)] = 0 ep(O) =
°
(aM at
t)
---+ t=O
'1 = _ (oep(t»)
at
E 1=0
f(uAo(T».
1.
Proof
INFINITESIMAL DEFORMATIONS
153
(2) has already been done. As for (1), by Equation (2)
(~qi(t)vpej -
o.
Oej) =
So
o = o(~ qi(t)opej) = L oqi(t)op (i P
-
L qi(t) /\ oOp (j. P
Thus, L o
~
= L L
y
+ LL
l'
The last term in this expression is zero, since
Now det(op ('f(z, Then (6) yields
t»
(6)
is nonzero if t is small, so op (j is invertible for small t. o
= 1[
[
Finally, for (3), we already know ry = -
"7r o(-,p( o(~ (; )' J
where
t~ = (O(~)
at
. 1=0
154
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
But
and the last term is zero, so
.
= _ (a~(t)) ot
Q.E.D.
1=0
We wish to find conditions on a cohomology class p for it to represent an infinitesimal deformation. We first check to see that the bracket [,] extends to cohomology. Let
E
f( oA o. p-l(T»,
tj;
E
f(oAo.q-I(T»,
so that
o
[
[,J
= O. Also if tj; = oa, then
= [ep, oa]
=
±o[
induces a map
HP(M, 8) ® Hq(M, 8) ~ Hp+q(M, 8) by using Dolbeault, and the facts just noticed,
HP(M 8) ,
f(oA o. P-I(T» - Dr(Ao. P-I(T»'
~
=-----=--.,.---
Now let (t l , •• " tm) be coordinates on By. Then any infinitesimal deformation is a linear combination of ones of the form
We claim: THEOREM 1.2.
If P E HI(M. 8) is an infinitesimal deformation, then
[p,p] =0.
2. Proof
AN EXISTENCE THEOREM FOR DEFORMATIONS I
155
By using the previous remark we need only check that [1],., 1];] = 0
for all A, v. We differentiate the equation 2cp(t) - [cp(t), cp(t)] = 0 twice at
t =
0 to get
o(iJ 2cp(t») 01;.
ai,
=
,=0
[ifcp CP(o)] Dt;, lv'
+
[cp(o) 8 cp J , ot;. t, 2
acp DCPJ +2 [ot;.' at,
Let
Then we get Q.E.D.
We remark that this is not a sufficient condition, and the higher derivatives give more information. It is quite difficult to compute [p, p] for a general p E Hl(M, 8), but in most specific examples we get [p, p] = O.
2. An Existence Theorem for Deformations I. (No Obstructions) We aim to prove the following theorem: THEOREM 2.1. [Kodaira, Nirenberg, and Spencer (1958)]. Let M be a compact complex manifold. Assume that H2(M, 8) = O. Then there exists a complex analytic family JIt ~ B" where B,
= {tJJtJ < e}
s;
em, m =
dim HI(M, 8),
such that: (I) Mo = w-I(O) = M. (2) The map To(B,) -+ HI(M, 8) given by (D/Dt) -+ (oM,/at),=o is surjective (in fact, an isomorphism). More specifically if {PI> ... ,Pm} is a base
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
156
for H1(M, 0), then we can define
(aMr) at. Proof
(1)
vi{
so that
= fJ r=O
forv=l,···,m. v
We will accomplish the proof in the following two steps:
Construction of a vector (0, I)-form
qJ(t) =
L qJk
1 '"
km t~1 ... t~~"
such that
qJ(O)
=
0,
oqJ(t) -l[qJ(t), qJ(t)] = 0, and = fl. E H1(M, 0). ( a~(t)) ot. 1=0
(2) Show that qJ(t) determines a complex analytic family by using the Newlander-Nirenberg theorem. First we survey the Newlander-Nirenberg theorem, which is sometimes called a "complex" Frobenius theorem. Let U !;; en be an open domain, and
qJ =
L qJ~ dza(a~/I )
a vector (0, I)-form on U. Let
L~ = (a~a) - ptqJ~(Z)(o~p), We want to consider solutions to the equations Laf(z)
=
°
(1)
on the domain U. The theorem [of Newlander and Nirenberg (1957)] is: THEOREM.
If L~ and L~ are (complex) linearly independent, and if
oqJ - t[qJ, qJ] = 0, then Equation (1) has n Coo solutionsiI(z), .. ·,f,,(z) such that
[1' ... , [n)) ¥ °
det(O(f1' "', in' a( z 1, ... , Zn, Z l'
•.. , Zn)
(that is, iI, ... ,f" define a differentiable coordinate system on U).
2.
AN EXISTENCE THEOREM FOR DEFORMATIONS I
La will be linearly independent.
REMARK
1.
If t is small, cp(t) is small and L~,
REMARK
2.
Linear independence is needed; for if
then
La.! =
IS7
°
implies f is independent of xa.
REMARK 3. If M is a complex manifold and cp is given satisfying the conditions of the theorem, then by using (the proof of) Proposition 1.2 we see M has another structure as a complex manifold which is described by the form cpo We say the almost complex structure cp is integrable, and hence associated to a complex structure.
In order to construct our form cp(t) we need to do some more potential theory. We want to define the Green's operator on
.!l'q = r(AO,q(T» = the space of vector (0, q)-forms. To do this we introduce an Hermitian metric product,
where the
* operator
gajJ
on M, and define an inner
has been defined before. We have the adjoint 9 of 0, + 09. Then the space of
(9cp, t/I) = (cp, ot/l); and the Laplacian 0 = 90 harmonic forms
w= ~
{cp I cp E .!l'q, OCP = o} Hq(M, 0),
defines a Hodge decomposition,
.!l'q = W EEl O.!l'q
:::I
W EElo.!l'q-l
+ 9.!l'q+ 1
into an orthogonal direct sum of subspaces. Thus, for cp 1'/ E IHlq,
t/I E .!l'q.
Since
E
.!l'q, cp = 1'/
+ Ot/l,
t/I E .!l'q, t/I = , + t/l1'
'E
W, t/l1
E
O.!l'q,
and
Thus (2)
158
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
LEMMA
2.1.
Proof
The decomposition in Equation (2) is unique. Surely 11 is unique. If tjl', t/I are both orthogonal to Wand
= 11 + ot/l', qJ = '1 + Ot/l,
qJ then
O(t/I' - t/I)
=
°
and t/I'-t/lEW. But t/I' - t/I .1 W so t/I'
=
Q.E.D.
t/I.
DEFINITION 2.1. Given qJ, the unique t/ll making Equation (2) true is denoted GqJ, and the mapping qJ -+ Gcp defines G : .!l'q -+ O.!l'q. G is called the Green's operator, and is a linear map. We write" = HqJ and call H the harmonic
projection operator. Then (3) PROPOSITION
oH
2.1.
= Ho = 0, 9H = H9
GH
= 0,
=
0, oG = Go,
= 9H = H9
is analogous.
=
HG
9G = G9.
Proof
oHqJ
=
0 since HqJ E W =
{t/I! ot/l =
9t/1
HOqJ = 0 since oqJ E o.!l'q .1 W. The proof that 0 HGqJ = 0 since GqJ .1 W. For GH = notice
°
HqJ
=
HHqJ
=
O}.
= HqJ + DGHII'
and uniqueness yields GHqJ = O. The proofs of the last two are similar to each other so we only prove the first of them. Recall
00
=
0(09
+ 90) =
090
and
Do = 092. Thus 00 = Do and
OqJ =
aD GqJ = ooGqJ
= HaqJ
= OGoqJ.
2.
AN EXISTENCE THEOREM FOR DEFORMATIONS J
159
Since oGq> ..l.. IHlq and Hoq> = 0, we use uniqueness of decomposition Equation (3) to see oGq> = Goq>. Q.E.D. To proceed further, we need to introduce the Holder norms in the spaces !f q • To do this we fix a finite covering {Uj} of M such that (Zj) are coordinates on V j . Let cP E !f q ,
).
q>"
1 = -q!
). I. q>j5oI"
_a
-al
50 q
dZ j
Let k El, k ~ 0; aE IR,O < a < I. Leth where n = dim M. Then denote
1\'"
11q>IIk+a
=
I q> Ilk +.
+
h
=
x;a- 1 + ixj.
is defined as follows:
m~x{ L )
dz/.
= (hi' "', h 2n ), hi ~O'L?~l hi = Ihl z]
Then the Holder norm
1\
(sup
jI.1 :sk
ID1 q>;a.I" •.(Z)I)
ZEU;
I
ID~ q>J"1 ... ".(Y) - DJ q>;ii.
sup
Iy - zl
J1,%IiV j Ihl =k
'" ,.(Z )I}
a '
(4)
where the sup is over all A., ai' ... , a q . We have the following a priori estimate of Douglis and Nirenberg (1955).
1Iq>IIk+a where k
~
~
C(II Dq>lIk-2+a + 11q>llo),
(5)
2, C is a constant which is independent of q> and
11q>llo
=
max
sup
j
ZE
A, al.·'
I
1q>7a., ... ".(z)l.
Vj
aq
REMARK. One can see that two norms defined as in Equation (4) for two different coverings {Vj}, {Vj} induce equivalent topologies on !f q •
PROPOSITION
dent of
2.2.
q> and I/J.
Proof
II [q>, I/J] IlkH s
C
11q>11k+ 1 +a 11q>11k+ 1 +a'
where C is indepen-
We leave the simple check to the reader.
We need to know the following strong kind of continuity for the Green's operator G:
160
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
2.3. I Gcpllk+a :$; C Ilcpllk-2+a, k Z 2, where C depends only on not on cpo
PROPOSITION
k and
r:1.,
Proof
We use Equation (5) to get
IIGCPIIk+a:$; C(IIDcpllk-2+a + IIGcpllo) :$; c(llcp - Hcpllk-2+a + IIGcpllo) :$; C(IICPllk-2+a + IIHCPllk-2+a + IIGcpllo)· The space IHI q is finite dimensional, so let {b.};= 1 be a base. Then m
Hcp =
L (cp, flv){Jv, v= 1
so m
Ilcpllk-2+a:$;
L \' =
Ilflvllk-2+a max I(cp, flv)1 v
1
:$; C1
0:: I fJvllk- 2+a) I cp 110
:$; C21Icpllk-2+a' Thus,
(6) and we need only prove
that is,
IIGcpllo < C Ilcpllk-2+a - 4'
(7)
Suppose (7) is not true. Then there is a sequence cp(v) such that
.
11m v
IiGcp(V)il o () I cp v II k - 2 + ~
=
+ 00.
By multiplying cp(v) by a constant we may assume that IIGcp(vlll o = I, and then 11q>(V)llk_2+a -+ O. Then Equation (6) implies that
iIGcp(Vlllk+a:$; K
(is bounded for k
z 2).
Write
G'rfIl(vl
=
9~! "L.. (GfIl(Vl)A_ 'r Ja I
..
_ dZ~'J
a.
II. ..• II.
d-~'J • to
(8)
2.
AN EXISTENCE THEOREM FOR DEFORMATIONS I
161
Then Equation (8) implies that each of (G cp (V»).ja.l ... a..
and all of its partial derivatives up to order k are uniformly bounded and equicontinuous. We are in a position to use Ascoli's theorem. We can choose a subsequence {cp(v,,) I n = 1,2, ... } such that Gcp(Y n ) and D~ Gcp(v n ) converge uniformly to 1/1 and D~ 1/1 for Ihl ::; k. For simplicity let us replace Vn by v. Then we get
Y .... 00
(self adjointness)
=
lim(DGcp(V), GI/I) Y .... GO
V .... 00
(since cp(Y) -+ 0) = O. Thus 1/1 = O. But we should have 111/1110 = I, since 111/1110 = lim IIGcp(v)ilo = 1. V .... 00
This contradiction proves the proposition. Let us now begin to construct the cp(t) of Part (a). We use power series techniques but we notice that we could also use the implicit function theorem for Banach spaces [compare Kuranishi (1965)]. We want to construct cp(t) = I:'=l cpit), where
and each cp" • ... "n, E r(Ao,I(T) such that
ocp(t) - ![cp(t), cp(t)] = 0,
(9)
m
CPI(t) =
I
"= I
I1v tv.
(10)
where {I1J is a base for W ~ HI(M, 0). We use a method due to Kuranishi. Consider the equation (II)
where CPI(t) is given by (10). We first show that (II) has a unique formal
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
162
power series solution cp(t). In fact, this is clear since CP2(t) = !.9G[CPl(I), CPl(t)] CP3(t) = -!9G([CPl(t), cpit)]
+ [CP2(t), CPl(t)]) (J 2)
PROPOSITION 2.4. IIk+a' Proof
For small Itl, cp(t) = 'L:'= 1 cpit) converges in the norm
Let
P
A(t) = 161'
L y"(tl + ... + tmY 00
,,= 1
" A \' I -- L..
.. "'m
t VI t
... t"on m •
As usual, Ilcp(t)llkH ~ A(t) means IICPvI" Vno IIk+a ~ AVI .. Vm' and cp"(t) = CPt(t) + ... + CPJt). Then (12) is equivalent to (13) We want to choose P and y. Suppose they are chosen so that Ilcp"- l(t)lIkH ~ A(t). Since .9 is a linear differential operator of first order, II~ 9G[cp, 1jJ] ilk +a ~ C 1 il G[cp, 1jJ] 11k+ 1 +a
~ C1Ck,all[CP, 1jJ]lik-l+a
:s; C 1 Ck ,,, C IIcpllk +" IltiJ Ilk+a' by Propositions 2.2 and 2.3, where Cl' Ck. a' and C are constants independent of cp and tf;. Hence by (13) I cp(" >c t) Ii ~ + a ~ C 1 C~. a C il cp("- 1 l( t) II k+ a II cpU' -
1 l( t) II k+ a
~ C 1 Ck , a C(A(t»2
~ CICk,aC(~)A(t) as in Section 3, Chapter 2. Thus choose p and l' so that C1 Ck,a C(Pjy) < I. Then IlcpP(t)llk+a ~ A(t). These constants are all independent of /1. So if P and yare chosen so that Ilcpl(t)llkH ~ A(t), which is clearly possible, then C1 Ck,a CWly) < I yields Ilcp(t)llk+a ~ A(t).
So for small Itl, cp(t) converges.
Q.E.D.
2.
AN EXISTENCE THEOREM FOR DEFORMATIONS
I
163
PROPOSITION 2.5. The q>(t) of Proposition 2.4 satisfies oq>(t) - ![q>(t), q>(t)] = 0 if and only if H[q>(t), cp(t)] = 0, where H: r(AO. 2(T» ~ 1HJ2 ~ H2(M, 0) is the orthogonal projection to the harmonic subspace of 22 = r(AO,2(T».
Proof If oq> = 1[q>, q>], then 0 = Heq> = -!-H[q>, q>], since Ha = 0, Conversely let H[q>, cp] = 0 and set t/I(t) = oq>(t) -1-[cp(t), q>(t)]. Then each /'). E lH]1 so 01}. = 0 and
2t/1(t)
= o8G[cp(t), cp(t)] - [q>(t), q>(t)].
Recall that any w can be decomposed w = Hw + DGw. Since H[q>, q>] we get 2t/1(t) = (u9G - DG)[q>(t), q>(t)]. Because D ='09 + 9'0 we get
= 0,
2t/1(t) = - 9OG[q>(t), q>(t)] = - 9Go[q>(t), q>(t)]
= - 29G[ocp(t), q>(t)]. This last equality is true because
a[q>, q>] = [acp, q>] - [q>, aq>]
= [ocp, q>] + [aq>, cp] = 2[oq>, q>]. Then
q>(t) = - 9G[oq>(t), q>(t)]
= -9G[t/I(t) + 1[cp(t), q>(t)], q>(t)] = - 9G[l/t(t), q>(t)] by the Jacobi identity. Estimating, we get
Choose It I so small that 11q>(t)IIk+~ C, Ck,~ C < 1. Then we get the contradiction 1It/1(t)IIk+~ < 11t/I(t)llkh unless t/I(t) = 0 for all small t, Q.E.D. With the assumption H 2(M, 0) = 0 we have completed Part (a) of our task. We should notice the dependence of q> on z and t. PROPOSITION 2,6.
q>(z, t) is COO in (z, t) and holomorphic in t.
Proof It is immediate that q> is C k since the series converges in II 11k+~' Coo dependence is not so obvious. To give the proof we refer to the regularity theorem for quasi-linear elliptic operators [Douglis and Nirenberg (1955)].
164
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
Since the '1. are harmonic,
O
= t09G[
Since
Thus
02
v= 1
otv at.
L --_ + O
t9[
o.
( 14)
Equation (14) is quasi-linear elliptic since m a2 L atv -+0 at.
v= 1
is elliptic and 9 is first order. We know that rp(z, t) is small for small t and that the coefficients of (14) are Coo. Under these circumstances the regularity theorem says that solutions of (14) such as
PROPOSITION
2.7.
{M,lltl < e} is a complex analytic family.
L
Proof Consider
= ~ o
.= a-vt L.,
1
L.,
a= 1
a-aZ
and a
3.
AN EXISTENCE THEOREM FOR DEFORMATIONS"
165
Equation (15) is satisfied if and only if
i.-r_", CPaP~-O aP- ,
::I-a'"
uZ
L..., P
z
a(
ai
(16)
=0
Y
1X=1,"',n
,
v = 1,"', m.
Hence, on some coordinate chart d/l j = U j x BE s; .A we have n + m independent solutions t l , " ' , tin' (z, t), "', (j(z, t) of Equation (15). So JI is a complex manifold such that the projection 11: : .A ~ BE is holomorphic of rank m and for each fixed t, 1I:- I (t) = M, is the complex manifold with complex structure given by M, . Q.E.D.
(J
This also completes the proof of Theorem 2.1. We remark that by the completeness theorem (see Chapter 2, Section 3) the family 11:: .A ~ BE is complete at 0 E BE .
3. An Existence Theorem for Deformations II. (Kuranishi's Theorem) We want to discuss the case in which H2(M, 0) does not necessarily vanish. Again fix an Hermitian metric on M and define 8, D, G, "', and so forth. Let {1]y I v = 1, "', m} be a base for IHJI ~ HI(M, 0). It is not necessary to know H2(M, 0) = 0 for Proposition 2.4 to hold. Thus we still have a unique convergent (even in the norms defined later in this section) power series solution cp(t) of
+ t.9G[
(\)
m
I
where 11(t) = y
1]y
t
y •
And this cp(t) satisfies
=1
ocp(t) -1[cp(t),
=0
(2)
if and only if H[ cp( t), cp(t)] = O. Let {{J;..I A = 1, ... , r} be an orthonormal base of 1H]2 and let (,) be the inner product in A2 = r(Ao. 2 (T» (note the change of notation from Section 2, A 2 ~ .£f2). Then r
H[cp(t), cp(t)] =
I
([cp(t),
(3)
;"=1
Hence H([cp. cp]) = 0 if and only if ([cp(t), cp(t)], {J;) = 0 for A= 1, "', r. Since cp(t) is a power series in t so is ([cp(t), cp(t)], {J;..) = bit). Thus bit) is holomorphic in t for A = 1, "', rand It I small (It I < e). Also biO) = O. Define an analytic set S as follows:
S = {tlltl < e, b;..(t) = 0, A = 1, "', r}.
166
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
Then S is an analytic subset of BE containing the origin and we have proved the following proposition: PROPOSITION 3.1. COROLLARY.
t [ep(t), ep(t)] =
ep(t) satisfies oep(t) -
°
if and only if t E S.
{M, It E S} is a set of complex structures on M.
We note that S may be singular; however, {M, It E S} can be made into a complex analytic family over S. The details are in Kuranishi (1965). We want to prove that if 1/1 is a small enough vector (0, I )-form such that ol/! - HI/!, I/!] = 0, then M", is biholomorphically equivalent to one of the M" t E S. This is the completeness theorem (an abbreviated version) of Kuranishi. We find it convenient to use the Soholev norms since some of the continuity properties that will be needed will be more transparent in these norms. For an open set U of IR n and complex COO functions f and 9 defined on D, the closure of U, we set
L J Dj(x) . Dag(x) dx,
lal sk
(4)
U
where we use the multi-index notation and D a = (ajax I t' ... (ajax"t". Then
(X
=
«(XI, ••• , (Xn), (Xi
~ 0,
Il/ilk = J <1.J)k = Ilfllt
I(XI = L7= I (Xi' (5)
The classical Sobelev lemma states that if V is a relatively compact open subset of U, then there is a constant c such that if x E V and k > nj2
c 11/11f+ lal' only on k, I(XI, V, u.
(6)
IDal(x)1 :::;;
where c is a constant depending (6) we have: LEMMA 3.1.
There is a constant c such that if k
~
As a consequence of
n + 2,
Ilfglli :::;; c 1I/IIf . Ilgllf. Proof
Let
It I :::;; k.
(7)
Then D((jg)
=
L
Drl· DSg.
r+s;::::.!
Then either Irl + nj2 + I :::;; k or lsi + nj2 + I :::;; k when r + s = t, since It I :::;; k nd k ~ n + 2 implies It I + n + 2:::;; 2k. Thus, there is a constant K so that ID{lg(x)l:::;; K
L (1I/IIfID g(x)1 + IDr/(x)llIgllf)
r+s={
S
3. for x
E
167
AN EXISTENCE THEOREM FOR DEFORMATIONS II
V, by Equation (6). Squaring,
IDtfg(xW :,;
K'
L+~Jllfllf)2IDsg(XW + (1Igllf)2I D'f(XW].
Now (7) follows easily.
Q.E.D.
By using a partition of unity, one defines Ilcpllk for any cp E A P = PeT»~. The estimates of the previous section are essentially the same and we list the ones we need.
rcA
0,
I [cp, t/I] Ilk::; ckllcpllk+ 111 t/lllk+ 1[for all large k. (k
~
2n + 2 where dime M = n).] (8)
IIHcpllk::; ckIlcpllk
(9) (10)
From now on k will be chosen so that the conditions of Lemma 3 hold.
PROPOSITION
3.2.
For fixed lJ(t), Equation (I) has only one small
(1lcpllk < e)
solution.
Proof
Let
T
= cp - cp(t). Then T
= !9G([cp, cp] - [cp(t), cp(t)]) = t9G([T, cp(t)] + [cp(t), T] + [T, T]) =
Estimating
IITllk
!9G(2[T, cp(t)] + [T, T]).
gives
IITllk::; D(IITllk Ilcp(t)llk + IITIID ::; D IITllk(llcp(t)llk + IITllk)' If I cp(t)lIk is small enough, the only way
o ~ x ~ Dx(llcp(t)llk + x) can happen with x close to 0 is for x = O.
L
Q.E.D.
The set N = {1J(t) = IJv tv Iitl < e} describes a small neighborhood of I . For small enough B there is a one-to-one correspondence between '1 E N and solutions cp of (I). This is proved as follows: If cp = lJ(t) + t9G[cp, cp], then H9 = 0 implies '1 = Hcp. Thus '1 is uniquely determined by cp. Given '1, call the small solution of Equation (I) (given in Proposition 3.2) cp. Then a correspondence F is defined by FIJ = cp.
oE !HI
168
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
LEMMA 3.2. Suppose Mt/I is given, If 91/! = 0, then I/! = cp(t) for some t if 11I/!llk < b for small b.
Proof
ol/! - tel/!,
I/!J
= 0,
hence
o I/! = 90l/! + 091/1 = t9[1/!,
I/!J
since 91/! = 0. Then
I/! - HI/!
= GOI/! =
tG.9[I/!, I/!J.
Let 11 = HI/!. Then I/! = 11 + t9G[I/!, I/!J; and by assumption 11I/!llk is small so that 11I1llk is small by Equation (9). Hence 11 = I1(t) for some It I < 13, and the remark .iust made shows that I/! = FI1 = FI1(t) =
°
In general 91/! =f. so we must try something else. When we say that Mob is hoiomorphically equivalent to M"'(t) we mean that there is a biholomorphic map f: M ",(t) --+ Mob and f: M --+ M thus induces a diffeomorphism f: M --+ M of the underlying differentiable manifold M. Conversely let f be any diffeomorphism such that its values and first derivatives are close to the values and first derivatives of the identity map. Cover Mt/I with a system {Uj , ('i(z)} of local I/!-holomorphic coordinates and let {Vj}, Vj ~ Vj be a covering of M so that f( Vj) ~ Uj . Then ('i(f(z» is a local holomorphic coordinate on Vj. Thus cp is determined by the equation o("'(f(z» =
L" cpP(z)op("'(f(z».
(11 )
P~1
We also know
o(a(z)
" I/!P(z)op("'(z), =L fJ~
I
that is,
~ ""'(z) = ~ ./,/1.(z) 8("'(z). :)-A (" L... 'I' A :)-P
uZ
fJ~1
uZ
(12)
AN EXISTENCE THEOREM FOR DEFORMAnONS
3.
II
169
Putting Equations (11) and (12) together we get
L i7("(f(z» P
afP(z)
+ L a,a(f~z» ajA ).
ofP =
of
L
y, (1
The matrix (o(a/of{1) is invertible since it is assumed that Thus,
of{1(z)
I t/llik is close to zero.
+ L t/l1(f(z» oJA(z) ).
PROPOSITION 3.3. Let M", be the complex structure induced from M", by the map f: M --. M. Then
Proof
We just notice that
(oyfP(z) is an invertible matrix.
Iit/lilk small
implies that
+ ~ t/I~(f(Z»oYr(Z»)
Q.E.D. ~"
Thus to prove Kuranishi's theorem in the case 9t/1 =1= 0 it suffices to show the existence of a diffeomorphismf such that 9( t/I 0 f) = O. This is our task. We need to digress for a moment to describe a way of indexing diffeomorphisms close to the identity by the use of geodesics. We shall be brief, and refer the reader to any text on differential geometry for missing details. Let an Hermitian metric (gap) be fixed on M. Then we have the Christoffel symbols
r")'Il =" giia(Og{1ii) L..., ;:) A ' /J
uZ
170
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
and if u(z)
=
L ua,(z)(ojoza,) is a vector field, we have the covariant derivatives oua, + "r" V;, ua, = -;, L. U , OZ
(1
;'(1
(1
_ OUa, V;,Ua, = OZ;" If z(t) is a curve in M, we define V t ua,(z(t))
dz\t) t
_
dZ'"(t) dt
= L V;.ua, -d- + L V"ua,-;,
dZ"(t) ua,(z(t» + L r~p(z(t» - - u(1(z(t)) , dt ;',(1 dt d
= -
when u(z) is defined along z(t). Then the geodesics of M are the curves z(t) satisfying the differential equation dza,(t)) _ Vt ( dt - 0, that is, d(za,(t» dz;' dz(1 -d 2 + L r~p(z(t» - (I) -d (t) t ;" (1 dt t
=
(14)
O.
Since M is compact, all metrics are complete, that is, the geodesics through a given point with given tangent direction are defined for all time t. Let za,(t) = za,(t, zo, ~) be the solution of Equation (14) satisfying za,(O) = zo, (dza,jdt)(O) = ~a,. Then the following facts are easily verified by using existence and uniqueness theorems of ordinary differential equations: (I) (2)
za,(t, Zo, ~) are Coo in (t, zo, z"(kt, zo, ~) = za,(t, zo, k~).
~).
We set /"(zo, ,) = za,(l, zo, ~). Then f is Coo in za.(t, Zo , ~). Differentiating this relation we get dza.
dt
n ofa. (t, zo, ~) = fl~J ~(1 o~fl (zo,
n
(zo,~)
_
and /"(zo,
o/"
lO + P~l ~(1 o~fl (zo, to.
So dz lZ
n
[
or
or]
~a, = dt (0, zo, ~) = fl~l ~p 0~(1 (zo, 0) + ~fl a~P (zo, 0) . This implies
t~) =
3.
AN EXISTENCE THEOREM FOR DEFORMATIONS II
171
The Taylor expanison then yields (15)
°
where O(l~12) is a term bounded by M 1~12 for some M> and for small I~I. Given a vector field ~ = = I ~a(z)( 0/ oz") on M we define a diffeomorphism f~: M~M by
I:
By Equation (15) ( 16)
(16)
We wish to calculate ((J = t/I 0 ~ and we use Equation (13). We abbreviate withf~ = z" + ~a + ra. Then (13) becomes o~P
+ or + L t/l1(f~)(dz). + oe + of).) ).
=
JI [M +
Oy ~P + 0), r P + ~
t/I~(f~)(ay e + (Iy f).)] ((JY.
Multiplying by the inverse of the expression in brackets [ - ] we get ((JY =
oC + L t/lHf~) dz}. + ... ).
=
J~Y
+ L t/lHz) dz). + RY(t/I, 0J.
Thus, (17)
where R(tt/l, t~) = t 2 RI(t/I, ~, t) if t is a real number and both R, R, are Coo functions of the parameters t/lp(z), t/I(l~(z», ~a(z), (o~"/ozp)(z), (a~a/azp)(z) in local coordinates. In AO we have RO, the space of holomorphic vector fields on M. Using the L2 inner product on AO we let FO be the orthogonal complement of RO. So ~ E F ° if and only if (~, '1) = for all '1 E RO, that is, F D is the kernel of the map R: AO ~ RO. Then for ~ E FO,
°
~
=
GO~
+ R~ =
GO~.
Since 9 is zero on AD, 9~ = 0, and O~ = 90~,
yielding ~ = G90~.
Now give AO, Al and their subspaces the"
(18)
Ilk topology.
172
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
PROPOSITION 3.4. There are neighborhoods of the origin U and V in Aland F 0, respectively, so that for any 1/1 E U there is a unique ~ = ~(I/I) in V such that (19)
Proof
Using (17) we see (19) is satisfied if and only if
0= 9(1/1
0
h) = 90~ + 91/1 + 9R(I/I, ~).
By (18) ~
= G90~ =
-G91/1 - G9R(I/I, ~).
Thus (19) is equivalent to
+ G91/1 + G9R(I/I,
~
~)
= O.
(20)
Now choosing neighborhoods U1 , VI so that R is defined on U1 x VI we can define a map by h(l/I, ~) = ~ + G91/1 + G9R(I/I, ~). By our previous remarks on R(I/I, ~), we see that h is continuous if VI' VI, F 0 have the I Ilk topology, since R is continuous as a map from U I x VI with the II Ilk topology to A 1 with the II IIk-l topology. In fact, h is even uniformly continuous and hence has a (unique) extension to a mapping h of the completion of the domain to the completion of F O• Then the partial derivative -ah a~
I ..F----+r, ~o "'0 (0,0)
where po is the completion of F O, is the identity map. Then by the implicit function theorem [see Lang (1962)] there is a COO function 9 on a small neighborhood of the origin in the space Al (completion of AI)such that Equation (20) is satisfied if and only if ~ = g(l/I) for some 1/1 E U. Thus given 1/1 E U, where U is a small neighborhood in A I, there is a unique solution g(l/I) = ~ of (20) which is sufficiently small. Then 0 + 9R(I/I, ) + 9 is an elliptic secondorder equation with Coo. Thus, if 1/1 E U £; A I, ~ = g(l/I) satisfies O~
so ~ is Coo, that is, ~
E
FO.
+ 9R(I/I,
~)
+ 91/1 = 0
Q.E.D.
Let us summarize our conclusions. THEOREM 3.1. (Kuranishi) (a) Let M be a given compact complex manifold, and let {'1v} be a base for W ~ HI(M, 0). Let ({J(t) be the solution of the equation
((J(t) = !'J(t) + !9G[({J(t), ({J(t)],
4.
STABILITY THEOREMS
173
where l1(t) = L~= 1 tv I1v' It I < p, and let B = {t I H[cp(t), cp(t)] = O}.
Then for each t E B, cp(t) determines a complex structure M t on M. (b) Let 1/1 be any vector (0, 1)-form satisfying 01/1 - H 1/1, 1/1] = O. Then 1/1 defines a complex structure M", on M. If II 1/1 Ilk is small enough, there is a unique E FO such that 1/1 0 f~ = cp(t) for some t E B, and hence M", is biholomorphically equivalent to M,.
e
4.
Stabi Iity Theorems
The main point of this section is to prove that if .I( = {M,} is a complex analytic family and Mto is Kahler, then M, is Kahler if It - tol is small. We first study elliptic differential equations depending on a parameter. Let P = {tlltl < (1, t = (tl' ... , tr )} be an open disk in cr. Let X be a compact differentiable manifold, and let f!4 be a differentiable complex vector bundle over X x P. Let B, = f!4lx x (') be the restriction of f!4 to X x {t}. Then we can consider f!4 = {B,} to be a family of vector bundles over X depending differentiably on t E P. Let L(f!4) = the space of differentiable sections of f!4, L(B,} = the space of differentiable sections of B,.
We are only interested in small deformations so we can assume that P is small and then there will be a finite covering {Xi} of X such that f!4/ x,xp =C" x Xi
X
P,
«(t,
that is, f!4 is trivial over Xi x P. Let x, t) be a local coordinate on f!41 Xi x P • Then the coordinate transformations on f!4 are written as
(t = v=L" 1 b~v(X, t)G· By an (even-order) differential operator E, : L(B,) - + L(B r),
we mean a map which can be written locally in the form (E r I/I)t(x) =
L" Etv(x, t, Di)l/I7(x), v=1
where I/I(x) = (I/If(x» is a section of Br and where E~(x, t, D i) is a polynomial of degree m =0 (mod 2) in Di = (ajaxf). In our applications we will only need m = 2,4. For Er to be well defined we must have
L Ei~(X, t, Di)brkt(X, t)I/I~(x) = L bi1.(x, t)E;.<x, t, Dk)I/I~(X), t,
V
t,
V
174
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
where )L
I/It(x) =
L btklx, t)I/IZ(X). Y=
1
We say that Er depends differentiably on t if all of the coefficients of Ef.(x, t, Dj)as a polynomial in Djare Coo functions of (x, t). We assume given a Riemannian metric on M and an Hermitian metric on the fibres of Br so that
L giAv(X,t) etC dX At
j
= L gk(1t(X, tK~ er dX k
\I
at
t'
is invariantIy defined. We have written these expressions in terms of local coordinates for X where
C; = L bj~(1(x, tKZ (1
and dX j = dxl ... dx'i
(n
=
dim X).
We assume that 9 iAv(X, t) are Coo in (x, t). An inner product on L(B r) is defined by
(1) We will consider only those Er which are formally self-adjoint; that is, (E r cp, I/I)r = (cp, Er I/I)r
(2)
for all cp, 1/1 E L(Br). Let us see what this implies for the coefficients of E r . Let E~A(X, t, D j) be the terms of order m in Ej~(x, t, DJ Writing out (2) we get
f A,f,Y L gjAfEf."(x, t, D;)cpi(x)l/Ir(x) + lower order terms = f L g;;.v cpt(x)E7:,Y(x, D;)I/IXx) + lower order terms. x
t,
x A, v. T
Integrating the first term of the left-hand side by parts and using the fact that m is even we get
This is true for all cp and 1/1 so we get
L giAvE~vl/J~ = L giYtE';';.vl/J~ Y, T
for alI 1/1. Hence, (3)
4.
175
STABILITY THEOREMS
Replace (a/(Jx~) in Ei~v with Y... Then EjT(x, t, y) is a homogeneous polynomial of order m in y with coefficients which are Coo functions of (x, t). Equation (3) becomes
if A i .« = Lv gjv, EiJ.V(x, t,y). We have proved: PROPOSITION 4.1.
Equation (2) implies (3).
We shall assume further that EI is strongly elliptic, that is,
(_l)m/2
L Aj).;;(x, t, y)w). w! > l,r
°
for any real y = (Yl' "', Yn) #- 0, and any complex w = (WI' "', wit) #- 0. We need to collect some facts about such operators. We quote the following well-known theorem which can be found, for example, in Palais (1965, p. 182): THEOREM 4.1. EI has a complete orthonormal set of eignenfunctions {eth}h"=1 ~L(BJ Let the eigenvalues be Ah(t). Then they are real and
[Completeness means that any t/J
E
L(B 1) can be written as follows:
00
t/J =
L= ajeth •
h
1
Furthermore we can arrange the e1h such that
and lim h _ oo Ah(t)
=
+
00.
The following theorem is proved in Kodaira and Spencer (1960), and we shall not prove it here. THEOREM 4.2. REMARK.
Each eigenvalue Ah(t) is a continuous function of t E P.
Ah(t) may not be differentiable. For example, let
E = (r:x(t)P(t)) 1
y(t)b(t)
be a Oth-order differential operator (just a matrix). Then
1)
/1.( t
+ c5(t) ± J(r:x(t) ~ b(t))2 + 4P(t)b(t) = r:x(t) -,--,----,--""":",,,--,---,---,---,--,--:---,---,2
176
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
which could fail to be differentiable when (tX - 15)2 the kernel of E r • Then
IFr={I/IIEtl/l=O}={I/III/I=
I
+ 4f3t5 = O.
Let IFr be
aherh }.
).h(r)=O
Let Fr be the orthogonal projection to 1F t , that is,
I
Ft 1/1 =
(1/1, eth)eth'
).h(t)=O
The Green's operator G r is defined by
Gtl/l =
I
1
1 (t)(I/1, erh)eth ·
"h(t)*O Ah
F t and Gr are related by the equation 1/1 = E t G r 1/1
+ Ftl/l·
We have already investigated the case P = a point, E t = D, Fr = H, 1Ft Gt = G. In the general case we have the following theorem:
= !HI,
THEOREM 4.3. dim 1Ft is an upper semicontinuous function of t. This means that given to, there is a small enough e so that dim 1Ft ~ dim IFro for It - tol < e.
Proof dim 1Ft Ah(to) we have
...
~
= d t is finite since An(t) -+ 00. In the ordering of the
A)to) < 0 = Aj+ I(tO)
= ... =
Aj+do(tO) < Aj +do +let)
~
By continuity, choose e so that Ait) < 0 and Aj+do+l(t) > 0 if Then in this disk d t ~ do. Q.E.D.
....
It - tol <
e.
Next we want to say what we can about the differentiability of F t and Gr. DEFINITION 4.1. Given I/ItEL(Bt) for each tEP, we say that I/Ir depends differentiably on t if there is a 1/1 E L(!!J) such that
I/It = 1/1 Ix x {t} . Let
«(j, x)
be a local coordinate on fll, where
(; = I b;kv(X, t)(Z· Then if I/It
= {I/I;j(x)} in local coordinates, we have I/It~(x) =
I
b;kvCX, t)l/I;keX).
If 1/1 E L(!!J) is such that 1/1 t = I/Ilx x {t}, then 1/1 = {I/I;(x, t)} and I/I;(x, t) = I/I;/x) so I/I~(x) are Coo in x and t. Conversely, if the I/I;ix) are Coo in x and t, it is easy to see that I/Ir depends differentiably on t.
4.
177
STABILITY THEOREMS
DEFINITION 4.2. Given a linear operator A, : L(D,) -+ L(D,) for each t E P, we say that A I depends differentiably on t if the following condition is satisfied: If 1/1, depends differentiably on t, then A, 1/1, depends differentiably on t. REMARK.
E, depends differentiably on t in this sense.
Given E" let f = define
[~,
p] be a real interval on the A-line such that 0 E f.
We
THEOREM 4.4. Let to E P. If Ah(t O) ¥ ex, f3 for all h, then there is an open set V around to such that F,{f) and GI(I) depend differentiably on t E V.
Sketch of Proof the figure below.
Consider a rectangular contour C in the C-plane as in
c
~
71-line
(j
Ot
Figure 10
Let,
E
C and form E, - C. Then we have II(E, -
01/111; = IlL (1/1, elh)(Ah(t) -
Oe'hll;
~ min IAh(t) _ (l2111/111~ h
~
kill/lll?,
where k is some constant (greater than zero), as long as t remains in a small compact neighborhood of to. This implies easily that (E, - 0- 1 = G/(O is defined and continuous. It is harder to see that G,(C) depends differentiably on (t, C). This is proved in Kodaira and Spencer (1960). We can write down G,(O:
178
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
Then integrating around the contour C we get F,(I)
f
1 . G,(O d( = - -2 7rl
C
because
IJ
d( 2ni c A - ( =
{-I,0,
irA E I if A¢ I
(A. is real).
We also have 1
J
d(
G,(I) = 2ni cG,(O T since
f
1 d( 2n:i c (A. - O(
=
{O, 1,
if A. E I if J..¢ I.
Thus GlI) and F,(I) depend differentiably on t.
Q.E.D.
4.5. If dim IF, is independent of t E V, where V is some open subset of P, then F, and G, depend differentiably on t.
THEOREM
Since the Ah(t) are continuous in t and dim IF, is independent of t, we can find a small open set U around I and small interval I = [ - c, + c] around 0 such that Fs = Fs(I) and Gs = G,(I) for S E U. Then apply Theorem 4.4.
Proof
Before we apply these theorems, we need to make some more definitions. 4.3. By a differentiable family of compact complex manifolds we mean a triple (.II, W, P) where viI, P are differentiable manifolds, w is a differentiable map of rank = dim P, and each M, = w-I(t) is a complex manifold. More precisely for each Xo E .II there should be a local diffeomorphism from IilJ j 3 Xo DEFINITION
to a domain in en x IR m , where w(x) = (tl, "', t m ) are coordinates around w(x), and such that for a fixed t, (ZI(X), ... , zn(x» is a complex coordinate on M,. On IiIJ j n IiIJ k>
where fj~ is differentiable in (Zk' t) and holomorphic in Zk for a fixed t. Let T(.II) be the complex vector bundle over .II whose transition functions on
4.
STABILITY THEOREMS
179
uti j n Wk are given by the matrix
(~:D· T(A) is the bundle of holomorphic tangent l'ectors along the fibres of .A. Let
T*(.;{t) be its dual and T(A) its conjugate bundle. Then we set !T\r, s) =
(~ T\.;/I)) /\ ( ~ T*cA»).
We call !T*(r, s) the bundle of (r, s)-forms along the fibres. Let r t be restriction to M t • Then we have rt(T(A» = T(M t)
and
rr<,r\r,
s» = T*(r,s)(MJ
Let C' S be the space of Coo sections of !T* (r, s). Then L~' S = rtC' S is the space of COO (r, s)-forms on M t. Any tjJ E L r , S can be written 1
IjJ
= r! s!
where dzj
L 1jJ"" ... ""
Tf''''jl,(z, t) dzj' /\ ... /\ dz'j' /\ dz1' /\ .. , /\ dzP>,
= (0/8zj)*. Our transformation law for (8/8zj) is iJ ) (-iJz%8)-- IP= -8z~ J( iJz% . n
8z~
I
Thus the law for dzj is n
iJz'"
dz~J = "L...a~ _ J dz P k'
P= 1 Zk
This implies that dzj is not the differential of zj. If it were, it would transform according to the law n
8z~
m
az~
dz'"j-L...;:,p - " _ J dz P _ J dt), k + " L...;:,).' P=I uZk ),=1 ut Next, let fJI be a vector bundle on vi{ and let B t = rt(fJI). Then L r , S(fJI) is the space of COO sections of fJI ® !Y*(r, s) and C, S(B r) is the space of COO sections of Br ® T,*(r, s)(M t ). If f!4 is given by the transition equation 11
(7 = I
v=1
and IjJ
E
L r , S(f!4) is given locally by IjJ
b7k.(Z,
tm
= (1jJ J ' .. " 1jJ), then
180
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
where
We have assumed an Hermitian metric on ~ depending differentiably on t; hence, if CPt> I/It are in L(Et) and are Coo in t, then (cpl' I/It)t depends differentiably on t. We can introduce the operators dt! ot! Ot, 9t , b t , !J. t acting on L~' S = L" S(Et). Then the operators
Ot = Ot 9 t + 9t Ol' 6 t = dtb t + btdt , Ot depend differentiably on t. We now state the main theorem of this section. w
THEOREM 4.6. Let.A - - - + P be a differentiable family. If M to = (i)-l(tO) is a Kahler manifold, then M t = (i)-I(t) is Kahler for It - tol small enough. REMARK I. A good problem would be to find an elementary proof (for example, using power series methods). Our proof uses nontrivial results from partial differential equations (Theorem 4.1). REMARK 2. Hironaka (1962) has given an example of a non-Kahler deformation of a Kahler manifold. Hence the theorem is only true for It - tol small. Proof (of the theorem) Assume P Kahler we have a Kahler form Wo on M 0
Wo =
=
iL gap(z) dz a
{tlltl < I} s;; IRm. Since Mo is 1\
dz P•
We extend this to an Hermitian metric on all of the fibres as follows: Let .. " tm ) be local coordinates on 0/1 j ~ Uj x P s;; en x IR m, such that (i)(ZIJ" ... tm) = (11 " ... . tm) Then on 0/1.J
(Z}, .. " zj, tl,
i
L gjap(z) dzj 1\ dz~ = Wj
a,p
is a Coo Hermitian form which is independent of t. Let {p iz, t)} be a partition of unity subordinate to {o/1j}. Then we define Wt = Pj(z, I)W/Z) = i gtiZP(Z) dz iZ 1\ dz P• j a,p
L
L
The metric Wt depends differentiably on t and Wo is the Wo we started with. Let our inner products (cp, I/I)t be defined with respect to the metric Wt where cP, 1/1 E L;' s. Then the operator 0 t = 9 t 0t + 0t 9 t is strongly elliptic and depends differentiably on t so our theorems apply. In fact, one can check that the principal part of 0 t is
4.
ST ABILITY THEOREMS
181
in a local coordinate where (gfa) is the inverse matrix to (grajJ) (see, for example, Chapter 3, Section 2). We define the following spaces: Z~;s = {q> I q> E L~,sol q> =
Zb;s
O},
= {q> I q> E Cr", a, q> = O},
Lqt
= "~ Lr,s t,
Z:J,
=
r+s=q
{q> I q>
E L~,
dl q> = O},
= {q> I q> = L~'" 0, q> = O}, 1Hl~ = {q> I q> E L;, LlI q> = O}.
1Hl~'s
The theorems of Dolbeault and de Rham yield ~ -
01 L~' s
1Hl;
~
Hq(M I , C),
I
where
!1~
Z~'s 0,
IHlr,s
~ 1 -
HS(M !1r) , ,
I'
(4)
is the sheaf of germs of holomorphic r-forms on M,. As usual, let
h;'s = dim
1Hl~'s,
bq = dim IHI;.
Then bq is the qth Betti number of MI (which is independent of t since all of the MI are diffeomorphic to each other). M 0 is Kahler, so 1::.0
= 20 0 = 2Do .
(5)
This was proved in Chapter 3, Section 5. We have for each t 0101
+ 0101
= 0 = 9101 + 91 81 ,
(6)
since Kahler is not necessary for this. However, if the reader will consult the proof of (5) he will find that in the Kahler case (for example, t = 0) (7) Now we define EI = 01019,91+ 91910101+ 91018101 + ,91019101+ 9101 + 910,.
Then (EICP, 1/1)1
= (9, 01q>, 91911/1)1 + (0101q>, 01011/1)1
+ (91 VI cP, 91011/1)1 + (01q>, 011/1)1 + (01q>, 011/1)1' PROPOSITION 4.2. EI is a strongly elliptic self-adjoint differential operator of order 4 acting on L~'s.
Proof
We clearly have (Elq>, 1/1)1 = (q>, EII/I)I'
(8)
182
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
Thus E, is self-adjoint. We let the reader check that the principal part of E, is " P(1. by L. gj gj ;) (1., p, y, b VZj
(1.
04
::I P ::I Y ::I-b VZj VZj VZj
Q.E.D.
in a local coordinate system.
4.3. Eo = Do Do + 8 0 00 + 80 00 and E, cP = 0 if and only if 0, cP = 0, cP = o.
PROPOSITION
8, 9, cP =
Using Equations (5), (6), and (7) we get
Proof
Do Do
= (9 0 00
=
+ 00 80 )(8 0 00 + ( 0 80)
90 00 80 00 + 00 90 00 80 + 00 80 90 00 + 9000 00 80
= 80900000 + 00 00 8 0 80 + 80009000 + 80 00 80 00 . This proves the first statement. For the second statement we use the obvious inequality 2
(E,CP, cp) ~ 118,8, CPII,
2 + 1101cpll, + 110Icpll"2
Q.E.D.
As before, let 1F~'s = {cp I E, cP = 0, cP E L~'S}. Let F, : L~'s orthogonal projection to 1F~'s and G, be the Green's operator, so 1/1 = E,G,I/I PROPOSITION
r-
zr,s - 0 0 d,-'II
4.4.
J
-+
1F~'s
be
+ F,I/I.
,s-l t.:p Fr,s where w"
t.:p W
means orthogonal
direct sum. Proof If cP E IF~'" then d,CP = O. If 1/1 E oIO,L~-l,s-l then, d,CP = 0 since dlo,O, = (0 , + or)o,or = O. Thus oIO,L~-l,S-l + 1F~'s s;;; z~;'. The sum is orthogonal since
(0 1 0, cP, 1])r for 1] E 1F~'s. Next take 1/1
E z~·s.
~,
= (cp, 8r 8, 1]), = 0
Then
= E, G, 1/1 + F,I/I = 0, or IX + 9, 13 + ij, Y + 1],
where 1] = Fr 1/1. Since d r1/1 = 0 dl (8, f3
+ ij, y) = o.
Let a = 8, 13+ ij, y. Then a E L~'s and Or : L~" = 0, a = O. We claim a = O. For
-+
(9) L~+ 1 ,s, Or : L~' s -+ L~' s+ I, SO
0, a
(a, a), = (8, 13
+ ij, y, a), = (13, 0, a), + (y, 0, a), = o.
Thus
183
STABILITY THEOREMS
4.
= O. Hence IjJ = 0,0,(1. + 1], and
0'
Q.E.D. dim [Fl·I ;::: b 2 - 2h~·2 where b 2 is the second Betti number of
LEMMA 4.1. M,.
Proof
[FI.I , Since
a (j L I
,
~ -
O ZI.I/ (j , L. O" d,
O
c dL ,I n ZI.1 d,
,-
ZI.I ) dim [FI.I > dim ( ~ n Z1.1 ,£ILl, d,
+ £ILl) = dim (ZI.I d, '. £I LI
, ,
By de Rham's theorem b 2 = dim (ZJ,/d,L,I). We claim there is the following exact sequence
o
~
ZI.I + d L I Z2 n Z2.0 Z()·2 d" I d,' il, 0, d LI ~ d e~oLI.o+ J LO. I ·
, ,
, ,
"
"
We must define 1[, and check that it has the correct kernel. Let IjJ E Z}" d, IjJ = IjJ = 1jJ2.0 + IjJI.I + IjJO.2. Then d,1jJ = 0 yields 0,1jJ0.2 = 0,1jJ2.0 = O. So we can map IjJ to 1jJ2.0 + IjJO.2 E + Z~:2. Let IjJ Ed, Li. Then
o where
Z;:o
IjJ = drC
+
This correspondence induces the map 1jJ2.0 + IjJO.2 = 0,0'1.0 + 0,.0.1. Then
IjJ _ d,(O'I.O where
~
To compute ker
1["
+ .0.1) = ~ = IjJI.I _ 0,0'1,0 _ 0,.0.1 =
suppose
1[,
IjJ =
~I.I
is of type (l, I) Then d,~ =
Thus ~
1[1'
EZJ;I 2
and this yields IjJ .
b - dIm (
d,1jJ = O.
Ed,L,1 + ZJ;I.
This exact sequence implies
) ZI.I d, + d" LI) . ( Z2.0 0, d I :c:;: dlm;;-T;O IL,
u,L,
. + dun
But Dolbeault's theorem implies
and
Z2.0 ) hO•2 = 11 2 •0 = dim ( _0_,_ " o,L/'o .
Q.E.D.
(
Z()·2 ) 0,
~ .
oIL,
184
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
LEMMA 4.2. Proof
dim 1F:,1 = dim 1F6'o for smalliti. Let us compute dim 1F6,1. We claim (10)
We use (5) and Proposition 4.3 for the proof of (10). Since
Eo
Recall that on a Kahler manifold
+ h6· 1 + hg,2 h6,l + 2hg· 2 •
b2 = h~'o =
Thus,
By the upper semicontinuity of dim IF/,l, for It I < s. Also by upper semicontinuity of dim 1Hl?,2, for It I < s. Thus, dim 1F6,1 COROLLARY. Proof
~
dim IFtl,l
~
b 2 - 2h?,2
~
b 2 - hg,2 = dim 1F6,l.
Q.E.D.
F t depends differentiably on t.
Use Proposition 4.5.
Now we shall finish the proof of Theorem 4.6. We have Wt =
i
L ga.p(z, t) dza. 1\ dzP,
a.,P
which depends differentiably on t and Wo is a Kahler form Mo. Hence, dwo = O. In fact, if the reader will consult Chapter 3, Section 5 he will find
4.
STABILITY THEOREMS
185
we proved that Wo E IHIb,l = IFb,l. Thus Fowo = Wo. Since I/It = Ftw t depends differentiably on t, I/It -+ Wo as t -+ O. The form Wt is of type (1, I) and satisfies at'" t = at'" t = 0 so d", t = O. Let
wt = 1(l/It + llit)· Then wt is a closed (l, I)-form which is positive definite for small t since Wo = Wo is positive definite. Thus wt is a Kahler form on M t for small Itl. Q.E.D.
Bibliography Adler, A., The second fundamental forms of S6 and IPn(iC). Amer. J. Math., 91, 657-570 (1969). Benzecri, J. P., Varietes localement affines. Seminaire Ehresmanll (May 1959). Borel, A., and J.-P. Serre, Groupes de Lie et puissances reduits de Steenrod. Amer. J. Math., 75, 409--448 (1953). Bott, R., Homogeneous vector bundles. Alln. Malh., 66, 203-248 (1957). Calabi, E., and E. Vesentini, On compact, locally symmetric K1ihler manifolds. Alln. Math., 71, 472-507 (1960). Dieudonne, J., Foundations of Modern Analysis. New York: Academic Press, Inc., 1960. Douglis, A., and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations. Comm. Plire Appl. Math., 8, 503-538 (1955). Ehresmann, C, Sur les espaces fibres differentiables. c. R. A cad. Sci. Paris, 224, 1611-1612 (1947). Fischer, W., and H. Grauert, Lokal-triviale Familien kompacter komplexen Mannigfaltigkeiten. Nachr. Akad. Wiss. Gottingen Math-Phys. KI 11,89-94 (1965). Frolicher, A., and A. Nijenhuis, A theorem on stability of complex structures. Proc. Nat. A cad. Sci. U.S.A., 43, 239-241 (1957). Gunning, R. C, and H. Rossi, Analytic Functions of Several Complex Variables. Englewood Cliffs, N. J.: Prentice Hall, Inc., 1965. Helgason, S., Differential Geometry and Symmetric Spaces. New York: Academic Press, Inc., 1962. Hironaka, H. An example of a non-Kahlerian complex analytic deformation of Kahlerian complex structures. Alln. Math., 75, 190-208 (1962). Hirzebruch, F., Nelle Topologische Melhoden ill der algebraische Geometrie. 2d edition. Berlin: Springer-Verlag 1962. ---,Uber eine Klasse von einfachzusammenhangenden komplexen Mannigfaltigkeiten. Math. Ann., 124, 77-86 (1951). Hirzebruch, F., and K. Kodaira, On the complex projective spaces. J. Math. pure et appl., 36,201-216 (1957). Hbrmander, L., Linear Partial Differential Operators. Berlin: Springer-Verlag, 1963. Kodaira, K., On the structure of compact complex analytic surfaces I. Amer. J. Malh., 86,751-798 (1964). --,On cohomology groups of some compact analytic vanetie" with coefficients in some analytic faisceaux. Proc. Nat. A cad. Sci. U.S.A., 39, 865-868 ( 1953). ---,On Kahler varieties of restricted type. Anl1. Math., 60, 28-48 (1954). ---,On compact complex analytic surfaces I. 41111. Math., 71,111-152 (1960).
186
BIBLIOGRAPHY
187
Kodaira, K., L. Nirenberg, and D. C. Spencer, On the existence of deformations of complex analytic structures. Ann. Math., 68, 450-459 (I958). Kodaira, K., and D. C. Spencer, Groups of complex line bundles over compact Kahler varieties. Proc. Nat. Acad. Sci. U.S.A., 39,868-872 (1953). ---,On deformations of complex analytic structures, I and II. Anll. Math., 67, 328-466 (1958a). ---,A theorem of completeness for complex analytic fibre spaces. Acta. Math., 100,281-294 (1958b). ---,On deformations of complex analytic structures, III. Stability theorems for complex structures. Ann. Math., 71, 43-76 (1960). Kuranishi, M., New proof for the existence of locally complete families of complex structures. Proceedings of the Conference 011 Complex Analysis in Minneapolis 1964. Berlin: Springer-Verlag, 1965, pp. 142-154. Lang, S., Introductioll to Differentiable Manifolds. New York: Interscience Publishers, 1962. Milnor, J., Morse Theory. Princeton, N.J.: Princeton University Press, 1963. Nakano, S., On complex analytic vector bundles. J. Math. Soc. Japan, 7, 1-2 (1955). New lander, A., and L. Nirenberg, Complex analytic coordinates in almost complex manifolds. Anll. Math., 65, 391-404 (1957). Palais, R. S., et al., Seminar on the Atiyah-Singer Index Theorem. Annals of Mathematics Studies No. 57. Princeton, N.J.: Princeton UniversIty Press, 1965. Serre, J.-P., Une theoreme de dualite. Comm. Math. Helv., 29, 9-26 (1955). Teichmiiller, 0., Extremale quasikonforme Abbildungen und quadratische Differentiale. Abh. Preuss. Akad. del' Wiss. Math.-naturw. Klasse, 22, 1-197 (1940). Wu, W. T., Sur les classes caracteristiques des structures fibrees spheriques. Actualittis Sci. indust., Hermann, Paris (1952).
Index
A
Algebraic, projective, II
Dolbeault's lemma, 74, 79 Dolbeault's theorem, 80 Dual form, 93
B Blowing up, 17 Bounded domain, 144 Bundle, vector, 62 conjugate, 65 dual, 65 quotient, 66 section, 66 sub, 66 tensor product, 65 Whitney sum, 65
F Family, complex analytic, 18 differentiable, 178 Fine resolution, 73 Form, differential, 67 G
Germ, function, 27 Green's operator, 158, 176 Group, discontinuous, 12
C H
Cauchy-Riemann equations, 3 Chern class, 64, 127 Completeness, 55 theorem of, 56 Conjugate tangent bundle, 66 Coordinate, local complex, 7 Covariant differentiation, 106-109 Curvature, Ricci, 118 tensor, 117
Hartogs' lemma, 24 Hermitian metric, 83 Hirzebruch, F., 15 Hodge decomposition theorem, 114 Hodge metric, 134 Holomorphic, function, I mapping, 10 I
D
Deformation, 19 infinitesimal, 36-39 small, 19 de Rham's theorem, 72 Derivative, exterior, 68 Differentiable mapping, 10 Differential operator, 173
Implicit mapping theorem, 7 Integral, 88 Inverse mapping theorem, 6
J Jacobian, 5
191
192
INDEX
K Kiihler, form, 84 manifold, 84 metric, 84 Kodaira, K., embedding theorem, 136 vanishing theorem, 125, 131 Kuranishi, M., theorem of completeness, 172 L
Laplacian, 97 Locally trivial, 39 Logarithmic transformation, 16 M
Manifold, complex, 7 Hodge, 134 Hopf,14 Kiihler,84 Meromorphic function, II N
Nakano, S., vanishing theorem, 132 Newlander-Nirenberg theorem, 156 Norm, 92 Sobolev, 166
o Oriented, 88 Osgood's theorem, 2 P
Partition of unity, 61 Poincare's lemma, 70 Polycylinder, 2 Polydisc,2 Positive line bundle, 131 Product, inner, 92 wedge, 67
Projective algebraic, 11 Pseudogroup, 8
Q Quadratic differentials, 106 Quadric transformation, 17 Quotient space, 12 R
Refinement, 31 Rigid,45
s Semicontinuity, upper, 45 Serre duality, 104 Sheaf,29 cohomology, 31 exact sequence, 57 fine, 61 homomorphism, 56 quotient, 57 section, 29 Sobelev lemma, 166 Stability of Kabler manifolds, 180 Stalk,29 Stokes' theorem, 91 Structure, canonical, 9 complex, 7 flat affine, 9 Submanifold, 11 Subsheaf, 56 Surface, Hopf, 23 ruled, 15, 25, 41 Surgery, 15
T Tangent bundle, 66 Tensor bundle, 67 Torus, 13, 21-23
v Vector field (holomorphic), 36
Errata Page v, line 9: "lines fundles" should read "line bundles". Page 1, line -9: Ck ... k n should read Ck1 ... k n • Page 2, line 1 after the figure: There should be a comma between and i. Page 2, line -5: ::::; should read ~. Page 3, lines 9, 11, and 13: Ckl .•• kn should read Ck 1 •.. k n • Page 3, line -3: oZv should read oZv. Page 4, line -13: i ::::; 1/ ::::; n should read 1 ::::; Page 7, line -15: ZU(p) should read zn(p).
1/ ::::;
Page 10, line 16: L:~:~l xf should read L:~:il Page 10, line 21: v = 1,2 should read 1/ = 1,2. Page 12, line 12: pI should read ]pl.
IWi I
n.
xf = 1.
Page 12, line -5: pEG should read 9 E G. Page Page Page Page Page
13, 13, 18, 19, 26,
line 2: Fm+s should be Fm+!. line 2: gm should read g, twice. line 10: 1m w -+ 0 should read 1m w > 0 . first line of the diagram: should be ~. line 5: z(f' should read
!
zr.
Page 32, line 13: TJl.l ... Jl.q should be replaced with (kT)Jl.l ... Jl.q' Page 33, line -9: Omit the first "f/ (in the subscript of II). Replace the second script "f/ with %'. Page 33, line -5: Replace the first occurrence of II~ with II~. Page 35, line 11: Replace all occurrences of Y with {j Page 38, line 5: Page Page Page Page Page Page
39, 41, 42, 43, 44, 55:
"lAO
should be
"lAV'
line -10: Mo should be Mb. lines 3 and 4: ;:::: 1 should be > 1. line 10: Zl should be Z2. line -6: OZI should be 0(1. line 10: -mglOzlb lO should be blO - mglOZl. Add the map cp to the top arrow of the diagram. 193
ERRATA Page 58, lines -13 and -6: E cO should be E Co. Page 61, line 3 after the diagram: hcp should be kcp. Page 64, line -13: n ~ Z should read n ~ 2. Page 64, line -6: hk(Z) = should read fik(Z) =. Page 80, line -3: There should be a space after the comma between cp and q. Page 81, lines 12 and 13: There should be a comma after 0; "The" should be "the". Page 85, line -11: The last term in this equation should read ~
-
2:"",6 8z"'8zil dz Q 1\ dz,6. Page 85, line -2: dz", should read dz Q • Page 86, line 10: 22: 9jQ,6 should read 22: 9jQ!3· Page 87, line -4: The second z in this equation is missing a subscript j. Page 93, line -13: 'lj;ApBq should be 'lj;ApBo; f.L1 should be f.Ll. Page 93, line -9: Aq should be Ap both times. Page 93, line -7: The first f3 in this equation should NOT have a bar. Page 95, line -5: cp 1\ *'lj; should read cp 1\ *1P. Page 95, last line: The left-hand side of the last equation on this line should read 8'lj; = instead of iJ'lj;. Page 100, line -10: The second equation on this line should read - >.
ofjkv(z) =
o.
Page 108, line 12: omit the bar over 0>.. Page 112, line -7: Proposition 5.4 should read "In the Kahler case
" Page 118, line 5: The last factor in the subscript of the R in the right-hand side of the equation should be 1/. Page 118, line 9: The subscript on the last R should be {3i1>... Page 120, line -4: {31 should be i31. Page 120, last line: (f)i should be (f)k. Page 124, line 10: x should be +. Page 126: line 5: {3q should be Bq; the term involving cp after the summation sign should be CPjB• . CP:·. Page 126. lines 11, 12, and 13: All superscripts Bq should be Bq; in line 13 there should be a bar over the entire expression ~ ",cp:•.