Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Series: Mathematisches Institut der Universit~t Bonn Adviser: F. Hirzebruch
603 Boris Moishezon
Complex Surfaces and Connected Sums of Complex Projective Planes
Springer-Verlag Berlin Heidelberg NewYork 19.77
Author Boris Moishezon Mathematics Department TeI-Aviv University TeI-Aviv/Israel
Library of Congress Cataloging in Publication Data
Moishczon, Boris, 1937Complex sumfaces and connected sums of complex projective planes. (Lecture notes in mathematics ; 603) Bibliography: p. Includes index. I. Surfaces, Algebraic. 2. Projective planes. 3. Manifolds (Mathematics) I. Title. If. Series: Lecture notes in mathematics (Berlin) ; 603. QA3.L28 no. 603 ~QA571~ 510'.8s r516'.352~ 77-22/36
AMS Subject Classifications (1970): 14J 99, 32J15, 57A15, 57 D55
ISBN 3-540-08355-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-08355-3 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
TABLE
OF C O N T E N T S
Page Introduction
1
Part I.
6
Topology of simply-connected algebraic surfaces of given degree n
§i.
A topological comparison theorem for fibers of holomorphic functions on complex threefolds
~2.
A topological comparison theorem for elements of linear systems on complex threefolds
35
§3.
Comparison of topology of simply-connected projective surfaces of degree n and non-singular hypersurfaces of degree n in ~pg.
60
§4.
Simply-connected algebraic surfaces of general type
66
§5.
Topological normalization of simply-connected algebraic surfaces
69
Appendix I.
Generic projections of algebraic surfaces into ~p3
6
72
§i.
A theorem of F. Severi
72
82.
Duality Theorem and Corollaries of it
82
§3.
Proof of Theorem 3 (§3, Part I)
99
Part If.
Elliptic Surfaces
116
§i.
Deformations of elliptic surfaces with~'non-stable '' singular fibers
116
§2.
Lefshetz
162
fibrations of 2-toruses
Kodaira fibrations of 2-toruses
196
Topology of simply-connected elliptic surfaces
222
Appendix If.
R. Livne.
A Theorem about the modular group
223
Bibliography
231
Subject Index
23~
Introduction
In [i] Wall proved the following theorem: If VI,V 2 are simply-connected h-cobordant such that
compact 4-manifolds,
which are
to each other, then there exists an integer k ~ O V 1 / k(S2×S 2) is diffeomorphic
(/ is the connected
to V 2 ~ k(S2×S 2)
sum operation).
It follows almost immediately is a simply-connected
from this result that if V
compact 4-manifold,
integer k ~ O such that V ~ (k+l)P ~ k Q
then there exists an
is diffeomorphic
to
LP ~ mQ for some ~,m ~ O, where P is ~p2 with its usual orientation and Q is ~p2 with orientation
opposite to t h e usual.
After the proof of his theorem Wall writes the following ([i], p. i~7):
"We remark that our result is a pure existence
theorem; we have obtained,
even in principle,
no bound whatever
on the integer k". As it was remarked in [2], the operation V ~ P (resp. where V is an oriented performing
(V /Q)
4-manifold could be considered as
of certain blowing, up of some point on V.
this blowing-up V-process
We call
(resp. ~-process).
W e say that an oriented compact simply-connected W is completely decomposable
4-manifold
(resp. almost completely decomposable)
if W (resp. W ~ P) is diffeomorphic
to ~P ~ mQ for some ~,m ~ O.
2 Let V be an oriented For
(kl,k2)
obtained
~ ~x~,
from
V
compact
simply-connected
k I ~ O, k 2 ~ O, let V(kl,k2) by
kI
J-processes
and k 2
!-manifold.
be a 4-manifold
q-processes.
L
Denote by
(V) = {(kl,k2)
completely
decomposable].
that
(V) ~ ~.
minimal
elements
g = x ~ I k I & O, k 2 ~ O, V(kl,k2) It follows
An important of
from the theorem of W a l l
geometrical
(V) (in any natural
problem
of
(V) in explicit
2-dimensional
form,
A certain
step
of some
say in terms of the
Betti number and of the signature
In the present work we show that possible when V admits a complex
is to define
sense).
for solving this problem could be the construction elements
is
of V.
such a construction
structure.
is
The main result
is
the following: Theorem A. Let V be a compact admits a complex to a certain polynomials
structure.
complex
simply-connected
4-manifold which
Take an orientation
structure
on it.
Let
on V corresponding
K(x),L(x)
defined as follows:
K(x) : K(~(Sx+~))-x,
L(x) : ~(~(5x+~)),
~(t) = ~(t2-6t+ll),
~(t) = ~ ( 2 t 2 - 4 t + ~ )
(K(x) = 30375x 3 + 6885Ox 2 + 52OO4x
where
+ 13092,
L(x) = 6075Ox } + 14175Ox 2 + iiO265x
+ 28595).
be cubic
Denote by b + (corresp. b_) the number of positive (corresp. negative) squares in the intersection form of V !
and let
|
k I = K(b+), k 2 = max(O,L(b+)-b_). Then the pair (k~,k~) ~
(V).
From the Kodaira classification of compact complex surfaces it follows that if V is a simply-connected compact complex surface, then there exists a non-singular projective-algebraic complex surface ~ such that ~ is diffeomorphic to V and one of the following three possibilities holds: (b) ~ is elliptic;
(a) ~ is rational~
(c) ~ is of general type.
our theorem is evident.
In the case (a)
In the case (b) we prove a much stronger
result: Theorem B
(see Theorem 12, §4, Part II).
Any simply-connected
elliptic surface V is almost completely decomposable. (That is, (i,O) E
(V)).
In the case (c) (see Theorem 5, §4, Part I) we use Bombieri's results on pluricanonical embeddings ([3]), results of [2] on the topological structure of non-singular hypersurfaces in ~p3 and the following: Theorem C (see Theorem 4, §3, Part I).
Let
Vn
algebraic surface of degree n embedded in ~ p N
be a projective N ~ 5, such that
V n is not contained in a proper projective subspace of ~pN.
Suppose that V n is non-singular rational double-points.
or has as singularities
Let h: ~
n
only
---~V be a minimal
desingularizat ion of V n (that is, ~n has no exceptional of the first kind s such that h(s) is a point on Vn). X n the diffeomorphic
type of a non-singular
curve Denote by
hypersurface
of
degree n in ~p3. Suppose
l(Vn) = o.
Then
i) b + (~n) < b+(Xn) , b_(~n) < b_(Xn) , ii) ~ n ~
[b+(Xn)-b+(~n)+l]P
is diffeomorphic
to X
n
{ [b-(Xn)-b-(~n)]Q
/ P.
Note that Theorem B together with results of [2],[~],[5] shows that all big explicit classes of simply-connected
algebraic
surfaces considered until now have the property that their elements are almost completely decomposable the "theoretical"
4-manifolds.
That is,
Theorem A gives much weaker results than our
"empirical knowledge".
The interesting
question
can move with such "empirical achievements" classes of simply-connected
algebraic
is, how far we
in more general
surfaces.
I prepared this work during my visits to IHES, Bures-sur-Yvette, West Germany,
France, and Sonderforschungsbereich,
in the spring of 1976.
The excellent
Bonn,
conditions
5
w h i c h I found in these I n s t i t u t e s w e r e v e r y i m p o r t a n t necessary)
for the a p p e a r a n c e of t h i s work.
(and
I am v e r y
g r a t e f u l to b o t h of these Institutes. The A p p e n d i x to Part I is e s s e n t i a l l y b a s e d on the advice of D. Mumford.
D. M u m f o r d told me about S e v e r i ' s
t h e o r e m and e x p l a i n e d its use for the proof of part 3) of T h e o r e m 3, §3, Part I.
The proof of part 4) of this t h e o r e m
is a l s o due to D. Mumford. The idea to use in the proof of L e m m a 4, §i, Part II, a n o n - r a m i f i e d c o v e r i n g is due to P. Deligne. ~.
N e u m a n n and D. H u s e m o l l e r read the m a n u s c r i p t b e f o r e
it w a s t y p e d and made m a n y u s e f u l remarks. I w o u l d like to express here my deep thanks to all of them.
Part I Topology
§i.
A topological comparison theorem functions on complex threefolds.
Lemma
1.
function where
c-ball
double-point
in U of radius
2-disk in A of radius
f: U -->4
point c on Uj
on f-l(o).
f(c) = O (~A) and c
Let
B
E and with the center
e
to S
'
, be a closed
and such that
{and therefore
= ~B £
be a closed
c, D e
E' with the center O E ~
f - l ( ~ E ) is transversal
be a
A is the open unit disk in ~, such
f has only one critical
is a rational
surfaces
for fibers of holomorphic
Let U be an open subset in ~3,
holomorphic that
of simpiy-connected algebraic of given degree n
E
Denote fs = f f-l(
It is clear that ~: f - l ( ~ , )
%:
N SE ~
: f-l(~
)NS '
f-l(~E, ) n S e ~E#×
'
) N S
e
'>~
~n •
E
> D E ~ is a fibre bundle~
fsl(O)
let
be some trivialization
of it. Let U
= f-l(7),
desingularization
T ~ ~,
h: L
of U o (that is,
bU o
be a minimal
U~o does not have an exceptional
curve of the first kind s such that h(s)
is a point on Uo) ,
T, E D ~ , .
Then there exists a diffeomorphism e: h-l(Uo n Be)
> UT, D B e
such that
(~ lh-i(UoNS e) )
(h-i IUonSe ) : ~o N s ~ oo~uT, N S
"
coincides with the canonical diffeomorphism fsl(O) --~ fSI(T') corresponding to the trivialization ~ of fs" Proof.
We use the theory of simultaneous resolution for
rational double-points (see [6],[?]).
It follows from this
theory that there exists a positive integer m and a commutative diagram of holomorphic maps
u <~
~i
i
A~...
coincides with canonical projection (U-c) x~ ~ "l
~il(~o)
: ,[l(Uo)
~>U
o
coincides w i t h
the function ~ has no c r i t i c a l values.
>~,
h: T0 '' > U
o
and
Let B = , [ l I ~ ), S = , ilIs~l ,
% : ~ ~-lI~n~ : ~ - l ~
NOW using
>~.
~S: ~ - l ( ~ ) n S
we can identify
~: f-I(DE.)AS E
trivialization of ponding to ~.
%"
~
= , -l
~ecause .~llc~ ~ ~ :
>#
with [f--I(DE,)~SE]X~,~-
} D E , x fsl(0)
~: ~-I(~)N~
we obtain a
> ~ × ~S I(O) corres-
In particular, the canonical diffeomorphism
~Zl(O)
~ ~SI(@), Q , D , corresponding to
fsl(O)
>fsl(~m)
It is clear that
corresponding to ~.
~
coincides with
Let
~-i~) is transversal to ~B = ~.
no critical points, % :
~-i(~)~
>~
Since ~ h
is a differentiable
fibre bundle and we can construct a trivialization ~: ~-l(~)N~---~ ~
~-l(~)n{
~-l(~)nF
x[~-I(o)AB] of ~B such that the diagram
, ~"
>9
x
[~sl(o)]
~ > ~ x [~-l(o)n~]
is commutative. Take
~ E ~-0
with (~.)m = 'I" and let ~:~BI(O)--~ 'f'Bl( 0''
canonical diffeomorphism corresponding to ~,
"I,O",B
= ~l ~-l(a,)~
: ~-l(~')n~--~
f-l(~')nB~"
N o w it is easy to v e r i f y that we can take ~ = ~I,E',B Q .E .D.
Definition
1.
Let W be a 3 - d i m e n s i o n a l
and let V be a complex the singular
complex manifold
subspace of W, dim~V = 2.
locus ~(V)
of V is canonical
We say that
if V is reduced and
for any p E S(V) we have one of the following possibilities: a) p is a rational
double point of V;
b) p is an ordinary
singular
exists a complex coordinate
point of V, that is, there
neighborhood U
of p in W such that P
V is defined in U
by one of the following equations: P
(i) zlz e = o, (ii) ZlZ2Z3 = O (triplanar 2 2 (iii) Zl-Z2Z} Theorem
1.
point),
(pinch-point).
Let W be a } - d i m e n s i o n a l
A = It E ~I Itl < 4], f: W --~ A Suppose that the singular
complex manifold,
be a proper holomorphic
locus S(Vo)
of V o = f-l(o)
map.
is canonical
and let R denote the union of all rational double points of Vo, S = S ( V o ) - ~.
Let
h: L
be a minimal
• > V°
hs: ~---~S
be the n o r m a l i z a t i o n
desingularization
of S,
of Vo, C = h-l(s),
10 hc: ~--~.C
be the normalization
neighborhood
of C in ~ o.
points of Vo,
of C,
Let Xl,X2,...,x V be all the triplanar
yl,Y2,...,y p
be all the pinch-points
~ ~ ~ 3) = h sI(x~) , ~ = 1,2,...,~, (Xl,X2,X Consider
~ copies of ~p2
PI,P2,...,PM in P~,
7: TC --~C be a regular
and let (~i:
of Vo,
~Ym = h-l(ym) , m = 1,2,...,p.
(with usual orientation),
~2 : ~3 ) be the homogeneous
say coordinates
~ = 1,2,...,~,
E
:
(~ ~ p~ ~
(x)
=
o}
si = where
(i',i")=
= 1,2,-..,~,
Let T. l = 1,2,''" disjoint;
,9.
Take
and
i = 1,2,3
T1,T2,T3
i).
of si, i = 1,2,3,
so t h a t T. N E. l 1
(see Fig.
L
neighborhood
for any i = 1,2,3
of s. in E.~ 1 1
(i',i-)
be a tubular
(1,2,3)-(i)},
T1,T2,T 3
are pairwise
is a tubular
neighborhood
T i~ N E ~ = ~, T~I O E~..I = ~ where ' '
= (l,2,3)-(i).
Then there exist: a)
a complex-analytic
b)
a differential
embedding
n°i
coincides with the canonical
C
: ~---~
projective ic: ~
line bundle
~: A
> ~;
> A such that map corresponding
11
11 Fi~. 1
.
7~
~7~\ ~ ~
~
.
.
.
~
°
3
E~ Tc
TC
12
to
hIc: C
> S, that is, ~.ic: ~
of degree two ramified in exactly p c)
>~
is a covering
points y'~l,°.o,~'p of ~;
3~ pairwise disjoint closed 2-disks d.1 with the centers xi, i = 1,2,3, ~ = 1,2,...,~, on ~S -
G
~(qm)
m=l
where
qm = ic((~'ic)-l(~m ))' m = 1,2,--.,p; d)
diffeomorphisms where
,~1 ~'-l(Sd~)--->ST:, i = 1,2,3, ~=I,2,..-,V ,
w' = n.~: A' --~ ~
and
~: ~'
~A
is the blowing-up
of A in the points ql,-..,qp, with the following properties: (ld)
each
~i' i = 1,2,3,
orientations induced on of
A'-W'
-1 ( d~)
(2d)
~.I
T~
by the complex structures
is an isomorphism of the following S2-bundles
> Si~ (~Ti~
~'I
: w'-l(Sd~) ~,-l(~d~)
' ~si
> 8d'~ and
corresponds to the canonical projection
> si) ; (3d)
image of e)
~.-i( ~d i) ~ ard ~Ti
and P~-Ti;
with the base SI: ~T.lz
D. = 1,2,...,v, reverses the
~(L'On'-l(sdi~)) = 8(T.~E?)
where L' is the strict
ic(~ ) in A';
an orientation reversing diffeomorphism ~: 8TC where
(union
3 ~ ~, i~J_l(Ei-(Ti~Ei )) is taken in P~),
> 8TC~,
13
V: L' N [ ~/ ~'-l(~di)] L=I i=l
>
~=l
8(T ~E )
=
is the d i f f e o m o r p h i s m w h i c h is equal to
i L,N~,-I(~d~) A
on the c o r r e s p o n d i n g
connected components, of C on A, where
3
~
~=i i=l
~
~,-i( di ) U,[
~=i i=l
~=i
3
(P~-
=l(;Ti)
connected components,
the o r i e n t a t i o n s of ~TC and ~ T ~ c o r r e s p o n d to the
orientations structures
f)
Ti)],
is the d i f f e o m o r p h i s m
w h i c h is equal to %i on the corresponding and
of TC and T ~ which are defined by the complex
of ~
and A';
o
open subsets U c C, U c C and a d i f f e o m o r p h i s m such that (I) and
> C
A
is a regular n e i g h b o r h o o d
= A' - ~
A
~: TC
_
= _
~)
where _~ = •
(2) the diagram
/%~I
:L U
is commutative;
>
/%
!: U
Do: U --> U _~ =
14
g)
diffeomorphisms ut: (~o-TC) U
(A-TC)
> Vt,
where
t ~ ~ - ( 0 ) , vt = f - l ( t ) . Proof.
Let
B~,Bm,
open s m a l l paiz-~ise d i s j o i n t
for any
~ = 1,2,..-,~,
m = 1,2,...,p,
c o o r d i n a t e 6 - b a l l s on W such t h a t
~ = 1,2,...,~ (corresp. m = 1,2,-..,p)
B~ (corresp. Bm) is x~
be
the center of
(corresp. ym) and V ° N B~ (corresp.
Vo N Bm ) is defined in B~ (corresp. Bm) by the local equation z I(~) z 2( ~ )z(3~ ) = 0 (corresp . (z~(m)) 2 - z~(m)(z~(m)) 2)
where
z [ ~ ) , z ~ ) , z ~ ~) (corresp. zi(m),z~(m),z~(m) ) are complex coordinates in B~ (corresp. B'm ). Let
T: T
that for
~S
be a regular neighborhood of S in W such
~ = 1,2,...,~
and
m = 1,2,...,p
~G'n~4¢, ~ ' n ~ ~ 4 ~, ~
is transversal to ~B~ and to ~B' m
and if p
s
= s-
[s n ( ( ~ i B ~ >
u (~
m=l
B~))],
p
then
g(g') = S' and
g ~.: ~'
Let A' = [t , ~{ It{ < ~]. coordinate t we can assume that
to ~g.
> S'
is a 4-disk
bundle.
Changing if necessary the ~ t ~ ~',
V t is transversal
15 Consider the ball B I.
We can assume that complex coordinates
Zl,Z2,Z 3 in B 1 are chosen so that f B1 = ZlZ2Z 3. z(1)z(1)z (1) = O 1
2
3
is a local equation of
f-l(o) has no multiple components,
V
o
[Recall that
A Blin B1 . f
fl = Zl(1)z2(1)z~l)
Since
is an
invertible holomorphic function in some neighborhood of x I. z I = fl" z 1(I) ' z 2 = z i) ' z3 = z
take
smaller].
Changing, if necessary,
We
and choose B 1 and
Zl,Z2,z3,t
(t is the complex
coordinate in 4) we can identify B 1 with the open ball
3 C(Zl,Z2,Z3) E ~3 i__Z~llZil ~" 2 < 3~ in ~3 and fiB 1
with the
function ZlZ2Z 3 . Let Z = V 1 N B 1 = C(Zl,Z2,Z3)E ¢31ZlZ2Z3 = i,
! 'zi'2 < ~ ' i=l
5 be a small positive number, ~ o = {(Zl,Z2,Z 3) E
¢3] IZil <_ 1+5,
i = 1,2,3}~ for j = 1,2,3
i3j = {(=x,=2,= 3) ~ ¢3tx+5 <__ Izjl <__ e, Izj, I <__ x, I=j,, I <__ l, where (j',j")= (1,2,3)-(j)},
K
3
= ~
J
nz,
~
= ~ o
o
nz,
~=
3 U j=O
~ J
Consider a 3-dimensional real euclidean space ~ 3 w i t h
coordinates
rl,r2,r3 (see Fig. 2) and let ~3+ = [(rl,r2,r3)EIR3 ri ~ O, i=1,2,3}
16
Fig. 2
J~
r~
17
M o = [(rl,r2,r})
~ IR3+ Irlr2r3 = i, r i _< I+B, i = 1,2,7},
for j=1,2,3
Mj = ~(rl,r2,r3)
E ~ + Irlr2r~ = l, I+B < rj <_ 2, rj. <_ I, rj.. < i,
} where (j',j") = (l,2,~)-(j)}, M = ~
j=O
M .
J
I d e n t i f y U($) w i t h [g E ¢ I Igl = i ] and l e t T z = U(1)XU(1)XU(1) subgroup of TZ~
with coordinates gl,g2,g3,
defined by the equation:
and T be the
glg2g } = i.
z i = rig i where ~ i E IR, r i > O, gi ~ ~'
Writing
Igil = I, we identify
~o with M o X T , ~j with MoXTj and M with MXT. It is clear that M with the vertexes:
o
is diffeomorphic
al = ( (I+B)2 i
a} = (I+8,1+D, I ~ 2 ) .
to the triangle in ]R~
1 ,1+8), , l+D, 1+5), a 2 = (i+8,~(1+8)2
Let us take on M
the orientation,
which
(1+5) on M o coincides with the orientation (al,a3,a2)
of the vertexes
Note that the map T
corresponding
to the order
al,a2,a 3 • ~.U(1)XU(1) given by
(gl,g2,g}) ---> (gl,g2) is an isomorphism
(of groups).
Using this
isomorphism and the canonical orientation on U(1)XU(1) (corresponding to the given order of factors and to the positive rotation on U(1)) we fix an orientation of T.
It is
easy to verify that the orientation on MXT which we now obtain coincides with the orientation of MXT = ~ corresponding complex structure on M.
to the
18
Consider (see Fig.
a new copy of ~ 3 with coordinates
3) and let
= {(Pl,P2,P3)IPI+P2+P3 = 3, R o be the triangle bI
pl,P2,p 3
:
Pi ~ O, i = 1,2,3},
in ~ with the v e r t e x e s
(2, 1
1 1 1 1 ~, ~), b2 = ( ~1, 2, ~), b3 = (~, ~, 2 ) ,
for j = 1,2,3 R
J
be a polygon in h which is formed by the following
straight
four
lines :
3) P], = 2pj,,; 4)pj,, = 2pj,, (j',j")
= (1,2,3)-(j),
3 R =
~j
j=0
Let T 3 P
hl,h2,h3, Denote
R. ]
be a new copy of U ( 1 ) ~ ( 1 ) ~ ( 1 )
with coordinates
H a subgroup i n T3p ' d e f i n e d by t h e e ~ a t i o n s T P
hl=h2=h 3.
= TP~. P
Consider
P = ~p2 with h o m o g e n e o u s
chosen so that
l~iI + 1621 + I~31 = 3.
coordinates
Let for j = O,1,2,3
gj = {(~1,~2,~3) E Pl(lqL,l~21,1~31) ~ ~j} 3
F=UF. 3 j=O
~i,~2,~3
|
j J
~
• jr
•
/*
•
J/.
.
J
\ \ \
Q
\
~o
20 Take on R the orientation which on R O coincides with the orientation corresponding to the following order of vertexes: (bl,b2,b3).
The map T p
[hl,h2,h3]
~ (~,
~U(1)xU(1)
given by
~--32)is an isomorphism.
Using this isomorphism
and the canonical orientation on U(1)×U(1) we fix an of Tp.
Writing ~i = Pihi '
identify
Pi i JR, Pi > O, h i ~ ~,
orientation lhil = l,we
~
with R×T . It is easy to see that the orientation P of RXTp (corresponding to our choice of orientations of R and Tp) coincides with the orientation on R ×T
= ~ corresponding to the
P
complex structure. There exists an orientation preserving diffeomorphism ~: R --bM
with the following properties:
~(R o) = M o, ~(R I) = M 2, Let
8': T 3 P
> T3
be a homomorphism defined by the formula:
8'((hl,h2,h3))=
hI (~,
Im 8' c T.
(gl,g2,g3)
8'((gl,
g~
If
h3 h2 h2 , ~ ) .
,i) = (gl,g2,
~(R 2) = M I , ~ ( R 3) = M 3
g~g2
E T ) =
Evidently
(that is, glg2g 3 = I), then
(gl'g2'g3
see that 8' defines an isomorphism = ~x~: ~(= RXTp) orientation.
ker 8 ' = H and
).
8: T
Thus
Im 8' = T.
--->T. Taking P > M(= MXT) we can check that J preserves
We
21 Let (i,j,k) be one of the triples (1,2,3),(2,3,1),(3,1,2). Identify ~3 with ~1×~2
by the rule:
(Zl,Z2,Z3) --~ (zi,(zj,zk)).
Taking the projective closure of ~2 we embed ~3 in ~l×~p2.
Let
homogeneous coordinates in ~p2 be ~o,~j,Dk and let ~J ~k z.j - ~o' Zk = ~o"
Denote
M'i = [(Zl'Z2'Z3 ) ~ 3
l, . z1~2~3= . . 2 < . Izil < 2,5, Izjl
5 ~ = {(Zl'~2'z3) ~ ~ 3 2 < Izil < 2,5, and let "':l M~I
z. = 0
zx = o]
> KI, i be defined by g
Zi(~i(X)) = zi(x), zj(~i(x)) = O, Zk(~i(x)) = O, x 6 Mi .
~. defines on M~ the structure of a fiber bundle with the base KI, and the fibers ~[l(y) = ~(Zl,Z2,Z3) The embedding ~5
( ~31zjz k = zi~y), izjl <_ i, iZkl <--i]. ~i x ~p2
above gives us a compactification ~'i: M'i"
~Kl,i
where
which we have constructed ~ : M. i 1
~ y ~ El,i,
~K 1
~l(y)
,i
of
is a non-singular n2o
rational curve in ~p2 defined by the equation ~jU k = z i - ~ . Cji (corresp. Cki )
be a subset of M
(corresp. ~o = ~j = O), and
i
defined by ~
o
Let
= ~k = O
Tji c S-~l (corresp. Tki c Mi--) be
22
defined by
l~k[
IDjl
~ ~. l~oI (corresp.
1
~ ~--.I~oI),
Yi = {(Zl'Z2'Z3) E • 3, Izil = 2, zj = O, z k = O},
ri
:~-i(~
i ) ' ~ji ' :r
i n Tji'
' rki
:r
1 Lmn: [(~1:~2:~3 ) E P II~mI = ~I~nl,
1 m' : (l,2,3)-(m,n)), l~m.I & ~, where m
L"mn
~r~i, 5~.:~q '
i n Tki' Hji =
=
1,2,3, n = 1,2,3,
mn
Let
~:
Cl,2,3}
and
,i [(J. J)-diagonal]
~: [(JX~)-diagonal] be a i-i map defined as follows ,((1,2))
:
(2,3),
~((i,3))
:
(2,1),
~((2,3))
:
m((2,l))
= (1,3),
m((3,1))
:
(3,l),
~((3,2))
= (3,2).
It is easy to see that
~i T . (corresp. 3z
(1,2),
~i Tk2 defines on
Tji (corresp. Tki ) the structure of a 2-disks fiber bundle over Cji (corresp. Cki).
Thus we can consider Tji (corresp. Tki) with this
projection as a tubular neighborhood of Cji (corresp. Cki) in Mi" Because Hji,Hki c M we can construct a new space
~=~u
H21 r21
%31r]l uHI3 r13 %23 :~3 %32:32 un12 r:x2
A direct verification shows that ~(Lmn) = H (m,n). diffeomorphisms
~mn: L'mn - - ~ H (m,n)
where
We obtain
~mn = ~ L'mn
23 Note that
L'mn (corresp. Lmn ) is a circle (corresp. 2-disk)
fibre bundle over a circle in P defined
by ~m' = O, I~mt = ~l~nl.
In the notations used above a fibre of the fibration L
mn
is given
by: h __mm h = b = const E U(1) " n Because
[~m[ = ~[~n [ "
l~m, [ = ~,
~(hl'h2'h3) = (h~" h3h 2' ~21) we have
~({~
= b})
=
{g3
h h3 --= {h2
=
b}
= 1
=
(g2 = b]
h2 e(~-33 = b}) =
i Cg e = ~}
h1 ~( --= [h 3
[g
8(
hi)
b})
=
h ~({h~_ = b}) = 1 Considering Hji,Hki as circle-bundles
= b]
1 i (gl = b]"
over Yi (with projection ~i)
we see that the fibers of Hji,Hki are given by the condition gi = const.
Hence
~mn
is a diffeomorphism
of circle-bundles
24 Lmn and H (m,n). corresponding
Thus we can extend ~mn to a diffeomorphism
2-disk bundles
~mn: L mn
of
> ~ ( m,n)" which
transforms the centers of fibers to the centers of fibers. ---I ~ mn
Considering
and
X:M
~-i ~ together we obtain an embedding
>p
.
Let Sm' =
{(~o:~1:~2 ) ~ P ~m' = O,
L'mn =
C(~I:~ 2 ~3) e p,
l~ml =
:
~ b m' = L
mn
1
l~m,
- ~' ~ n
U L' U L (union zn P) mn nm
We can consider ~ ~m' of Sm, in P.
I~nl] ,
m'
-< ~m -< 2~n]'
e
as the boundary of a tubular neighborhood
A fiber of the corresponding
is defined as follows: h I f b = f f - m (we can identify
b
2-sphere fibration
with a point of Sm, ) then the
n
corresponding of L
mn
fiber F b is equal to the union of the fiber Lmn(b )
over b, the s u b s ~
I~m,] - ~,
of P, defined by
~m
arg 7-- ~ arg b,
~n 1 ~l~nl &
I~I
~ 21~nl,
and the fiber Lnm(b ) of Lnm
over b. Now let (re,n)
be one of the pairs (1,3),(2,1),(3,2).
Take on Sm, the orientation corresponding
to the positive rotation
25 h m of ~--. Then the orientation on ~ m' = ~ m ' defined by the n complex structure on ~m' induces on every Fb an orientation which on Lmn(b ) is opposite to the orientation corresponding to the complex structure of the complex projective line ~m = ~ n " But that means that the orientation on ~ m , , ~(P-~m,), defines on F b
considered as
an orientation which coincides on
Lmn(b ) with the orientation given by the complex structure of the line {~m = ~ b~n]' This "complex" orientation on Lmn(b) defines on L ~ ( b ) = ~Lmn(b ) an orientation which corresponds to the positive rotation of Since
h
n
~((hlJh2'h3)) = ( ~ ' ~h~2' ~i) m
we n have h that
defines on
~13(L13(b)) (corresp. ~21(L21(b)). corresp. ~32(L~2(b))) l the same orientation as the positive rotation of g2 (corresp. 1 corresp. I___) does. gl' g3 Let
,: ~
> ~ be given by
A direct verification shows that ki = kl~i: ~i
> ~%(i)"
,(1) = 2, ,(2) = l, '(3) = 3. k(~i) = ~ , ( i ) "
Let us fix on ~
which is given by considering ~ m ' structure of P-~m'" of
~i
> 7i
Let
m' the orientation
as ~(P-~m. ) and by the complex
We see now that
ki
induces on the fibers
an orientation which coincides with the
orientation given by the complex structure of these fibers. can check also that if
o
ki: 7i
> s,(i)
We
is the diffeomorphism
26 of bases
corresponding
the positive in ~3).
orientation
Take on
~. l
and by the complex orientation by
structure
3
~: ~
~ k: Z
FIj) u (U
j=0
We obtain that k. is an l 3 3 Define k': U ~. --9 ~ - / ~ 3 i=l i m'=l m'
UI,[P- W = l ~ m ,
where
].
Using the above we
~ ~(1) Z = Z n ,
l/~),
J=1
F~ = I:(z1,=2,= 3) ~ ~312 <_ Izjl <__2,5, (j',j")
on Yi
in the zi-axis
' ~ p which we constructed ~ ~ P,
3
induces
which is given by ~. c ~M. l i
of M.-~M.. l l
P = ( ~ Mi) i=l
k~ l
Yi as a circle
diffeomorphism. 3
~ Mi'
obtain an embedding
9 (l) = (u
(we consider
and let
Mi
to k i then
the orientation
reversing
i' H i = li
embeddings
canonically
=
Izj, I <_ l,
Izj,,I <__l,
(1,2,3)-(j)}.
Note that we can consider 9 (1) as a part of the regular neighborhood
~ of S in W and /k of the subcomplex C (1) in P which 3 ( ~ (C~iUCki)) i=l J
where
Em' =
[(~i:~2:~3)
as a regular
neighborhood
is equal to
3 U ( ~ (Em,-~m,NEm,)), m'=l
~ P ~m' = O},
(j,k) = (1,2,3)-(i).
in P
27 Now we can do the same constructions = 2,3,''',v, as we did for B 1. = 1,2,..-,~
a set
for each B~,
We obtain then for each
ZZ c W, a manifold with boundary P~,
annuluses K~,l, K~,2, K~, 3 on S, fibrations i = 1,2,3, cross-sections
M~,i --->KL,i,
C~,ji , C~,ki , (j,k) = (1,2,3)-(i),
of
these fibrations, tubular neighborhoods T~,ji , T~,jk of these cross-sections and an embedding ~ :
Z~
~ P~
which have the
same properties as /%
A
Z, P, KI,I, K1,2, K1,3, Mi
> KI, i, Cji, Cki, Tji, Tjk,
constructed above. We can define also T(~)C (~) c p~, Z = 2,3,...,~, way as we defined
T(1),C (I) and assume that
in the same
~ ~ = 1,2,..., ~
T (~) is a part of the regular neighborhood ~ of S in W and consider p~-k~(Z~) as a regular neighborhood of C (~) in P~. Consider now the ball B1.
We can assume that there are
complex coordinates Zl,Z2,Z 3 in B 1 such that f B{ is defined in !
B 1 by the equation f = z~ - z3z ~. (z~(1)) 2 B{.
-
Since
z~(1)(z~(1)) 2
=
0
[Recall that
is a local equation of V O
D
f-l(o) has no multiple components f
f2 = (z~C1))2_z~(1)(zi(1))2
is an invertible holomorphic
B 1' in
28 function in some neighborhood of YI" zI = ~
zi(1)
smaller].
z2 = ~
J
z~ (I) •
We take !
z 3 = z3(1)
and choose
Now l e t ~ =
and
we can identify B 1
fiB 1
with the function
~(Zl,Z2,Z3) E G3llZll < 2, lz21 < 2,
Z = {(Zl,Z2,Z3) ~ ~3 z2-z3z 2 2I : i},
--
~O
where ~O,WI,~2
1
Iz31 <
7},
T = Z A ~.
Identifying ~3 with ~2x~l we embed ~3 in ~p2x~I. ~I n2 Zl = ~o' z2 - --
and !
Changing if necessary Zl,Z2,z3•t
with [(Zl,Z3,Z3) ~ ~3 l~iizil2 < 3} ,= 2 2 z2-z3z I •
i
B1
We take
are homogeneous coordinates in
~
~p2, and consider z 3 as a coordinate in ~i.
Let
O
C( o,
Cz =
{(~0:
z3) '
hi: n2; z3)
2
1
~ ~ ~o = o].
Note that D1 ~ 0 on C Z . Hence we can define C z also by the O
equation ~i = O.
It follows from this that the set
~C = C(~O: ~l: ~2; z 3) 6 Z IUol ~ ~IDI I]
is a tubular
neighborhood of C Z in Z. It is clear that ~C n ~ c ~ _ T. 1 have l~o(X) l > ~l~l(X) I,Iz3(x)I ~ 3"
Take x ~ T - ~C N T. Hence ~o(X) ~ O,
IZl(X) l < 2, Iz2(x) I <~1+Iz3(x)(zl(x))21 < V~ < 2.
We
29
We obtain x E T.
~C N T' = T - T.
Thus
Let
Cz3 ~cr J
K=
~K = {x E EC z3(x) ~K: ~K ----~K
<__~],
<__ I z31
E K],
is defined by projection
(nO: 71: n2; z 3) Take z 3 E K and consider lnol < llnll,
> (z3)-
o~21 = no, 2 We have ~2 II2-Z3H
3 ~Kl(z ).
it is clear that 7 1 %
O, so denoting
Uo 72 u =--nl , v = ~71, we obtain
2 o 2 1 v - z 3 = u ; m~l _< ~.
v =
I >_ ~
<_ ~; o
nKl(z3 ) is an unramified of two 2-disks d+,d_
map.
From
we see that the projection Hence
-i
o
~K (z 3) is a disjoint union
where
d+ = C(u,v)
~ ~2
d_ = C ( u , v )
~
¢2
tul <__ ½, v = lul <__ ½, v = -
2~]
It is clear that these 2-disks are transversal we can consider
them as fibers of the tubular
of C z corresponding
to the points: o
and
of
+g
z 3 = z3,
~ = O,
V =
z3 = z~,
~ = O,
V = -z ~ .
. to C z and
neighborhoods
C
3O (Here
+f
and
Taking
-~
mean simply some choice of branches
for each of the constructed
is, its intersection C K = C Z ~ K.
2-disks
P: ~C ---->Cz is defined
so that
Note that the projection has the following
property:
P ~K
PK"
~
>~i
~:
~l(z3)
if z3 ~ 0 and a pair of transversal
non-singular
projection
~.: ~pl
b ~i
T c T by
in ~ which
k': T
as we did for B I.
and am (
rational
of
~pl
curve
curves
as composition
of
x ~i with
= ~ G mm
~ : ~ m m l(ym)),
in ~ of a
is defined by the equation 7
O
= O.
for each B'm' m = 2, p,
We obtain then for each m = 1,2,-..,p
2-disk d m on S with center m
rational
~T. W e see from
neighborhood
N o w we can do the same constructions
~
~
and blowing-up
~C ~ ~ = T-T that T - T is a regular
with
C K where
a E ~'-i(o).
Denote the embedding
subcomplex
(that
projection
is a non-singular
This means that we can consider x ~i
>
/--).
( ~ ( ( 7 0 : 71 : 72; z 3 ) ~ )
if z 3 = 0.
some center
its center
with CZ) we obtain a map PK: ~K
We can assume that the canonical
of
(where
Ym" a manifold
~m: ~p1Xd
m
> d
> ~plxd m is the G-process a non-singular
complex
m
Tm' a map
an
open
Wm: ~m
is the canonical
>dm projection
with some center
curve C m c ~ m such that
31
~m - -
: C Cm
> d
m
unique branch
is a ramified
m
point over Ym' a tubular
of C m in Tm' an embedding Tm
= ~m ~' m
covering
where
T
~' : T m m
of degree two with
neighborhood
> T--m with
Pm: ~m
~ ' (Tm) = T m - ~m'
= ~ n B' A Vl, and an annulus m
m
> Cm
K
m
c d
m
with
center Ym such that
~m ~ l ( K m ) ~
Using Lemma Vt
t E A'
=
~m
" (Pro ~m-l(Km)D~m)
1 and the assumption
we can identify
can consider Denote
m
h-l(Vo ~ )
TC = h-l(Vo ~ )
that ~ with
V;-VI~
as a regular
is transveral %-h-l(Vo~
neighborhood
and let the corresponding
to
).
We
of C in % . projection
be
• : TC --->C. Consider
the annulus
W e can assume that
h-l(K~,i ) on ~,
x~£i is the center
of
~ = 1,2,...,~,
hs I (K~, i ).
Let
i =1,2,3. ~dio be
the closed 2-disk on ~ with center x~ i such that ~dio N h; 1 (K~,i ~ ";o . Denote center
dio = hs(dio ) . Ym such that
S " = S - [( ~J d.
~=i 10
7" = 7 TC": TC" of C" in % ,
Let
be the closed 2-disk on ~
dmo N K m = ~d mo .
) U (m=~ldmo ] > C".
~: ~
dmo
Let
"
'')
q~ = qlN~
: N
'
>C"
> C" be the projective
We can assume that TC" is a subspace Let N ~ = N-TC",
Denote
C" = h - l ( s q: N
~
TC" = T - I ( c " )
"
be the normal bundle closure
of N (C
'>C",
with
of q: N ---> C".
and ~" = q : N
.. = ~ S"
32
C ~ be the cross-section at infinity of q: N C-=
>C", that is,
A N-N. Since V 1 is transversal to ~T, ~IvI~-I(s,,):VIN~-I(s")--~S"
is a differential fibre bundle. assume that
(Changing if necessary T we can
~ ~--l(s,,): ~-l(s")
>S"
is a ~-disk bundle).
Using
the same arguments as in ([lO],Lemma 2.1) we obtain that the typical fiber of
~IVlN~-l(s,, ) ,
is diffeomorphic to slxI (I=~xE]R lO<x<_l).
Using our identification of embedding
TC" c N
Vl-VlDT~
> ~N~.
We can assume that
71 ~(Vln~'l(s $I)) = (~I~N~)'~~'I ~ : A
--% - h - l ( V o ~ ~ and the
we obtain a diffeomorphism
~--: ~(V 1 N ~-I(s"))
and
with
~
let A~ = [Vln~-l(s")]u~~ ~ ,
> S" be defined by ~(x) if x E V 1 n ~-I(s" )
-=(x) = Because ~ :
N~
~~(x) if x ~ N ~
> S" is a fibre bundle with typical fiber
diffeomorphic to the disjoint union of two closed 2-disks, we see that ~ : A
~
is an S2-bundle.
It is easy to verify that we can do all our identifications and constructions so that for any (L,i), ~ = 1,2,.-.,9, i = 1,2,3, I L M~,i K~,i-~dio
> K
~,i - ~dio'
~ = 1,2,..., ~, i = 1,2,3,
33 will coincide with
) : (--)-I C ,ji n C~,ki
~ i- dio)-K
i_Sd~o '
(where (J,k) = (1,2,3)-(i)) will coincide with
(~)-l(K~,i-~d~o)
n C~
and for any m = 1,2,..-,p
~m I ~ i (Km_dm O ) : ~m-I(Km-dmo ) --~ Km-dmo will coincide with ~
--
: (~)-l(Km-dmo)
~ Km-dmo ,
and
I ( ~ ) -i (Km_dmo )
C m N ~ i ( Km-dmo ) will coincide with
C -- N ( ~---1(Ks_ ) dmo) •
A .. "'T p where we identify We define A as union of A ~ , % ' "'''Pg'TI' a point x E ("=)-I(K,, i -~d~io )(c A ~) J ~ = 1,2,'--,~, i = l,e,3, with the corresponding point
x' E M~, i K~,i -~d'~lO(c p~) and a point
Y ((~)-l(Km-dm~(~)
with the corresponding point y'6~ml(Km-%o)(~). U p Let ~ = C ~ U [(~=l ~ C ( ~ )) U (m=~lCm] (union in ~)~
^ TC = N ~ U [ ( ~ i ( ~ - ~ , ( ~ ) )
U (
~m) ]
(union in ~).
m=l Because each
M~, i
>K~,i,
~ = 1,2,.--,~, i = 1,2,3, is a
trivial S2-bundle and C ,ji,C~,ki are "horizontal" cross-sections in M~, -- i
• >K~,i, we can extend
n~ : A ~
~ S" to an S2-bundle
P
~:
A~
> ~ - ~/ d m=l mo
and
C-- to a 2-manifold C ~ in A ~
for any (~,i), * = 1,2,...,~, i = 1,2,3,
where
C = N (~)-l(~i~o) is
34
equal to disjoint
union of two cross-sections
Using t h e c a n o n i c a l m
: ~
>d
m
embedding o f
L-I(Km )
we extend A ~ to a differential
m
(A'-A = =
~ml(dmo)), m=l
of ~
the map ~
over d. . lo
> Km i n manifold
to a differential
(dmo) and C ~ to a 2-manifold The construction where
~: A
~
of ~
L' in A' : ~
m
m
--~ d
m
ic: ~
~A
canonical
map corresponding
dmo)
shows that we can write
such that to
map
(L'NA~ = C ~, L ' N ~ l ( % o ) = C m ~ m
is an S2-bundle,
embedding
A'
%~.
~' = ~.~,
there exists a differential
~Oic: ~
> ~
hlc: C --~S,
coincides with
~: A'
~A
is the
l
P
monoidal
transformation
qm = ~-l(hsl(ym)) of
of A with center
N ic(~ )
and
L'
equal to
~ q_, where m = 1Ill
is equal to the strict
image
ic(~ ) in A' It is easy to verify that we can construct A, C and TC
from A' and L' by the same way as in the statements of our theorem. properties
Now the existence
formulated
of 7: ~TC
in the statements
diffeomorphisms
ut: % - T C
of the theorem)
easily
U~ (A-TC)
follows
c),d),e) /% b ~TC with the
e) and f) and of ~ V t (see the statement
from our constructions.
that above we identified V I - V I ~ w i t h __~-h-l(Vo ~ ) v/k p V 1 N ~ = [V 1 a ~-I(s")] U ( k/Z,)~=l " U (m=~iTm)).
g)
(Recall
and that Q.E.D.
§2.
A topological comparison theorem systems on complex threefolds.
Lemma 2.
Let
~: X --~Y
Yg be a closed oriented embeddings
for elements
be an oriented
g
surface
of linear
differential
S2-bundle,
of genus g, 71,72 be two smooth
of S 1 in X such that Yl N 72 = ~, 7i is a cross-section
of ~ over n(7i)
and
transversally at one 2 point of Yg. Let d be a closed 2-disk in Y - ~ ~(Yi ) and X be g i=l obtained from X by the surgeries along 71 and 72 . Let Z be the image of ~-l(d)
in ~.
n: X
-> Y
(a)
-(Y1),~(Y2 )
Then there exist
and a 2-disk ~ c y
g-i
~: X~
~Yg-1
intersect
g-i
a differential
map
such that
is a differentia~le
S2
-bundle,
-i (b)
Z = ~
(~)
Z ---> d
and
~J~-l(~):
corresponding
to
~-i(~)__>~
coincide with
~ - l ( d ) : ~-l(d)
> d.
of the S2-bundle over S 1 and 2 considering a tubular neighborhood of ~ ~(7i) in Y we see that i=l g only the case which we have to look at is the case g = I. Thus Proof.
assume g = i.
Using the triviality
Let p be the center
of d, S 2 = ~ p ) P !
obtained
from X by the surgery along S 2p"
Let
and X' be !
Yl,Y 2 be the images I
of 71,72 in X' and X' be obtained Using uniqueness
of tubular
from X' by surgeries
neighborhood
prove only that X' is diffeomorphic
along
(of S 1 in S 4) we have to
to S 4.
!
71,72 .
86 First of all we shall prove that there exists a diffeomorphism ~: X'
> (S1XS3)l~ (SIxs3)2
such that for i = 1,2
(connected
~(yi ) = (slxai)i c (slxs3)i J al,a 2 E S 3.
o Y'l = n(Yi )' i = 1,2,
Let
sum of copies of SIxs 5)
o o q = Y1 A Y2
and
B
be a small
There exist differential
closed 2-disk in YI- p with center q. embeddings: ~i: I X I such that
~l(iXi)
i = 1,2
> YI- p - int B, n ~2(ixI) = ~,
~i(zx~z) c ~B,
~i(ixI) = Yi-Yi~B,
i = 1,2 (see Fig.
4).
Denote A i = ~i(I×I), A = A I U A 2 U B. We can assume that n-l(A) structures
of
sirs 2 on ~-I(~A)
c X',
Let
T: X'-n-l(A)
to the equality
[~-Z(*(Zx=I)) U X
= SIxD ~.
be a projection
from
corresponding
We have
•-i (,i(~z~)] x z ~~[(Ixs 2 )UsOxs2(sOxD3 )]xz l
Z(union in X ) ~$3
It follows
and the
= ~(SIxD~)( = slxs 2) are the
> ~A
X'-n-l(A)
= SIxD 3
obtained correspondingly
~-l(~A) --->~A and from ~(X'-~-I(A)) same.
X'-~-I(A)
from this that ~T-l(~i(IXI))
is diffeomorphic
to S3XI.
Xl.
U $-l(~i(~IXI))
(union in X')
x
f~
II
38 Let X" be obtained from X' by the surgeries along u
s3), i = 1,2.
We have 2
,
4 u
X" = X' - ~ [~-I(Ii(IXl))UT-I(Ii(~TXI))]U j D I O i= 1 iu
4
--4
j, D l l U j ~ f 2 0 "~ zu
u
4
~21D21
where D4..~], i = 1,2, j = O_~i~. are four copies of the 4-dimensional ball D 4, and ~ij: ~(D4ij ) - - ~ ( X ' - U [~-l($i(Ixl))uT-l(li(~Ixl))]) i=l smooth embeddings such that
are
8ij(~(D4ij)) = [~-l(ii(IXj) ) U T-l(li(~TXj))]. We can write X" = X'i U X'2 where Xl = ~l I(B) , 'l
2
U~l[D311xT]
X2 = [X'-~-I(A)-U T-l($i(~IXI))] i=l
U820[D~oXI ]
D 3ij' i = 1,2, j = O,1, are four copies of D I,
B'
3 x~i)
ij : (Dij
2
> ~[x'-~-l(A) - t.J ~-l(*i(~I×I) )] i=l
are smooth embeddings such that 8'ij(D ~jXk) = T - lli(k,j ), and 7: ~X{
i = 1,2,
> ~X 2 is a diffeomorphism
j = O,I,
such that
k = O,1,
39
~1
2
2
(~B-i__Ul*i(IX~I)) = ~(X'-~-l(A)
-
u'-l(*i(~IxI), i=l
2
i#l
U *i(I×al) ~ l~-l(;B-•= ,i(ix;i) : ~-I(~B- i=l
>
2
--~(x'
- ~-l(a) - U ~-l(*i(~I×I)) i=l
is the identity map,
D(n-l(~i(IXj)) and
V
i = 1,2,
= ~D~ijXI ,
j = O,1,
t E I
n(--l(~i(txj)) = ~D~j × t. We see that we can consider X'~ as D2×S 2, X'2 as sl×D 3 and ~ as natural identification diffeomorphic
of ~(D2×S 2) and ~(SI×D3).
Thus X" is
to S 4 .
Our construction of X " shows that we can get X' from S 4 performing
surgeries along two embeddings of S ° in S 4 (see
Fig. ~ ), say ~ I O Ii ) and (b2ob2 ~ , such that 71' and ¥2' are obtained as follows Let bij,cij
(Fig. 5). 4 D.., i = 1,2, lj
4 be points in ~(D..),
with ci°
in S 4 Cil'
paths in S~ ×I 1 connect
j = O,i, be small balls with the centers
-
5. be smooth disjoint paths connecting
4 ~z [ )D.., i=l j~=O 13
4 = S ~ × j, ~D.. lJ i
which are cross-sections
Cio with
Cil in
S~1 xI.
Then
of
£. be smooth i
ST xI i
Y1t = DiUel,
> I
and
Y2i = 52ue2"
40
Fig. 5
I< ~z
~o
2O
41 We
see from this that there exists a d i f f e o m o r p h i s m
~: X'
> (SI×s~)I ~ (SIxs~)2
~(~)=
(slxai)i c (SIxs~)i ,
such that i = 1,2,
From SI×s } = ( S I x s ~ ) U S I x s 2 ( S I x s ~ ) on SIrs ~ along slxa,
al,a 2 E S 3.
it follows that
a 6 S ~, transforms
gives that surgery on X' along
SirS } in S 4.
a
surgery This immediately
Y1' and ¥2' transforms X' into a connected
sum of two copies of S 4, that is, into S 4.
Q.E.D.
T h e o r e m 2.
[E]
Let W be a 3 - d i m e n s i c n a l
be a complex
(analytic)
compact complex manifold,
line bundle on W, ~O and ~i be two global
(holomorphic)
cross-sections
(~i) ° of
is a complex submanifold V 1 of W, the singular
~i
of the zero-divisor
(~o) ° = V o of
locus S of all ordinary complex curve. following system
of [E].
~o
singularities
Suppose that the zero-divisor
is canonical and the of V ° is an irreducible
Suppose also that V 1 is transversal
sense:
V
(Zl,Z2,Z3)
hood U x of x in W,
locus
to V ° in the
x E V 1 N V ° there exists a local complex coordinate
on W
with the center x such that in some neighbor-
V 1 is defined by the equation z} = O and V ° is
defined either by the equation z I = O or by the equation ZlZ 2 = O. Let
h: ~
b V°
be a minimal
the number of pinch-points
desingularization
of V o.
Denote by p
of Vo, by 9 the number of triplanar
of Vo, by b the i n t e r s e c t i o n
number SoV 1 (in W) and by g(S) the
genus of S (that is, the genus of a n o n - s i n g u l a r
model of S).
points
42
Suppose that p ~ O and that nl(Vl) = ]TI(~) = O. i)
if b ~ O, then V 1 ~ P is diffeomorphic # (2~+p+2g(S)-l)P ~ (~+2p+b+2g(S)-2)Q
2)
if b = O then V 1 ~ P ~ Q
Then
to and
is diffeomorphic
to
# (2~+p+2g(S)-l)P # (~÷2p÷2g(S)-l)O Proof. ~
: ~
~W
Let B = V ° N Vl .
We construct a modification
(following Hironaka's idea) as follows.
If B = ~ then ~ = W and
~
is the identity map.
B N S = ~
then ~
center B.
Now let B N S ~ ~ and
If B ~ ~ and
is the monoidal transformation of W with the B N S = [Xl,X2,...,Xb].
We can
take a coordinate neighborhood U k of W with the center in Xk, k . .1,2, . . V 1 N Uk
,b, with coordinates z 3(k) = O and
is given by
zl(k)'z2(k) = O (see Fig. 6).
Let
in U k given by the equations
~ Ukl
center ~ k ) w h e r e 3 " : ~"
~k)
B (k)i , i = 1,2, be complex curves kl: Ukl --->Uk
of U k with center B~ k), of Ukl with
B~ k) in Ukl and b be the monoidal transformations of W" = W-k__~ixk b
b W'
is the strict image of
Evidently~22k!(~ik~k)_l(Uk_Xk
coincide with
~ • I~"-l(Uk-X k)
to
}W"
~"
is given by
be the monoidal transformations
with the center B - k=~iXk .
~':
V o N Uk
such that
zl!k) = O, z~ k ) = O, ~
be the monoidal transformation ~k2:Uk2
z ~k) ,z ~ k ) ~•zk )
)
Using this identification we add
the disjoint sum
~(~ik%k:
U2k ~ U k )
and
48
~ Vl
ig.
6
~2
B N
S-._/7
Vo
B
44
obtain
~:
~
Denote complex
>W. ~ = ~-I(B).
line bundle
on local equations cross-sections
codim~
on ~ defined by ~, of B we get from
~o,~ 1
i = 1,2 ("strict
Clearly
of
images"
[El.
[~] =
~
Let
= 1 so let [~] be a ~W[E]-[B].
~o" ~ W ~ l ~. = l
~
Dividing
some global
-I(v.-V.NB) l l
of Vo,VI) , ~ = ~ - I ( s - S N B ) .
'
Clearly
(~i)o = ~i' i = 1 , 2 It is easy to verify isomorphic
(see Fig.
6) that L ~ I
to V 1 (and then diffeomorphic)
singularities
only ordinary
[ is the locus of ordinary S (that is, g(~) = g(S))
singularities singularities
and that ~o
has as
and rational
double
of L ,
p and ~ are for ~ •
and if o
then
h: V ~
}~
o
diffeomorphic
to
~ is isomorphic
the same as for V
resolution
O
of singularities
from ~ by b ~-processes
(that is, ~
of
~:
~ ---~W
rational
(b) S 1 intersects
is
double-points, with
h-l(s)
Thus we assume B = ~. with no indeterminacy use the same notations
triplanar
in a single
points and we can apply Theorem
i.
of Theorem
does not
of V o and
point.
Now f = ~o/~i is a meromorphic
as in the formulation
curve of
(a) h(Sl)
and pinch-points
transversally
for
also what kind
we can do in the case when an exceptional
the first kind S 1 on ~ exists with the properties: contain
of
shows that it is enough
only the case B = ~ and to understand
of modifications
to
V / bQ).
The construction us to consider
points,
O
is a minimal
is obtained
= ~, ~i is
function We shall
i.
Using
45 part f) of Theorem 1 we can find a non-singular closed 2-disk d
P
in U such that if ~
point p E U, small
denotes ~o(dp~.. then we have '
P
a commutative diagram:
t-l(dp)li-l(dp)> ~o dp d
P
P
Identify now V 1 with -TC
and
A-TC
E -TC UD (A-TC) (that is, we shall consider
as subspaces of V 1 and D as the identity map).
can construct two embeddings ~:
X[O,I])xS 1 ~
A-TC
9((dpX[-1,O])XS1)D()TC
b V -TC
and
such that
= ~t(dpXOXS 1) = _'1"-l(dp) =q-l( "~_ dp) =
~ (dp~OXS I ) = A~/((dpX[O,1])×S *x A.A. V
~: (dpX[-l,O])XS 1
I)
[')
~TC
and A ~
x £ dp, x' ~ dp, y 6 S I
'r~'(xxoxy)
= x,
'r~(x'xo×y)
= x',
~'(xxOXy) = ~'(~o(~)XOXy). Let
Cp = ~do, %
S 2=p_ (dp×(-l))
= ~d%,
U (Cp×[-l,O])(union d
Sp+ = (CpX[O,l]) C XO p
=
Define
~S~
S 2p
-" =
~c XO p
U ( =
X(1)) (union in dp×[O,l]). ~S 2 p+
S 2p- U(r/o Ic l
and
((7oi
c
)× id)(CpXO) P
xid).S 2p+ and let
P
P
in
X[-1,O]),
We have =
C *~ p
X O.
Yp' = (~(dpX[-l,O]XS I),
We
46 Y" = ~( d~ riO,l] xs 1 ), P P tubular neighborhood ~-manifold obtained
= Y' U Y". P P
P
in V 1 of
=
Clearly we can consider Y
~(p×OXS I) = ~-l(p).
from V 1 by surgery along e: S 2 X S I P
We have a map
e(x,y)
Y
>~Y
Let
~
P
as a
be a
T-l(p).
defined as follows:
P
?(x,y)
if
X 6 S 2p+
*(x,y)
if
x ~ S2 . p-
and
We can decompose ~ as follows:
Now
= X1 U X2
where
X 1 = (S 2_ x D2)U e S2 xsl(~o-TC-Y'p), px2
We can construct ~": A-TC-Y
P
=
($2+ p
x
D2)Ue S2p+XSl(~'
"diffeomorphisms"
~A-TC
such that
-
Tc
V -TC,
~': V -TC-Y' P ~' (corresp. ~")
identity outside of some small neighborhood
2 1 )(corresp. ~(Sp_XS
A. - y.p ) .
i s the
of
z.$(Sp+XS 2 1) , _~ , ( ~ ( S ~ _ × S 1 ) )
= ,(dpXO×S 1)
(corresp.
~"(~(S~+XS I) = ~ ( ~ ×orS1)) and there exist P 2 2 /% "diffeomorphisms" _~': Sp_ > dp, _~": Sp÷ > dp V x E S2 S2 SI p-' x' ~ p+' y E ®'()(x,y))
NOW let
=
C•
"We can write
) ( _~ ' ( x ) x 0 x y ) =
-C-dp, -
?"
,
such that
we have w ( ,^ ( x ' , y )
--~ Tc" = ^C-dp,
~ -TC" = (dpXD 2) % , ( ~
=
=
A~ . ,(= '-, ,(x , , x o x y , .
~
~ - l ' c " { ~ , ~^'c
-TC),
=
~' r- 1 . ^{ c ' ) .
~A-TC" 14 = (% .XD 2,) %
^ ~ (A-TC),
47 where ~': d p X s I
~ -T-l(dp),
are some trivializations of Let
e':
p-
XS 1
~d X s I p
~":
- i ( dp )
(~p) > ~-I -and
> dp
> dP .
> ~(d ×oxsl)(corresp. e": S +XSI-~;(~XOXSI)) P P
be equal to ('' ~(S~.XSI) )'(e S2p XS l)(cOrresp"
(~"I~S2p~XSI))°(elS2p÷XS I))"
We have diffeomorphisms X1
> (S~_×D2)Ue,(~
-TC),
X2
~ (S~+ xD )Ue.(A-TC),
and because of the commutativity of the diagrams, S 2 XS I p-
e' ~ #(d XOXS I) p
pr[ S2 p-
S2p+XS 1
I~_ ~' --
~ d
and
e"
> ~ ( 2 xOXS I) P
pr i
[~
S2 P+
p
~'
>
dA
P
A
we can identify X 1 with ~ --TC" and
X2
with
A-TC'.
We see that
A
there exists a diffeomorphism diffeomorphic to
7": ~TC °
7
~TC"
such that
~
~--TC" U~. A-TC:
small neighborhoods of
is
We see also that outside of some A-I. A . Tjl(dp) and ~ ~dp} 7" coincides with 7.
In particular, we have that ~" reverses orientations coming from
48
orientations
on TC" and T C ' d e f i n e d by complex
structures
of
and A'. Suppose that an exceptional the properties
curve of the
mentioned above exists on ~
be the corresponding can assume that
contraction.
p = h-l(s)
Without
.
and
Let ~: ~
~V
loss of g e n e r a l i t y we
~ S 1 (= C N Sl)
Let C = ~(C), C" = ~(C')
first kind S 1 with
and S 1 N TC = - l ( p ) .
TC" = ~(TC').
W e see that
S 1 n TC" = ~, that is, we can identify TC" w i t h T~" and construct S1
(c ~ -TC'(C ~))
contraction,
in ~.
we see that
Let
~: ~
~
>~
is diffeomorphic
Now let us make the following remark. connected
4-manifold,
i: D3XS 1
d A i(~D~XSI))
= ~d = i(a×sl),
= (S2xD2)UiI~(D3XsI)M-i(D}XSI)
smooth 2-sphere last c o n d i t i o n and if
in M
~N: ~
diffeomorphic Applying
} ~ to M ~
Let M be a simply-
and if
then N = (aXD2)Uilaxsld.
equal to -i.
follows that M is d i f f e o m o r p h i c
is the c o n t r a c t i o n
Suppose
in M-i(D}XS I) such
a ~ ~D 3 = S 2
with s e l f - i n t e r s e c t i o n
it easily
to (Z -T~')U~.(A-TC').
b M be a smooth embedding.
that there exists a smooth 2-disk d embedded that
be the corresponding
is a
From the to M ~ P ~ Q
of N to a point then M is
P.
this remark to the case M = Vl, N = S 1 we see that --
is diffeomorphic
to
V 1 ~ P.
We use now the notations ~ and ~
and ~ in the case when there exists S 1 with the properties
for
49
and C in the case w h e n such S 1 does not
formulated above and for ~ exist.
Our aim now w i l l be the prove the following
Statement
(*):
(~-T~')U~(A-TC')
is diffeomorphic
to
~ (2~+p+2g(S)-l)P # (~÷2p+2g(S)-l)Q. Before proving the Statement follows
(*) let us show how our T h e o r e m
from it.
Case I). b ~ O. is diffeomorphic
W e have ~ = Z
to
V
~
and Statement
(2~+p+2g(S)-l)P
(~) says that
~ (~+2p+2g(S)-l)Q.
k
Going b a c k to our a r g u m e n t s w i t h
~
: ~
of proof of the theorem) we see that ~ initial notations to
~
~
(b-l)Q.
>W
(in the b e g i n n i n g
is diffeomorphic
Because
in our
~ i is d i f f e o m o r p h i c
to V 1 ~ P we get a d i f f e o m o r p h i s m
vI / p ~ ~
# (2~+~+2g(s)-2)p # (~+2p+b+2g(s)-2)Q.
Case 2). b = O.
Let S 2 x S 2
= ~ , ~ = V 1 ~ $2 xx S 2
v I # s 2 x s2 ~ ~
be S 2 x S 2
and Statement
by
~,~
means
p ~ 2.
> S.
We have
# (2~+p+2g(S)-l)p ~ (~2p+2g(S)-l)Q.
are n o r m a l i z a t i o n s
hlc: C
P ~ Q.
(~) says that
Note that p is the number of b r a n c h points where
or
Hence
for C and S and ~
p is even and the condition
We see that
has odd intersection
for the map ~
form.
~+2p+2g(S)-2 Suppose
> O.
S2 x S2
>~
• > is induced
p ~ O really
Hence, V 1 ~ $2 ~X S 2 is a c t u a l l y S 2 X S 2.
50 T h e n if V 1 has odd i n t e r s e c t i o n (see [8]) that intersection
V 1 ~ S2XS 2 ~ V 1 # P ~f Q
form then
w h i c h contradicts write here
V 1 # $2~
on S'2 w i t h i
n ~
j
(~). ~ di, C' = I?'-I(s')DL', i=l We choose
, w h e r e ql'" ' ..,q~
the following
= ~ and
transversally
smooth circles ~i'
properties:
(2) if 1 < i < g ---
such that the
~" = 7' e': C'--~S', i = 1,2,..., 2g,
are a l l the b r a n c h points of (i) if
then ~
and in a single point.
make our choice
and we get
/ (2~+p+2g(S)-~)P # (v+2p+2g(s)-1)o.
Suppose g > O.
on S' - ~ l ~
form
W e see that always we can
S2 ~ V 1 ~ P # Q
Proof of S t a t e m e n t Let S' = ~ - ~ ~=l
and if V 1 has even
V 1 ~ S2xS 2 has also even i n t e r s e c t i o n
our remark above.
vI # P # Q = ~
g = g(S).
form we have by a result of W a l l
~"
lj-il ~ g,o then
intersects ~l+g
i
W e claim that we always
following a d d i t i o n a l
can
property holds:
|
(3) There exist smooth circles l
is a c r o s s - s e c t i o n
1 _< i _< g
n"
i = l,--.,2g,
~"-l(~i) :
~i~ intersects ~ ~ + g
proceed by induction. el,-..,e2g
of
~i'
~..-I
transversally
w i t h the properties
l
)
>~i'
and for
and in one point.
Let s o = ~, ~' = ~ and suppose that o (i),
(2) are chosen
when k ~ 0 there exist smooth circles ~l' such that ~
(~i
on C' such that
is a c r o s s - s e c t i o n
of
IT"
such that
i = 1,2,.-.,k,
~"-l(~i)
: ~"-i(~ i)
on C' >~i
We
51 and for any pair
(i,j),
i,j = 1,2,...,k
intersects
~[ transversally J
conditions
are empty).
with
and in one point
Consider
a k + I.
lj-il = g, ~
l
(for k = O these
Suppose
that n"-l(~k+ I)
is connected. Recall that p ~ O and let q' = q{. Take a E ~k+l' 2g a ~ ~ ~., and a' E ""-l(a). Let i': I ~ C' be a path in C' i=l l i#k+l which
is the lifting
i'(O) = a'
of ~k+l to C' satisfying
Because
~"-l(~k+l)
Note that ~l'''''~2g
is connected we have i(1) ~ a'.
is the so-called
and we have that S' -
the condition
is connected.
canonical
basis
of S'
Hence we can find a
i= 1
i~k+l 2g smooth
path
6
y C S' i=l i~k+l
y n ~k+l = a.
(see Fig.
7).
connecting
l
a with q' and such that
Take a point b Q ~k+l'
b ~ a,
2g
b~
~i' and let
B(a,b) be the arc on ~k+l connecting
a and b
i=l i~k+l 2g such that
8(a,b ) n (
~J~i ) = ~. i:l i~k+l
We can find a smooth path
y(a,b)
2g on S' - ~ i i=l
m=l
i~k÷l connecting path
a and b and such that Ya,b N ~k+l =
Y(a,b)
the domain
U 8(a,b ) is homotopically
on S' with ~ V =
Y(a,b)
trivial
U 6(a,b)
Ca,b],
the closed
on S' and if ~ i s
then ~ N
(~/i<)
= q.
52 Fig. ?
53 (We choose b and Y(a,b) sufficiently close to a and Y). ~k+l = ~k+l-6(a,b) ~':
I
U Y(a,b).
Because ~" is of degree two we getth~tif
> C' is the lifting of ~k+l to C' satisfying to condition
~'(O) = a' then
~'(i) = a', that is,
It is clear that basis of S'
is not connected.
is again a canonical
Because we did not change ~ I ' ' ' ' ' ~ K we see that we
~"-l(~k+l)
~k+l,2
n"-l(~k+l)
~l'''''~k'~k+l'~k+2 '''''~2g
could from the beginning assume that Thus
Denote
~"-l(~k+l)
has two connected components,
is not connected.
say ~k+l,l and
and if k+l > g we consider the point Xk+l = n..-l(~k+ 1 N ~k+l_g ) n ~'k +l-g
and take for ~'k+l the one of ~k+l,l' ~k+l,2 which contains ~ + l " This finishes our induction. with the properties
Now fix some
(1),(2),(5). ~'i c ~"
Let
~
3 that
N ~
~
= ~ (E i - Ei n Ti). i=l
~Ii
Ei N T.I = [X ~ E i ~ i " ( x ) Let
Yi,i' = {x E
If g = O
3 = P~ - ~ T. i=l l
We know
We can assume that
--< ~i,(x) --< 2~i.(x),
E~ ~i ,(x)
|
/k
(that is, ~i N dp = ~).
we take s o = ~' = ~o o Consider P~, ~ = 1,2,..-,~.
0
~l,...,~2g
We can assume also that
, V i = 1,2,...,2g,
~l,-..,~2g,
1
<_ ~i..(x),
(i',i") = (1,2,3)-(i)}
~i'(x) =
Im ~
O, i"=(1,2,3)~i',i ),
aii , be a point of P~ with ~i(aii,) : O, ~i,(aii,) = ~, ~i..(aii,) = i, ~i'
= (~ ) - l ( a % i" " ii ')"
54
It is clear that all aii , are on different connected components of ~C'
Now ~'l'''''~2g is a part of some canonical basis of C', say
' "'" "~2g'~2g+l ' ' ~l' ' "'''~2gc ' where gc = g(C')(= g(~)
non-singular
model for C), ~i A ~' = ~ 1 ]
i > 2g, j > 2g,
where
~ is the
when i > 2g, j < 2g or
lj-il ~ O, gc-g and for 2g < i < 2g+(gc-g ) --
e~ l
!
intersects ~i+(gc_g ) transversally and in one point. Evidently 2g c C' - U ~ is connected and we can find disjoint smooth paths l i=l 2gc /~ Bi, i = 1,2,5, on ai~q ( i ) with
C' - ~
~ a~(i)i,
i=l
where
~
l
- d
8i~
=
(~i~'~
connects
l
is defined by
8 ~l N ~C' = (~i~(i) U ~ ( i ) i )"
y~i~(i) ) ~
~
8
~: (1,2,5) --~ (1,2,5)
~(i) = 2, ~(2) = 3, ~(3) = i, and Let
such that
p
i@(i) 2g c v 3 Take u O E C' - ~ ~' - ~ ~ ~ j=l J ~=i i=l z
Y~=(i)i"
,(i) a~( ~ i )i and connect
uO
with all
!
ej, j = l,.-.,2g C • and smooth paths
6~l'
tj, j = 1,2,.--,g,
= 1,2,-..,~,
where
the end-point of
t
is
J
when
2g+l,...,2g+(g-g),
~' n ~' ]
~ = 1,2,.--,~,
by disjoint
ti, i = 1,2,3,
~gc ~ ~ int(ti~ ) c C' - ~ ' - ~ ~ 6i, j=l ~j ~=l i=l
int(tj),
J
~J+(---g)gc
i = 1,2,5,
when
J~
j < g and it is -
j > 2g+l, the end-point of t. belongs to 8.. /~.
We obtain a bouquet are "very close" to
--
l
of circles on C t j U ~ j'
l
such that all elements of
j E (l,''',g)U(2g+l,.-,2g+l+gc-g)
55 or
tj U ~ j+g
or to
(j e (1,2 , ...,g), or
tjUc~j+gc_g(jE(2g+l,...,2g+gc-g))
t i~ U 8~, t i = 1,2,3, ~ = 1,2,...,~.
It is easy to verify
that there exists a deformation retraction of C" /%
/k
consider C" as a regular neighborhood of ~ i n TC" as a regular neighborhood of ~ i n
on
and we can
A
C'.
Thus we can consider
Using ~ : ~
> C we define
O
=
and we see that we can consider T~" as a regular
neighborhood of ~
in ~.
We will use now the following result of R. Mandelbaum (see [5])Theorem (R. Mandelbaum).
Let MI,M 2 be compact differential n 1 t-manifolds, M 1 is simply-connected, ~ = V S be a bouquet of k=l k n circles,
i :
~
> int(M )
J
=
~),
Tj: T~j
J
l j: ~ j
},~
j = 1,2,
be "smooth" embeddings,
•
be a map inverse to ij: ~
>~j be a regular neighborhood of ~ j
Suppose that a diffeomorphism satisfies the condition:
~: ~T~I
• ., ~2.~ = X2Xl_~ I.
>~T~2 Let
-->
(c Mj),
in M j, _jY = T j I ~
j.
is given which 1 Sk2
,
k
=
1,2,
"'',n,
1 is isotopically be disjoint smooth circles in M 2 such that each Sk2 equivalent to
i2(Sk). Then (MI-T~I) U~ (M2---~2) is diffeomorphie to 1
M1 ~ ~2' where
M2
is obtained from M 2 by surgeries along
s1
k2' k = 1,2,.--,n.
In our situation this theorem gives the following: Let
~j, j = 1,2,...,2gc,
~i' i = 1,2,3, ~ = 1,2,...,~,
be !
smooth circles in A which are isotopically equivalent to ~j, J = 1'2'''''2gc'
i' i = 1,2,3,
~ = 1,2,...,V, correspondingly.
56
Then
A
A
(V-TC') U~.(A-TC') is diffeomorphic to A A
obtained from
by surgeries along
~ ~,
where
~
is
~j, j = 1,2, ....,2gc,
8i, i = 1,2,~, ~ = 1,2,---,~. Let
be a 4-manifold obtained from A by surgeries along
8~, i = 1,2, ~ = 1,2,...,~. Lemma 3. M1
We need the following
Let M1,M 2 be two compact differentiable t-manifolds, . S 2 XD2 ---> int(M2) > int(Ml) , ti:
be simply-connected 2 t : D3XS 1 l
. 2 xD2), i = 1,2,3, be smooth embeddings, T i = ti(D3xsl ) , T~ = ti(S 3 3 T.NT. = ~, T~ n T' = ~ for i ~ j, i,j = 1,2,3, T = ~ Ti, T' = ~ T i , l J l J i=l i=l a12,a13 6 ST1, al 2 ~ a13, a21 E 8T2, a31 E 8T3, aL2,a~3 E ~TI, t
~
!
!
w
a~2 ~ a13 , a21 { ~T2, a31 E 8T~,
!
y2,y3 (corresp. y2,Y}) be smooth
disjoint paths in M 1 (corresp. M2) such that for j = 2,3 int yj c (int MI)-T
(corresp. int Y'j c (int M2)-T' ) and
yj (corresp. ~j!) connects Let
~i: 8Ti ~ > ~ T i ' '
alj with ajl (corresp. alj' with ajl )' .
i = 1,2,3
be diffeomorphisms respecting I
S2-bundle structures over S | on 8Ti,~T i diffeomorphisms
~ T i'
and such that for j = 1,2 Define
~: T
M = M I-T U~ M2-T' surgeries along
S2 X S1
t i ~(D3xSI)=s2×sI:
t'i ~(S2×D2)=S2xS I: S 2 x S 1
alj =
> T' by and s2 =
M
(which come from ~ ~Ti,
and projection S2XS 1
>S I)
~](alj), ajl = ~(ajl).
T i = $i" and let be a 4-manifold obtained from M by
8Y2Y2
y2U$
!
and
!
s 3 = Y3U~Isy3Y3.
•
57
Let
(S2XD2)i , i = 1,2,3
be three different
copies of
S2XD2, (S2×S1)i = ~(S2XD2)i. Then there exist diffeomorphisms
> (s2×sl,
i
i = 1,2,3,
of S2-bundles
over S 1
such that if
M2(~) = ((((Me-T')U~I(S2XD2) 1)U~2(sexD2)2)o 3($2XD2)3) then
~
is diffeomorphic
Proof of Lemma
to
M 1 ~ M2(~).
3.
Because M 1 is simply-connected embedded
in M~,such
that int(D~)(i
we can find 4-disks = 1,2,3)
contains i
~j, J = intersects point.
Consider
7. = S~-T.. l l l circle
D 14 and D J4 transversally D4 i'
=1,2,3
We can consider
embedded
of boundaries(we
diffeomorphism
of S2-bundles
~T. l
>S 1
and projection
as a tubular
e~:z ~'l ---> (S2XSI)i
diffeomorphism
%
neighborhood
of a
e. : 7. ---->(S2XD2)i. l l
is the corresponding
use ~T i = ~ i )
~T i
then
in a 4-sphere $4 and let i
in S~ and we get diffeomorphisms l
We have also that if
where
l
Tl and if
and each of them only in one
as embedded T
Di, i = 1,2,3;
>S 1 and
then e i' is a
(S2XSI)~ r
-->S 1
came from ~T i = ~ti(D3×sl ) = ti~(D3xS1 ) = ti(S2×S1)
S2XS 1
as in the statement
> S I.
Define
of the Lemma.
~i =
ei.(~i )-I and M2(~)
58
Let
~
4 4 = ei(Si-Di).
In an evident way we consider
Di, i = 1,2,3, as subspaces i n M2(_~ ). We see that we can identify 3 L! 3 ./i M with M I- ~ D U ~ (M2(~)- ~ D where f is defined i=l a i=l l by
f ~D~i = fi: ~D~i 3
4
MI-i__~/IDi
~D~.
and
l
f
i
= e I~D ~ i
l
Identify
3
and
with the corresponding subspaces of M.
M2(@)-~D~
=
i=l
i
Taking a ball D 4 c MI, with 3
4
int(D ~) ~ [ ( ~ / D ) U ~2 U ~3 ] i=l l when M 1 is diffeomorphic diffeomorphic
we reduce the proof to the case
to S 4
and we have to show that M is
to M2(~).
Let M = (MI-DI)Uf ~4 i(Me(~)-D~41. We see that we can consider M as a b-manifold obtained from M by surgeries along two O-dimensional spheres,
say
$2,$3,° o embedded in ~ ,
and consider
from M by surgeries along two 1-dimensional on M in the following way: bjl,bi2 ~ M
M
as obtained
spheres s2,s 3 appearing
If S J° = [bjl,bj2), j = 2,3,
then there are smooth disjoint paths y2,Y3 on M
such
that "~j connects bjl with bj2 , j = 2,3, and s2,s > on M are obtained canonically from
~2 and ~3o
that M is diffeomorphic
These considerations
to M2(~).
immediately show
(We use that if a,b ~ S ~ and
is obtained from S 4 by surgery along S ° = {a,b} then ~ is diffeomorphic
to SIxs 3 and if
~
is obtained from S by surgery
along some el×c, c E S 3, then ~ is diffeomorphic Lemma 3 is proved.
to S 4).
59 We return to the proof of Statement ~I is diffeomorphic from
~i by
to A' ~ 9P.
(~).
Let A~IIbe a 4-manifold obtained
surgeries along ~i' i = 1,2,..-,2g
their images on ~i).
Lemma } shows that
(we identify ~'l with
From the construction of A' we know that
A' ~ A ~ pQ, that is, ~I is diffeomorphic -
to A ~ pQ ~ vP.
I
~i' i = 1,2,-.-,2g~ be the images of ~i' i = 1,2,...,2g,
Let in A and
AII be a ~-manifold obtained from A by surgeries along ~i' i = 1,2,...,2g.
Clearly
is diffeomorphic
to A ~
pQ / ~P.
But from Lemma 2 applied inductively we get that AII is an S2-bundle over S 2. (see [8]) that
Because p ~ O we have from a result of Wall
~
is diffeomorphic
~i' i = 2g+l,.-.,2gc,
(corresp. ~ ,
to (P ~ Q) ~ pQ / ~P.
~ = 2gc+l,.-.,2gc+~ ) be the
,
images of
along
~-2g
~i' i = 2g+l,--.,2gc,
2gc+V ) in A~II.
Let
~
(corresp.
~
C, ~ = 2gc+l,... '
is a 4-manifold obtained from A~II by surgeries
ai, i = 2g+l,-..,2gc+V. Because
AII
is simply connected and
has an odd intersection
form we have from results of Wall (see [8])
that ~ is diffeomorphic
to ( P ~
Q) ~ pQ ~ v p
~ [2(gc-g)+9 ] (p ~ Q).
From the classical Hurwitz formula we have: 2gc-2 = 2(2g-2)+p, that is, 2(gc-g ) = 2g-2+p.
(2~+p+2g-l)P ~
We see that ~ is diffeomorphic to
(~+2p+2g-l)Q.
This finishes the proof of Statement Theorem 2.
(~) and also the proof of Q.E.D.
§3.
Comparison of topology of simply-connected of degree n and non-singular hypersurfaces
We shall use an old classical geometry.
result
projective surfaces of degree n in ~PJ.
of projective
Because we need it in slightly modified
give the main parts of its proof in the A p p e n d i x P. ~9).
The result
Theorem
3.
singular
any i = 1,2,---,q at a i is equal
form, we shall
to Part I (see
is the following:
Let V be an irreducible
only isolated
algebraic
points,
say
the dimension
to three.
Then
algebraic
surface
al,.-.,aq.
of the Zariski
for generic
in ~pN with
Suppose
that for
tangent
space of V
projection
~T: V
> ~p3
we have the following: (i) There in W(V)
exist open neighborhoods
such that V
,
i = l~2,...,q,
of
w(~i)
i = 1,2,...jq
: ~-l
"I -l(ui)
(Ui)
(2)
q TI(V) - ~ ( a i ) i=l
(3)
If
N >_ 5,
subspace
singular
in
is biregular.
has only ordinary
V
corresponding
locus of
Su(V)
>U i
V is not contained
of ~pN and
~p2 in ~pS,
curve
Ui,
~(V)
> ~pN
singular
points.
in some proper
projective
is not the V e r o n e s e
embedding
of
to mQnomials of degree two, then the q - ~(ai) is an irreducible algebraic i=l
~(V) and
~-I(sTr(V))
is irreducible.
61 (~)
If N > ~
and V is not contained in some 3-dimensional q projective subspace of ~pN then ~(V) - ~ ( a i ) has pinch-points. i=l
Theorem 4.
Let V n be a projective algebraic surface of degree n
embedded in ~pN, N > 5, such that V
is not in a proper projective n
subspace of ~pN.
Suppose that V
is non-singular or has as n
singularities
only rational double-points.
be the minimal desingularization hypersurface
of degree
n
~V n
be a non-singular
Then
b+(~n) < b+(Xn) , b _ ( % ) (in particular,
Let X
h: ~ n
n in ~P~.
Suppose ITI(Vn) = O. O)
of V . n
Let
< b_(Xn)
b2(~n) < b2(Xn) ) ;
O a) ind(~ n) > ind(Xn); I)
Vn ~
[b+(Xn)-b+(V--n)+l]P ~
is diffeomorphic 2)
to X
n
~ P;
~n ~ [b+(Xn)-b+(Vn)+l]P~ is diffeomorphic
More precisely,
let
[b-(Xn)-b-(Vn)]Q
[b-(Xn)-b-(Vn)]Q
to [b+(Xn)+l]P ~ U: Vn
in ~P3, V'n = ~(Vn)" 9 = 9(V~)
[b_(Xn)]Q.
>~p3 be a generic projection of V n (corresp.
triplanar points (corresp. ,pinch-points) locus of ordinary singularities
i
p = p
V'n, S = S(Vn) be the
of Vn, d = d(S(V~)) be the degree
62
of S, and in the case when S is irreducible (that is,V where ~p212] is the image of Veronese embedding ~p2
n
~ ~p212]
>gp5
corresponding to monomials of degree two, see Theorem 3 (3)), let g = g(S(V~)) be the genus of (non-singular model of) S. Vn = ~p212]( c ~pS), let g = g(S(V~)) = -2. Then
0)
i) b+(Xn) - b+(~n) = =
~( n 2-6n+I1
=
-M
+ ~
)-l-b+(~n)
+ d(n-~)
=
=
= 2v + p + 2g-2, ii) b_(Xn) - b (Vn) : = n~l(2n2-~n+3) - b_(Vn) =
= -2V + ~p + d(2n-4) = =
oa)
~
+ 2p
+ dn
+ 2g
- 2;
Ind(~n) - Ind(Xn) = ~(dn-1) + ~
1)'-2)'
% ~ [-M+d(n-4)+l]P =
V
n
+ 7p
2 6 +yg
~ [-2M~p+d(2n-~)]Q =
/ [2M@p+2g-l]P / [M+2p+dn+2g-2]Q
Xn { P ~ k ~ Pn Q~' n wher e
kn = ~(n2-6n+ll),
~n = ~ ( 2 n 2 - 4 n + 3 ) .
If
63
Proof. of T h e o r e m generic
If V = ~P212] we can directly verify that all statements ~ are true.
projection
corresponding hypersurface
Thus suppose that V ~ ~p212].
~: V
>~P}
n
map of V n to its image.
~': V
X
n
space for a rational
b = S.X n = dn ~ O and we obtain
/ P is diffeomorphic / [2~+p+2g-1]P ~
C = ~'-l(s),
generic
~
projection ~
Considering ~': V' n
in V
n
a partial
>~p2
),
divisor on V (that is, ~
~ = ~'~',
>~P2 D
= h(~)
be a
be the r a m i f i c a t i o n
F = O be the equation of V' in ~p3. n derivative
of F c o r r e s p o n d i n g
we can easily see that
line in ~p2
from T h e o r e m 2, I)
section of Vn, E = ~'-I(E
divisor on ~), ~': V' n
"is linearly equivalent"). projective
Theorem 2--.
[~+2p+dn+2g-2]Q.
of V' on ~p2, n and
([9]).
to
be a canonical
is a canonical
locus of
is
of the Zariski
double point is equal three
Let E' be a generic plane
where ~
n
to S = S(V~).
Evidently, n
does not
n
double points of V n and X
N o w T h e o r e m } shows that we can apply to V ~ , X n c ~p}
that X
be the
Let X n be a non-singular
It is well k n o w n that the complex dimension tangent
>V' n
n
of degree n in ~P3 with the properties:
contain the images of rational transversal
and let
Consider
Because
we have
~
(n-1)E ~ C+D
K~p 2 ~ -3Z ~ -3E+D.
to (~ means here
where
~
is a
Thus D ~ ~ + 3 E
C ~ (n-1)E - D ~ (n-4)E - ~ .
and
64 Let gc (corresp. C (corresp. generic
E).
Because
projection
ordinary
double
gE) be the genus of (the non-singular C has 3 ~ ordinary
of E in ~p2 and E' is of degree
- 6~ = ( n - 4 ) E ( ( n - 4 ) E - ~ )
(n-4)2n - ( n - 4 ) E . ~
C~+E)E
points,
E' is a
n and has d
points we have
2ge-2 = ( ~ + c ) c =
double
model of)
= 2gE-2 = nCn-3)
- 6~ =
- 6~,
-2d
and ~.E
= n(n-~)-2d
2gc-2 = ( n - 4 ) 2 n Let
~,~
canonical degree
(n-4)[n(n-4)-2d]-6~
= 2d(n-~)-6v
be the normalizations
of C and S and p: ~
map corresponding
~' C" C ..... > S.
to
two and has p branch-points 2gc-2 = 2(2g-2) 2g-2--
We see that
1 ~(2g c-2)
9~+p+2g-i
-
+ p
A
2
.
Because
> ~ be the p is of
we have and
= d(n-4)
= -v+ ~ + d ( n - 4 ) + l
- 3V
-
Z . 2
and
v +2 p +dn +2 g -2 = -2~ ~ p +d (2 n-4 ). Note also that in [2] it is proven that X n ~ P is diffeomorphic to knP ~
~n Q
where
that the statements
kn = ~(n2-6n+ll)' O'),
1')-2')
Zn = ~ ( 2 n 2 - 4 n + }
of our theorem
)"
are true.
We The
see
65 statement
Oa)'
can be v e r i f i e d now by direct c a l c u l a t i o n
(using the
formula 2g-2 = d ( n - ~ ) - 3 v - ~ ) . N o w we have o n l y
to verify
But it i m m e d i a t e l y and
p ~ 0(2).
follows
the
0).
from O') using
0 ~ O (Theorem 3 (4))
(If b+(Xn) = b+(VL) we have g = O, p = 2, v = O .
T h e n d(n-~) = -i and d = I, n = 3. always
statement
But for a surface V 5 we can
find some ~p4 in ~pN w h i c h contains V3)o
QoE.Do
~4.
Simply-connected al~ebraic surfaces of general type.
Theorem 5.
Let V be a simply-connected non-singular
(complex)
algebraic surface of general type, Vmi n be the minimal model of V, c = c(V) = ~
(self-intersection
of the canonical class),
man C~ = ~ ,2
pg = pg(V)
(geometrical genus of V), b+ = b + (v), b _ = b _ (V),
and k(X), L(X), K(X), L(X), ~(X), ~(X) E ~[X] be polynomials of degree three defined as follows: ~(x) =
}(x2-~+ii); ~(x) = x~--!(2x2-~x+3);
~(x) = K ( 9 ( ~ + ~ ) l - x ; k(x) : K(2X÷l),
L(x) = t ( 9 ( ~ + ~ ) ) ~
~(x) : L(2X÷l)
Let m be a positive integer and
%
= V ~ [~(m2c)-2pg-l]P # [max(O,~(m2c)-lOpg-9+~)]Q,
li!
if if
c>_ 6 c = 2
if
c=
or
c ~ 3
and
or
c = 3,4,5
i.
Then (i)
~ is completely decomposable;
(2)
V ~ [K(b+)]P ~ [max(O,L(b+)-b_)]Q = = v # [k(pg)]p #
[max(0,~
is completely decomposable.
pg ~ 4, and
Pg ~ 3,
67 [Remark.
Note
that
~(b+l = 3037563+ + 6885062+ + 52004b+ + 13092; ~(b+) - 6o75ob3+ + I~175ob2++ ~65b+ + 28595;
k(pg) = 2~30OOp3g+ 2511OOp~ +211738pg + 164321; 2,(pg) = 486000p3g+ 1296000p~+llSLm3rS~pg+ 341360. ] Proof. and
Using
b + = 2pg+l
V ~ Vmi n ~ aQ
(Hodge Index Theorem)
is enough to prove our Theorem assume V = V
has no base points,
rational
only
a = c-c = b_-b_(Vmin) ~ 0
it is easy to verify
that it
for the case V = Vmi n.
Thus
min"
Let m ~ O be an integer
ponding to
where
Im~l
the regular
is birational
double-points.
corresponding
to
desingularization
such that the linear
~m"
Let V
map and
--~m: V ~m(V)
= ~m(V)
>
system ~pN(m)
I~I corres-
has as singularities
only
and h: V --->~ be the map
We see that h: V --->~ is the minimal
of V and we can apply to V the Theorem
4.
Note
m
that the projective b_ = IOpg+~-c
degree
of V is equal to (mK.mK)
(because b2(V ) = b++b_ = 2pg+l+b_
1 and l+pg = ~ ( c + 2 + b 2 ( V ) ) ( N o e t h e r We get from Theorem Now results
(Hodge Index Theorem)
Formula)).
~ that ~
of E. Bombier~[3]
2
= m c and
m
is completely
decomposable.
show that we can take m = 3
68 if c ~ 6 or c ~ 3 and pg ~ 4, cases.
m = 4 if c ~ 2 and m = 5 in all
This proves the Statement
The Statement only that
(2) follows from (i).
32c ~ 9(5~+4)
42c ~ ( 5 b + + 4 )
for
(I) of Theorem 5. We have to remark
(because c = 5b++~-b_),
c ~ 5 (because b+ h 1 and
~2c ~ 80, 9(5b++4) h 81) and
52c ~ g(Sb +4)
for
c = I. Q .E .D.
§5.
T o p o l o g i c a l n o r m a l i z a t i o n of s i m p l y - c o n n e c t e d a l g e b r a i c surfaces. R e c a l l that w e say that an a l g e b r a i c complex function field R
is t o p o l o g i c a l l y normal if there exists a n o n - s i n g u l a r m o d e l V = V(R)
of R such that V is almost c o m p l e t e l y d e c o m p o s a b l e
(see [4]).
Let R',R
two variables.
be two a l g e b r a i c complex function fields of
We say that R' is a s a t i s f a c t o r i l y cyclic e x t e n s i o n
of R if there exist n o n - s i n g u l a r models V' and V for R' and R c o r r e s p o n d i n g l y and a regular map
f: V'
r a m i f i e d c o v e r i n g and the r a m i f i c a t i o n
>V
such that f is a
locus of f in V is non-
singular and linearly e q u i v a l e n t to (deg f).D
w h e r e D 2 > O and
IDI is a linear system in V w i t h o u t b a s e - p o i n t s and fixed components.
(This is a small m o d i f i c a t i o n of the c o r r e s p o n d i n g
d e f i n i t i o n in [4].) We define
nl(R ) = UI(V) w h e r e V is any n o n - s i n g u l a r model of R.
D e f i n i t i o n 2. two v a r i a b l e s w i t h
Let R be an a l g e b r a i c c o m p l e x function field of ~I(R) = O.
W e shall say that R' is a
t o p o l o g i c a l n o r m a l i z a t i o n of R if R'
is a s a t i s f a c t o r i l y cyclic
e x t e n s i o n of R and R' is t o p o l o g i c a l l y normal. It w a s proven in [4] that
for any R w i t h
~I(R) = O there
exists a t o p o l o g i c a l n o r m a l i z a t i o n of R w h i c h is a q u a d r a t i c e x t e n s i o n of R. form.
N o w we shall give to this r e s u l t more e x p l i c i t
70 Theorem
6.
variables
Let R be an algebraic of general
of R embedded coordinates integer
nl(R) = O,
in a projective
(Zo, .-- ,zN) .
function
field of two
v be a non-singular
model
space ~pN with homogeneous
Let n = deg V, m = [ ~ ]
+l([~]
is the
part of _~n).
Suppose z O = O. of
type~
complex
that V is not contained
Let
~2m(Zo:
in the
..... :ZN) be a homogeneous
Zo,.-.,z N such that the corresponding
is non-singular.
The theorem
[iO], Sections a)
If E
m
v.El
follows
from Theorem
is a hyperplane
section
section of V
R' = R ( ~ )
and
4 and 5, and from the following
2g(Em)-2
b)
f = (~2m(Z°:''':ZN) ~ ) V zo normalization of R.
given by
form of degree 2m
hypersurface
Let
R' is a topological Proof.
hyperplane
4, results
.
Then
of [4],
remarks:
of V of degree m then
= (~+mEl)mE 1 = m~El+m2n
~ m2n+m
(because
> o).
For m =
+l
m 2 n + m + 2 h ~ ( n 2 _ 6 n + l l ) - b+.
Theorem
7.
variables
Let R be an algebraic of general
P2 = P2 (R)(2-genus)'
type,
complex
function
~I(R) = O, pg = pg(R)
c = P2-Pg-l.
Denote by
field of two
(geometrical
genus),
71
I
6([3V3c]+i) .16c~ 8([~----j+l)
a = a(R) =
150
if
c h 6
or
c ~ 3
if
c = 2
or
c = 3,4,5
if
c = I.
Let ~2a'8~ be two regular and a correspondingly
pluridifferentials
such that
and
pg ~ and pg ~ 3
of R of degree 2a
~a ~ O and on some non-singular
model V of R
is a non-singular
on V.
R' is a topological
Let
zero-divisor (~2a)o of ~2a ~2a f = and R' = R(y~). Then
normalization Proof. [4],
of
R.
The theorem
follows
[i0] and from the following Let V be a minimal
be a non-singular
from Theorem
of
Hl(v,
Let n = I.2 c. [2~])
we have
Then
lim'~I
~
= O (see [Ii]).
2g(Em,)-2 h
(m')2n+m'
of
remarks:
(non-singular)
element
5, results
a model of R, m' = ~ and Em, where
i = 3 if c ~ 6 or c h 3
and pg ~ 4, i = 4 if c = 2 or c = 3,4,5 and pg ~ 3 c = I.
curve
and i = 5 if
= p2_Pg -i = c (because As in the proof of Theorem and
(m')2n+m'+2
6
h ~(n2-6n+ll)-b+" Q .E .D.
APPENDIX TO PART I GENERIC PROJECTIONS OF ALGEBRAIC SURFACES INTO ~p3 §i.
A theorem of F. Severi. The following theorem was proved by F. Severi in 1901
(see [12]) in non-singular case. fact.
We need a slightly more general
Our proof is very close to Severi's arguments.
Theorem (F. Severi).
Let V be an irreducible algebraic surface
with only isolated singular points embedded in ~p5 such that V is not contained in a proper projective subspace of ~pS.
Let K(V) be
the variety of chords of V (that is, the algebraic closure in ~p5 of the union of all projective lines in ~p5 connecting two different points of V). Then 4 ~ dim~K(V) ~ 5 and
dim~K(V) = 4
iff V is a projective
cone over some algebraic curve or the given embedding
V
~ ~p5
coincides with the Veronese embedding of ~p2 corresponding to monomials of degree two Proof.
(that is, V = ~p212]).
It is clear that K(V) is irreducible and dim~K(V) ~ 5.
Suppose that
dim K(V) ~ 3.
Let E be a generic hyperplane
section of V corresponding to some hyperplane ~P~ c ~pS.
Since
V ~ ~P~, we have K(E) ~ K(V)(K(E) is the variety of chords of E). Hence
dim~K(E) < 3.
Because E is non-singular, we obtain from
dim~K(E) < 3 that generic projection of E in ~p2 is also non-singular.
73
Denote by E' the image of this projection isomorphic
to E).
Let
[Z]E' be the r e s t r i c t i o n
line b u n d l e on ~p2 c o r r e s p o n d i n g Then dim~H°(E',~[~]E , ) = 3. in some ~p2 c ~p~. subspace of ~pS.
It follows
the case
Denote by P
x
dim~K(V)
= 4.
dim~K(X)
Then using
= 4
d i m ~ x <_ 2.
point "chords"
Suppose
we have that a generic
of V contains an infinite number of points of V, that is,
V = ~p2.
Contradiction.
Consider As above,
a generic
dim~K(E)
irreducible, K(E) = K(V)
Thus
hyperplane ~P4E of ~p5 and let E = V A ~pE~.
= 3.
B e c a u s e K(V),
K(E) = K(V) N ~P~.,
d i m ~ x = :_~.
N ~pE4
and
K(V)
N ~PE4
and
K(E)
We can assume that x is generic on K(E).
be all the chords of E passing m ~x D ~p4E = ~ ~i" i=l
We can assume that x is a generic Suppose m > i.
Then
through x.
Let
It is clear
point on a generic
for a generic
of E there exists another
are
dim(~(K(V)C~PEd) = 3, we have
~l'''''~m
"chord"
Let x be a generic
the cone in ~p5 w h i c h is union of all
d i m ~ x = i.
of E.
in a proper projective
Contradiction.
It is clear that
that
line in ~p2.
from this that E is contained
of V passing through x.
"chord"
E' is
on E' of a complex
to the projective
Hence V is c o n t a i n e d
Now consider of K(V).
(clearly,
"chord"
point x of a generic
~'chord" of E passing through x.
74 Let a,b be two different generic generic
point of the "chord" k(a,b)
our supposition of E, w h e r e
points of E~ and e be a
corresponding
it follows that there exists another
c,d 6 E, c ~ d, such that e E k(c,d).
generic we can assume that c is a generic generic
projection
double points~ k(a,b)
N o w let
c ~ k(a,b), }~p3
E' = ~a(E)(that
b' = Ua(b)'
point of E.
only ordinary
be projection
is, the closure of ~a(E-a)).
4 in ~PE containing k(a,b)
c' = ~a(C),
of E in ~P3 w i t h the Let M be
U k(c,d), S' = ~a(S),
d' = ha(d). a ~ k(c,d),
we have that b' and c' are generic on E', b' ~ c',
"chord"
Because
d ~ k(a,b).
Since b and c are generic on E, c,d ~ k(a,b),
c' ~ d'.
"chord" k(c,d)
dim K(E) = 5 and a,b are generic on E we have:
na : E
the two-plane
From
Because e is
of E in ~p2 has as singularities
N E = [a,b~ and
center a,
to a,b.
We have now that M' = k(b',c')
b' ~ d',
(which is a generic
of E') meets E' in a third point.
Because E is not in a proper projective E' is not a plane curve in ~pS.
It easily
dim K(E') = 3, that is, K(E') = ~P~. ~p2 has the following has as singularities with all the branches
property:
subspace of ~PE4,
follows
A generic
from this that
projection
of E' in
if E" is the image of E' then E"
only the images of the singular as images of the branches
points of E'
of the singular
75 points of E' and finite number of ordinary can be proved by stability arguments in the n o n - s i n g u l a r
case].
K(E') = ~p3 and a generic
double points.
in the singular
chord of E contains three different
~i
is a generic
meets V only in two points of E) we see that C two.
The c o n s t r u c t i o n
the g r a s s m a n i a n map.
Since
(because
of
~
contains C x
~x ~ ~y
contradicts
x,y E K ( V )
[C(t),
as a generic element
corresponding have
system
to
Chow variety). and
Cx = C
Y
has projective
degree in
we have that
dim~T ~ 2.
corresponding
Cx,Cy are
now the m a x i m a l
t E T'} of curves on V w h i c h (T' is a subvariety of the
If for generic x,y E K(V) we w o u l d then C x is a projective
line.
This
deg(Cx) = 2.
But that means that the natural rational map degree one and
"chord"
Let T be the image of this
e q u i v a l e n t and let us consider
algebraic
From
gives us a rational map of K(V)
dim K(V) = 4 and d i m ~ x = 2
algebraically
C x = rxNV.
Z 1 is also a generic
of 2-planes of ,~pS.
It is clear that for generic
irreducible
x
Let
"chord" of V, that is, it
(which must be a curve)
x
points.
But this means that x is a non-singular
point of ~x, that is, ~x is a plane in ~pS. and
case as w e l l as
We get a c o n t r a d i c t i o n w i t h the facts:
Thus we obtain m = i.
x N ~p~ = ~i
[This
T
dim~T' ~ 2.
Now let us consider
the different
possible
cases.
~T'
is of
76 i.
Generic C
x
contains
some of the singular
V has only a finite number of singular
points of V.
points and T' is
irreducible we see that there exists a singular such that every C(t),
t E T', contains
la. Generic Cx is irreducible. point
a ~ V
elements
is contained
of [C(t)].
a is non-singular
Let
Since
Because
point a
o
of V
it.
dim~T' >_ 2
then a generic
in infinitely many irreducible CI,C 2
be two of them~
on V ; a n d hence
C2NC 1 9 a,
for any C(t ) w h i c h is "close"
to C 2 we have that C(t ) intersects C 1 in some point a' w h i c h is "close" to a.
That means that there exists a n o n - e m p t y
Zariski open subset U 1 in T' such that corresponding
C(t ) intersects C 1 in some point a{t) ~ a O.
By the same reason we have analogous C 2 are of degree two and let ~i,~2 C 1 and C 2 correspondingly. there exists a hyperplane ~i and ~2" of V. divisor
for any t E ~I
Let
El2
Evidently (of Weil)
for C 2.
be 2-planes
Since ~P~2
U 2 c T'
C 1 and
containing
a ° E C 1 N C 2 c ~i N P2 in
~p5
be the c o r r e s p o n d i n g
w h i c h contains hyperplane
section
El2 = C I + C 2 + D , w h e r e D is some n o n - n e g a t i v e in V.
Because
4
V ~ El2 c ~PI2
we have that
there exists a non-empty Zariski open subset U c T' such that for any t E U
the c o r r e s p o n d i n g
shows that we can choose such irreducible,
different
C(t ) is not in El2.
A l l this
C 3 E [C(t)] that C3 is
from CI,C2,
not a component
of D and
77
i n t e r s e c t s C I in some a' ~ a o a n d C 2 in some b' ~ a o. El2 N C 3 m
ao,a'
,b'
If a'
.
=
W e have
b' we have from C 1 ~ C 2 that
i n t e r s e c t i o n index of C= w i t h CI+C 2 in a' is greater than one. J
4
In any case, we ge2 ~P12"
lb. G e n e r i c C
x
C} h 9-
is reducible.
p r o j e c t i v e line in ~pS.
Contradiction
T h e n each c o m p o n e n t of C
Take g e n e r i c
say ~i"
~P~ ~ ~i
x
must be a
S u p p o s e that there exists o n l y a finite
number of such lines w h i c h contain a o. have a common line,
(deg C(t ) = 2).
T h i s means that all c(t )
T h u s every chord of V i n t e r s e c t s ~i"
and the c o r r e s p o n d i n g E.
W e o b t a i n that
any chord of E is passing t h r o u g h some fini%e number of points of E N ~l
and thus it c o n t a i n s three d i f f e r e n t points.
contradicts
dim~K(E) = 3
This
and the fact that a generic p r o j e c t i o n
of E on ~p2 has as s i n g u l a r i t i e s only o r d i n a r y double points. We get that there exist an i n f i n i t e number of p r o j e c t i v e lines of ~p5 in V c o n t a i n i n g a o.
The e x i s t e n c e of the C h o w v a r i e t y of
p r o j e c t i v e lines of ~p5 w h i c h are c o n t a i n e d in V gives us an i r r e d u c i b l e 1 - p a r a m e t r i c a l g e b r a i c system of such lines w h i c h are p a s s i n g t h r o u g h a o.
W e o b t a i n that V is a cone over some
a l g e b r a i c curve. 2.
Generic C
2a. G e n e r i c C
projective
x x
does not contain a singular point of V. is reducible.
lines
As a b o v e we have an i n f i n i t e number of
o f ~p5 i n V.
The c o r r e s p o n d i n g
Ch~' variety
78 shows that only a finite number is, not in some infinite lines.
of them can be "isolated",
irreducible
If generic C x would
contain
algebraic
line ~l" as above lines
This leads to a contradiction (Case Ib).
~ix,~2x,
systems. in V,
Thus
system of such
such an "isolated"
then we would have that every C(t ) 6 [ct]
contains
x
~2x.~2x ~ 0
lies in the non-singular
numbers
part of V).
a certain
is a union of two
which both are in some irreducible
Cx-d ~ O (intersection
line,
by the same arguments
We get that generic C
~lx.~ix ~ O,
that
algebraic
and for any curve d
are defined because Because
C
Cx
is a plane x
curve we have
2 2 2 (Cx) v = ~Ix+2~ix~2x+~2x
~ix" ~2x = 1 and
Take now a different
generic
C . Y
If
C
Y
contains
we get that all C(t ) E {C(t ) ]
have a common
impossible
Thus
(as we saw above).
are different then
from
~lx,~2x
"isolated".
Llx.~ly = ~ix~2y = ~2x.~ly = ~2x.~2y = O.
see that these equalities irreducible
algebraic
give that
system as
~ix and ~2x
Lly.
Thus
~ix or ~2x
component which
C Y = ~ly+~2y,
and not
~ 2.
where If
C
x
is
~iy,~2y .C
y
= O
It is easy to are in the same
~ix.~2x = ~ix.~ly = O.
Contradiction. We get a hyperplane hyperplane
Cx.Cy ~ O
and let a E Cx A Cy.
containing
C x and Cy.
section.
Because
Let
As above, we take
Ex,y be the corresponding
C x and Cy have no common component
79
we
have
(of
that E
A. W e i l )
= C
x,y
o n V.
2 = E
x.y
.C
x
+C +D y
As we
where
is s o m e
saw above,
= C2 + C C x xy
x
D
non-negative
C x . D ~ O.
+ D.C
> x--
We
divisor
have
2 + 1 = 3.
Contradiction.
2b.
Generic point
C
is i r r e d u c i b l e .
x
a E V
that
C(t ) containing We
can assume
o f V. curve of V
We
there
a.
also
Since
exists
Take
two
C
containing
non-negative
x
an infinite
different
that C 1 and C 2 are
s e e t h a t C I . C 2 > O,
d on V,
d i m T' > 2
.D > O.
As
C 1 a n d C 2.
divisor
above,
(of W e i l l
number
ones,
for a g e n e r i c
of irreducible
s a y C 1 a n d C 2.
in the n o n - s i n g u l a r
is, C x 2 > O,
that
We
we have
and
part
for a n y a l g e b r a i c
take a hyperplane
section
have
El2 = CI+C2+D ~ where
o n V,
and
El2
D is a
2 = El2"Cx = 2C2x + D.Cx. We
get C 2 = 1 x
connected intersects
and
D.C x = O. thus
with
a contradiction
Now
D either
[E] =
with
D.C 2 = O.
complex
[CI]+[C2] , where
hyperplane
contains
C 1 or C 2 in some
Now C 1 and C 2 are Cartier corresponding
suppose
sections.
that
In a l l
on V and we
line bundles
[CI] , [C2]
We
is t h e
have
the
is
cases we
or
have
D = O a n d E l 2 = C I + C 2.
divisors
[El
El2
C 1 or C 2 a s c o m p o n e n t s
points. Thus
D ~ O.
line bundle following
can consider o v e r V.
Clearly
corresponding
exact
sequences:
to
80
(i)
o
>H°(V,~v )
> H°(V,~v[Cl ])
>H°(C~,~ ±
(2)
± u1
O --->HO(V'~v[CI])---> H°(V'~v[CI+C2 ]) ~ H ° ( C ' 2'~c2[C I r +C2]c2)
C 1 and C 2 are both rational non-singular (CI+C2)C 2 = 2
(3)
[C~]~ )
uI
2 From C 1
curves.
I,
we get
dim~ H°(CI,~cI[CI]Cl ) = 2,
dim~ H°(C2,~c2[CI+C2]c2)
o dim~ H (V,~v) = i, we obtain from (i),(2),(3)
Because
=
fact that V is not in a proper projective
= 3-
and from the
subspace of ~p5
dim~ H°(V, ~v[CI]) ! 3 6 & dim~ H o (V, ~V [E])
= dim H°(V, ~v[CI~2 ]) <
i dim~ H°(V, %[Ci]) + 3 i 6. This shows that
dim~ H°(V,
Sv[E]) = 6
and
dim~ H°(V, Sv[CI]) = 3.
2 Because C 1 = 1 and C 1 is irreducible we have that global crosssections of [CI] have no common zero and we can define a regular map
f: V
>~P2
corresponding
see that f is surjective. that zero-divisor
to H°(V,
~v[CI]).
We can find a ~ ~ H°(V, ~v[CI])
such
i of ~, say Cl, is an irreducible algebraic curve j
in V.
It is easy to
!
Because the degree of C 1 is two we have that C 1 is
contained in some 2-plane of ~pS. and we can find a hyperplane
Cl2 = 1 gives
Cl' a C 1 4 !
section E 1 of V containing C 1 and Cl,
81
E 1 = CI+CI+D' , D' >_ O. we get
2 2 = ' = 2 E 1 = (CI+C2) = 4, EI.C 1 EI.C 1
From
EI.D' = O, that is, D' = O, E 1 = CI+C {.
bundle over ~p2 corresponding
Let [E'] be the line
to 2~, ~ is a line in ~ p 2
From
I
E 1 = Cl+C 1
we get that [E] = f~[E'], and
H°(v, ,viii _= f~H°(~p2, ~p2[E ]). But
dim~ HO(v,~V[E])
= 6
and also
dim~H°(~P2,~p2[E'
]) = 6.
We have
(4) Let
H°(V,
f[2]: ~p2
corresponding
~v[E]) = f~H°((~p 2, ~ p 2 [ E ' ] ) .
>~p5 to [E']
be the Veronese
embedding
> ~p5 our original
i: V
(because of dim~ H°(V,~v[E ] = 6) corresponds and (4) give us the following V
i
commutative
of ~p2 embedding which
to [El.
E-- [f~E']
diagram:
> ~p5
~p2 which shows that V = f[2](IP2).
Q .E .D.
~2.
Dualit[ theorem and C o r o l l a r i e s
Definition. in ~pN,
Let V be an irreducible
dim V = k,
of it. algebraic variety embedded
Tx for n o n - s i n g u l a r
x E V
be the k - d i m e n s i o n a l
projective
subspace of ~pN tangent to V at x, ~P Ne
projective
space for ~pN and V ~ be the algebraic
of the set of all t such that the c o r r e s p o n d i n g ~pN contains
T for some non-singular x
the dual projective v a r i e t y Duality Theorem.
x E V.
be the dual
closure in ~pN~ h y p e r p l a n e H t in
W e call V ~ (~ ~P N~)
for V.
If V is as above,
then N w
(i) V ~ is an irreducible
proper
(2) there exists a proper
subvariety
corresponding
;
subvariety S(V ~) in V ~ such that for
any t E V ~ - S ( V ~) the c o r r e s p o n d i n g T for some non-singular x
of ~P
h y p e r p l a n e H t in ~pN contains
x E V, and for any such x the
h y p e r p l a n e H e in ~pN is tangent to V e at t; x
(3) V is dual to V~; (4) in the case dimlY ~ = N-I we can take S(V ~) so that for any t E V w - S ( V ~) and for any non-singular hyperplane quadratic
section singular
Proof. of V, ~'=
the
E t = H t N V has at x a n o n - d e g e n e r a t e point.
(i) Let V
[(x,t)
x E V with H t m Tx
E Vsm
sm
be the set of all n o n - s i n g u l a r
X CpN*, Ht = rx~'
J
fl: ~'
points
>Vsm'
!
f2: F'
> ~ P N~
be the maps induced by canonical
It is easy to see that
fl: F'
~Vsm
projections.
is a fibre bundle over
83
V
w i t h a typical fiber i s o m o r p h i c to ~ p N - k - i and that V w is the sm I
a l g e b r a i c closure of f2(~") in ~P N~ dim~'
= k + N - k - i = N-I
dim~V ~ ~ N-I and (2) Let fl: ~
> V,
Take a point
~
W e o b t a i n that
and that r' is irreducible.
Hence
V w is irreducible.
be the a l g e b r a i c closure of f2: ~
~ V~
(Vo,to)
E ~'
~'
in
Vx~P
N~
!
, m
be the maps induced by fl and f2" such that t o is n o n s i n g u l a r
in V w and df 2 is s u r j e c t i v e in (Vo,to).
Let (Xl,...,x N) be some
a f f i n e c o o r d i n a t e system w i t h the center in v o, y I , . . . , y N be the r e s t r i c t i o n s of
X l , . . . , x N on V x (in some n e i g h b o r h o o d of Vo).
W e can a s s u m e that the a f f i n e c o o r d i n a t e s are chosen so that there exists an open n e i g h b o r h o o d U of v o in V
sm
such that
Y l ' ' ' ' ' Y k are local c o o r d i n a t e s of V in U and Y k + I ' ' ' ' ' Y N are r e g u l a r functions of Y I ' ' ' ' ' Y k in U. letters If
Yk+I'''''YN
We shall use capital
instead of Yk+I'''''YN"
v' E U then any h y p e r p l a n e of pN passing through v' N
is defined by an equation:
a° +
~ aiYi(V' ) = O. i=l
a° +
a.x. = O i 1 I l
where
This h y p e r p l a n e is t a n g e n t to V ~ at v'
iff it contains the f o l l o w i n g v e c t o r
(with the o r i g i n in v'):
(dq(v'),""',dyN(v')).
84 We have
N
i_~laidYi (v ' ) = o or
N ] lajdyj (v') +
~y.
k
(v')
~ a i ~ o~(v')dyj i=k+l j=l
N
=
O,
~)y.
(aj + ~ a i j-=l i=k+l
=
(v))dyj(v')
O.
We obtain the following system of equations: aiYi(v' ) + a = O, i=iLo N
aj +
~y.
l L_ ai ~ayj ( v ') = Oj i=k+l
1 < j <_ k,
or
N
~y
k
a° =i=k+l / ai[]=IL ~ y ~ ( V ' ) y j ( v ' ) N a~ = 3
-
i=k+l
Yi(v')]
~y
ai
(v) • ~~ ,l, ,
1 _<
j _ <
k.
These equations define an N-k-l-linear N ~
.
subspace
~(v')
of
parametrizing all those hyperpla es: of ~pN which are tangent
~P
to V at v'
It is clear that one of ai, i = k+l,...,N,
zero at t O .
We can assume that aN(to) ~ O.
is not
Taking U smaller
!
we can choose an open neighborhood U N of t o in ~P N~ such that aN
~
O
in
U' N'
U' N
N V~
is
a
non-singular
open subset of V ~ and
85
df 2 is surjective in any point (t',v') E ~"' t'
i
E UN N V * • v '
~ U.
as follows:
Take
affine
with
coordinates
in
U'N d e f i n e d
a.l si
= ~N '
i = O,1,.-.,N-1.
Now we write the equations
o
N -1
k
i=k+l
j=l
{i] in the following form: ~Y i
~YN
j=l ~YJ
N-I ~Yi k~ i= +i x ~yj
s. = J
k
~YN ~
j
1,2,..,k.
Taking differentials we have N-I
So= i=k+l X
/~ ~ y
k
j=l ~Yj
L=I
X
k
N-I
I
~2y.
m=l i= + I ~ Y ~ Y m
2Y N
C3] m= 1
N-I ds.
]
=
~y.
k
N-I
~2y.
~2YN
-
i=k+l ~Yj
x
L 1
= +I si
j = 1,2,...,k.
Let U' = U' D V*. N
Any tangent hyperplane in a generic point
of U' is defined by a non-trivial
linear combination of
dSo,...,dSN_ 1 equal to zero for any choice of dYl,...,dYk, where dSo,--.,ds~_ 1
satisfy (3).
86
One such combination is k
{4p~
ds ° +
yds + > j--~l 3 3 i=k+l i
The corresponding equation
N-I
hyperplane
in ~P
N*
has the following
:
k {5]
O 1
s°-s°(t')
N-1
+ j~IV y j ( v , l ( s j _ s j ( t . ) ) +
~
Yi(v')(si-si(t'))=O-
i=k+l Since k
YN(v') ~ -So(t') we can write
yj(v')sj(t')
j=
{5] in the following k
C63
N-1
i--~+l ~
)si(t ),
form:
N-1
s + ~lyj(v')sj + ~ Yi(v )si+Y~
= o o
j_
i=k+l
The point in ~pN which corresponds
to this hyperplane is v'
There exists a non-empty Zariski open subset V *1 in V* such that V 1. is non-singular, (df2)z is surjective.
(3)*)
Let
V I* _c f
P') and for any z E f
(V
,
We see that we can take S(V*) = V*-V~.
k* = dlm~ " V* , r~ = {(x,t), x E ~ p N
t ~ v *I (V~ is the
same as above),
the hyperplane H* in ~pN* corresponding to x x contains the k*-dimensional projective subspace Tt,V. of ~pN* The proof of (3) which we give here is due to F. Catanese.
87
tangent to V* at t}. Let
rI
It follows from (2) that f~-l(v~) ~ rl.
be the closure of
~'i
in
~pNx~pN*.
We obtain ~ _c ~i"
Now the same arguments as in (I) show that ~i is irreducible and d i m ~ 1 = N-I. pr~/1
Because
dim ~ = N-I
we get
~ = ~i"
is the dual algebraic variety for V*.
Evidently
Thus we proved that
V is dual to V*. (4)
Using the same notations as in (2) we can assume that
the local equation of E t in some neighborhood of x has the following form : k
N-I
I S (t)yj + / si(t>Yi+Y N = O. j=l j i=k+l
Denote
N-I ~2Yi ~2yN bj~ =i=k+ll si(t) ~yj~y~--(x)+ j ~ ( x ) ,
We must prove that ]~kl~j~ll = k.
Suppose that it is not true.
Then there exist not all equal to zero constants that
k
j,~ = 1,2,-..,k.
N-I
Cl,...,c k such
O-2y
I Cj( I si(t) ~2y~ X) + N ) j=l i=k+l ~yj;yz( ~yj;y (x) = O . k
For i = k+l,-..,N
denote
Now we obtain from [5] that N-I
[7]
X-
ci
i=l
ds
1
=
o.
~y.
ci = j=~l~cj ~-~j(x)l
88
Because
dim~ V
= N-I
and
it follows from {~] that at t.
t
Proof.
If
point on V ~,
S l , . . . , S N _ 1 are local p a r a m e t e r s of V ~
But this c o n t r a d i c t s
C o r o l l a r y i.
is a n o n - s i n g u l a r
[?].
Q.E.D.
dim~V ~ = N-I,
then f2 is of degree one.
We use the same n o t a t i o n s as in the proof of the
D u a l i t y Theorem.
W e can a s s u m e that S(V ~) ~ f2(~-~.) and
V ~ - S ( V ~) is non-singular. x i = fl(zi),
i = 1,2.
tangent to V ~ at t.
Let t E V~-S(Ve),
We have that Since
Zl,Z 2 E fjl(t),
X l , X 2 E V s m and H ~xI,H~2 are
dim~V ~ = N-I then
H~ = H~ xI x2
x I = x 2, z I = z 2.
C o r o l l a r y 2.
and Q.E.D.
If
dim V > O,
dimly ~ = N-1
and
V
has o n l y
i s o l a t e d singular points then there exists a proper s u b v a r i e t y S'(V ~) ~ S(V ~)
such that for any
t E V e - S ' ( V ~)
the c o r r e s p o n d i n g
h y p e r p l a n e section E t of V has only one singular point w h i c h is an o r d i n a r y q u a d r a t i c singularity. Proof. al,...,a N
E x t e n d S(V ~) as in the proof of C o r o l l a r y
Let
be all the singular points of V and N
S ' ( V ~ ) = S(V')U(~J(H~IhV~*). V i ~= l •= dim~V ~ = N - 1
and
Hai m V ~ - -
dim~V = O
1.
(V is dual to V~).
i = 1,2,''',N
H" ~ V" because ai
w o u l d mean that V ~ = H ~
and
a i
Thus S ' ( V ~) is a proper s u b v a r i e t y
89
of V ~.
Let
x E V
we have
we
sm
t E V~-S'(V~), that
(x,t)
see t h a t x is u n i q u e
that x is an o r d i n a r y corollary exists
3.
x E ~
N V
either
a cone
sm
of d i m e n s i o n Proof.
in ~ p N
part
subspace
a rational
algebraic
~x)
Duality
Then
algebraic
for a n y
fixed T
As
"
of E t.
Because
in C o r o l l a r y
Theorem
dimlY
~ot
Q.E.D.
< N-1
iff
there
of p r o j e c t i v e
generic
~ c M and depend
surface,
with
for x E V s m
of ~P
N ~
g: V
dim~g(V)
system
is e v i d e n t .
that
map
dim V ~ = 2 = N-I.
lines
for a l l
on x a n d V is singular
locus
the
f2fll(x)
is an N - 3 - d i m e n s i o n a l
Denote
>G
f2fll(x) where
subspaces
= 2.
means
That x~V
N > 3.
• PN-3(x)
is d e n s e
in V ~
exists
an a l g e b r a i c
sm
} on V ~.
we have
if"
part.
= ~PN-3(x).
We
is t h e G r a s s m a n i a n of ~ P N~ .
that we
Because
"only
G
projective
{~PN-3(x), Thus
.
Consider
have
a 2-dimensional
If N = 3 w e w o u l d dim~V e < N-I
that
have
and
for g e n e r i c
t E V~
sm
there
• PN-3(X) 9 t.
1
(~) w e g e t
on E t.
system
does
x
point
one. "If"
Suppose
= 2.
or an a l g e b r a i c
of a l l N - 3 - d i m e n s i o n a l
xEV
dim~V
D
t
singularity
corresponding
It is e v i d e n t projective
From
quadratic
that
the
(and H
on E t.
Let
such
be a singular
E ~
on V a 1 - d i m e n s i o n a l
[~ , ~ c M]
have
x
Let
~P~
curve
D t in V
be a generic
sm
such
that
4-dimensional
V
x E Dt,
projective
g0
~p N~
subspace of
c o n t a i n i n g t,
kx = ~p~ A ~pN-3(x),
x E Dt.
V' = V e D ~pt, Let L be a m a x i m a l p r o j e c t i v e
subspace of ~P N~ c o n t a i n e d in all ~PN-}(x), that
dim~L ~ N-~.
from this that
We can suppose that
x ~ D t.
It is clear
~P~ DL = t.
It follows
{~x } is a 1 - d i m e n s i o n a l a l g e b r a i c system of
p r o j e c t i v e lines on V' passing t h r o u g h t.
W e can a s s u m e that t
is n o n - s i n g u l a r on V ~ and (because of B e r t i n i ' s Theorem) V' is i r r e d u c i b l e and t is n o n - s i n g u l a r on V' non-singular
cone and thus V'
W e get that V' is a
is a 2-plane in ~P N~.
V ~ is an N - 2 - d i m e n s i o n a l p r o j e c t i v e subspace of ~P is dual to V ~, V is a p r o j e c t i v e line in ~pN. Thus
dim~g(V) < 2.
If
and V is a 2-plane in ~pN. have to consider now is
dim g(V) = 0
.
Because V
Contradiction.
then V e = ~PN-~(x)
W e see that only the case w h i c h we
dim~g(V) = i.
T h e r e exist n o n - e m p t y
Z a r i s k i open subsets U of V s m and U 1 c g(V) in all points of U,
N ~
W e see that
U 1 = g(U) and
such that g is d e f i n e d
{g-l(z)NU,
z ~ Ul~ is a
1 - d i m e n s i o n a l a l g e b r a i c system of a l g e b r a i c curves on U. C z = g-l(z), x ~ Czo Tx m C z
lines
W e have that ~PN-~(x)
That m e a n s that
Tx
is the same for all
is the same for all x E Cz~ and
x ~ Czo
Because (9))
z ~ U I.
Let
dimly w < N-I and V is dual to V ~ (Duality T h e o r e m
there exists a 1 - d i m e n s i o n a l a l g e b r a i c system of p r o j e c t i v e {~U' ~ E M}
on V.
T a k e generic x E U and let
~ (x),Cz(x)
91 be some elements Suppose
that
projection
~(x)
~ Cz(x)"
of x.
Let
the tangent
Because
and
LCz,Z6U I]
Let -: V
TX
plane of ~ at x.
is the same
i
hyperplane
section
of tangency
of
is generic
on ~
a 2-plane
greater
than one.
= O.
Now suppose
~
Hence
than one.
that is
the order Because
in ~P3.
Thus V is
for all x' E ~ (x)NV '
that V has only isolated Take generic
~pN-2(~)
line ~
in ~P Nw.
of ~pN dual to ~
points
Because
does not contain
and x ° £ B.
that an element
through x o does not contain Take
singular
sm
singular
and V
dim~V ~
We can assume that N - 2 - d i m e n s i o n a l
Let B = ~ P N - 2 ( L ~ ) N V
a cone we can assume
~ (Xo) ~ Vsm.
contains
Tx, is the same.
A V ~ = ~.
of V.
U C--z(x) c ~--x N ~.
Contradiction.
•
the corresponding
)' C--z(x)= ~(Cz~9"
to T_ x
we have that ~ is a ~ - p l a n e
~ (x)C Cz(x)
in some
That means that the
of ~ corresponding
dim~g(V)
x.
we have that T-is the same X
of ~ on C--z(x).
divisor
in ~pN and
is not a cone.
= ~(~(x)
T_ and ~ at x is greater x
We see that
points
through
be a generic
We have ~ ( x )
~ x' E C Z
Cz(x) with the multiplicity
subspace
>V
~ = W(x), ~ ( x )
V ~' in some neighborhood
we have
passing
of V in ~p3 such that u is an isomorphism
neighborhood T--X be
of C ~ , ~ M ]
projective
singular
Because V is not
~ (Xo) of [~ , ~ ~ M] passing points
x I E ~ (Xo), x I ~ Xo.
of V, that is, There
exists a
g2
h y p e r p l a n e H such that H ~ ~pN-2(~w) Let E be the hyperplane
and H D Xl"
section of V corresponding
E D ~ , E is connected and evidently E 4 ~
Since x 2 E V s m we have that H m Tx2. to H is contained
to H.
point x 2 E ~ o
Thus the point t(H)
in V ~ and V ~ n ~
Q.E.D.
4.
Let dim~V = 2, dim~V e = N-I.
not contained in some 3 - d i m e n s i o n a l T h e n for generic x,y E V, Proof. ~p4.
E ~p Ne
4 ~.
Contradiction.
Corollary
Because
(otherwise,
deg V = deg E = i) we get that E has a singular
corresponding
Then H D ~ .
Let
~: V
From C o r o l l a r y
projective
Suppose that V is subspace of ~pN.
dim~(~ x n ~y) < i. >V
be a generic
3 and dim V ~ = N-I
projection
of V in
we see that d i m ~ ~ = 3.
This shows that we can assume that N = 4. Suppose that
for generic x,y ~ V
means that there exists a } - d i m e n s i o n a l • P~ containing
Tx and Ty.
Let
We see that for generic x,y C V, generic t E V ~. For all generic
dim~(TxNTy ) = i. projective
subspace of
~pl(x) = f2(f~l(x)), ~pl(x)N~pl(y)
That
~ ~.
x E Vsm. Take
There exists a generic x E V with ~pl(x) ~t. y E V we have that ~ p l ( y ) N ~ P ~ x ) ~ ~.
union of ~ p l ( y ) ~ p l ( x )
for fixed x a n d generic
If the
y w o u l d be a
finite number of points on ~pI(x) we w o u l d have that there exists a point b E V ~ such that b ~ ~pl(y)
for all generic
y E V.
Let
93
~
be the hyperplane
for all generic
y
of ~p4 c o r r e s p o n d i n g
E V.
Thus
V c ~.
to b.
We have T
such that
Considering Corollary
Va
the map
E U, a ~ ~pl(x)c~pl(y)
f2: ~
>V ~
and using
with the singular
open subset
for some y E V
points of V ~.
Q.E.D. Let
~(V) be the a l g e b r a i c
Tx for all x E Vsm.
in some 3 - d i m e n s i o n a l
projective
Suppose dim V = 2 and V is not subspace of ~pN. dim V ~ < N-I°
Proof.
follows
The
"if" part i m m e d i a t e l y
the "only if" part.
Using generic
and C o r o l l a r y 3 we see that w i t h o u t Suppose
system of 2-planes of ~ p 4 g: ~ map
closure in ~pN of
Then dim ~(V) < 4 if and only if
assume N = 4.
~: ~
> G
dim~(V)
in ~!V).
induced by
~ G. ~: ~
from C o r o l l a r y }.
projection
Let
= 3.
[~x,XEVsm}
~ G
under
map of ~ in ~p4. >Vsm fj
T' = f(Vsm ) and T be the algebraic
4
is an algebraic
Let G be the G r a s s m a n i a n
~: ~
in ~P
loss of g e n e r a l i t y we can
be the canonical ~ P 2 - b u n d l e
~ ~p4 be the canonical f: Vsm
But
part of V ~ in 0nly a finite number of points.
C o r o l l a r y 5.
Consider
.
we w o u l d have that ~ p l ( x ) ( 9 t) intersects
Contradiction.
the union of
sm
dim~V ~ = ~ and
1 we see that all a E U are singular
taking t non-singular,
c Hb
Contradiction.
We get that there exists a non-empty Zariski U c ~pl(x)
Y
f: ~
of all 2-planes
over G and We have a regular
be the ~ P 2 - b u n d l e b~
over V s m
be the induced map~
closure of T' in G.
It is clear
94
that ~(V)
is the algebraic
~(V) = g("-l(T)). {~P2(t),t~T]
on
closure of g~(~)
We have an irreducible ~(V)
in ~p4 and
algebraic
such that for generic
system
t E T, ~P2(t)
is the
tangent plane of V in some generic x E V. Consider
two different
Case
dim~T = I.
i.
a 1-dimensional ~t = Supp Ct, C~t(x) ~ x. y E ~t(x)
z E ~(V)
f(~t)
For generic x E V
~P2(tl)
Let
be another
~P2(tl)
Then
for all t' E T "close"
= Tx
point z of ~(V)
for some generic x E Vsm
element of [~P2(t),t E T] passing through z. to t 2 the c o r r e s p o n d i n g
in some n o n - s i n g u l a r
"close"
to z.
W e get that for generic
~P2(tl)
in some point z(t) which be such that
passing
t I E T the
contains a n o n - s i n g u l a r
~P2(t2)
x,y E V,
But that
In this case we have that for a generic
and
~p2(tl)
Let
there exists a
tangent plane is the same.
We can assume that for generic
with such property.
intersects
on V.
< N-I.
dim T = 2.
corresponding
sm
divisors
there exist infinitely many different ~P2(t)
through z.
y E V
W e have
is a point in G we have that for all
the c o r r e s p o n d i n g
Case 2.
for t ~ T'
system of positive
~t = ~t A Vsm.
dlm~V
cases.
Let C t = f-l(t)
algebraic
Because
means that
possible
point z' on t 6 T,
is n o n - s i n g u l a r
~P2(t) = • . y
~P2(t')
~P2(t)
intersects
on ~(V).
We obtain that
• N T contains a n o n - s i n g u l a r x y
~(V)
point of
Let
for generic ~(V).
Because
95
dim~(V) we get
= }
we have
dim~(~xnTy)
h i.
From C o r o l l a r y
dimlY ~ < N-I.
Corollary
6.
Let
V has only isolated 3-dimensional dim~[~x~V]
Q.E.D.
dim~V = 2, dlm~V
singular
projective
= i.
= N-l,
suppose
that
points, V is not contained in a
subspace of ~pN and for generic x E V
T h e n for generic x E V,
of some projective
4
Tx N V
is the union
line of ~pN passing through x and of finite
number of points. Proof. algebraic
It is easy to see that there exists an irreducible
system of algebraic
rational map
f: V
for generic x E V,
curves on V
[Ct, tET}
• 7T
such that
~x(V)
is the union of Cf(x)
and a
f(V) is dense in T and and of finite
number of points. Consider
two different
Case 1.
dim~T ~ i.
Dt = f-l(t ) n Vsm. Yl ~ ~Y2 Ct ~
~ • Y~Dt Y
t E T and let
dim V w = N-I
we have that
yl,y 2 E Dt, Yl ~ Y2"
we see that C t is a projective
generic x E V. passing
Take generic
Because
for generic
possible cases.
Because line.
There exists an element Ct(x)
through x.
Take
of {Ct,tET}
(If not, we w o u l d have a line ~ c V such
96
that all
~x D ~, x E Vsm.
projective
subspace
Because Ct(x)
of ~P N~.
is a projective
Ct(x ) c Tx N V = Cf(x) Ct(x) = Cf(x)
Then This
Hence
and Cf(x) 9 x.
Suppose
/ ~ C t ~ ~ and let tET
we have
we have
d i m V~ = N - l ) .
+ (finite number of points).
dim~T = 2.
hyperplane
contradicts
line we have Tx D Ct(x) and
Case 2.
x E V
V ~ is in some N - 2 - d i m e n s i o n a l
T ) a. x
Hence
of ~P N~ c o r r e s p o n d i n g
V~ = H~'a
a E /~C~. tET ~ V~ c H~ -- a to a.
Then
where
H~ a
for generic is the
B e c a u s e dimlY ~ = N-I
But V is dual to V ~ and we get dim~V = O.
Contradiction. Thus
t/~ETCt = ~.
Because V has only isolated
we see that for generic
C t c Vsm.
corollary
2 gives us
that for z ~ V ~ - S'(V ~) the c o r r e s p o n d i n g
hyperplane
section E z
has only one singular singularity.
t ~ T,
singularities
point x(z) which is an o r d i n a r y quadratic
It is clear that
Cf(x(z))
we have that E z is in a 2-plane of ~pN. 3-dimensional Thus
projective
Cf(x(z) ) ~ E z
Cf(x(z) ) is non-singular, In particular,
subspace of ~pN.
c Ez.
If Cf(x(z) ) = Ez
Hence V is in a Contradiction.
and b e c a u s e E z is connected we see that Cf(x(z) ) g x ( z )
and ( E z - C f ~ ( z ) ) ) C f ~ =
we get that for generic t E T, C t is n o n - s i n g u l a r
and irreducible.
It easily follows
from this and from dim~T = 2
I.
97
that C 2t > O.
C 2t > 1.
Suppose
Then we have two possibilities:
l) For generic x,y E V, x ~ y, Cf(x) different points. Corollary
N Cf(y) contains two
In that case we have dim~(~xA~y) ~ 1 and
4 gives us a contradiction w i t h dimlY ~ = N-1.
2) For generic x,y E V, x ~ y, Cf(x)N Cf(y) and Cf(x)
is ~angent to Cf(y) at a(x,y).
there is a projective tangent projective We get
is one point a(x,y)
But that means that
line ~ c Ta(x,y) which is the common
line in ~pN of Cf(x) and Cf(y) at a(x,y).
T ~ L, ~ D L x y
and
dim~(TxN~y ) > i. • --
gives us a contradiction with dim~V e = N-I.
Again Corollary Thus
C 2t = i.
Take generic x ~ Vsm and consider the linear system of all hyperplane Eu
Each
Cf(x ) and let D u = E u ~ ( x ) . Evidently each D u 9 x.
y E Cf(x)
N Vsm, y ~ x.
{Eu](N > 3)) x,y ~ Cf(x)
non-negative
Because
[D u] is infinite
there exists a Du(y ) 9 Y"
But
Take
(as
Du.ef(x) = 1
and
N Du(y), x ~ y, give us that Cf(x) _c Du(y)(ef(x)
irreducible).
have:
sections of V having singularity at x.
[Eu~
is
We can write Eu(y ) = 2Cf(x)+D' where D' is some divisor on V.
if D' ~ O then Cf(x)
But Eu(y).Cf(x)
= Du.Cf(x)
Because Supp Eu(y ) is connected we A D' ~ ~.
Hence
+ Cf(x).ef(x)
= 2.
Eu(y).ef(x)
> 2.
Cont=adiction.
g8 We get D' = O and Eu(y ) = 2Cf(x). t 6 T
there exists a hyperplane
We showed that for any section E(t) of V such that
E(t) = 2C t. Because dim~T = 2 we have infinite number of different C t passing through x.
Take one of them, say Ctl , Ctl ~ Cf(x)-
Consider the corresponding E(tl) = 2Ctl. at x.
Thus
E(tl) m ~X n V.
We get
Then E(tl)
Ct I D Cf(x).
is singular Contradiction.
Q.E.D.
~}.
Proof of T h e o r e m } (§}; Part I I. This theorem was formulated Proof of T h e o r e m
}.
on page 60.
Using generic
~p5
projection V
we
can assume that N ~ 5. (I)
Let K a , i = 1,2,.-.,q, be the algebraic closure in ~pN 1 of the union of all projective lines ~(ai,b ) in ~ p N connecting a i and a point b 6 V-ai. projections V
>~P}
Denote by (M
M
the parameter
is irreducible).
It is clear that
dim~Kai ~ ~ and we can find a proper a l g e b r a i c such that for any
~ E
M-S o
subvariety S O in M
we have that ~ is regular on V,
is locally b i r e g u l a r at each ai, i = 1,2,''-,q, Vi
= 1,2,-..,q,
n-l(~(ai) ) = a i.
Let ~o be a small open Because
M-~
o
space of all
This
(classical)
and
finishes
(I).
neighborhood
of S O in
is compact, we can find open n e i g h b o r h o o d s
a. in V such that l "-l(~(~i) ) = ~ q l
~ E
M-~
and
o
and ~I~, : ~rU i l
V
i = 1,2,'--,q
M.
~. of l
we have that
,(U~i ) is a b i r e g u l a r
map.
Let
~'=v_ U~'. i=l i
(2) We can assume N = 5a r g u m e n t s work.) is a pro]ective
(In the case N = 4 almost the same
A projection line ~
~ ~ M is defined by its center w h i c h
in ~pS.
Denote by ~(V) the a l g e b r a i c tangent
planes
closure of the union of all
~ of V where x is a n o n - s i n g u l a r x
point of V. q Let T ( V s m ) be the tangent bundle of V s m ( V s m = V - ~ J a ~ ) and i=l ~
100
~: T(Vsm) integers
> ~(V)
< 4, then a = O, b = O, and if
g(V)
different
points
z E Txi(z)} .
Choose
Define two
x (z)l
for any z ~ ~
g(V)
N ~(V)
E V s m , i = 1,2,'--,b,
in a different
there exist b
such that
Clearly M ° is open and dense in M ~ .
Let
{Zl'''''Za}
i = l,...,b,
a hyperplane
Mo = [~ ~ Mo,
I.
= ~o
N ~(V)
Take
of Vsm with
Txij9 zj.
H ° in ~p5 with xij ~ Ho, i=l,...,b , j=l,...,a,ar~let We shall prove the following
Let x o be an arbitrary
{xij , i = 1,2,-.-,b,
j = l,...,a}.
There
element exists
Uxo of x ° in Vsm , a neighborhood i
and an open dense M
of an open
Mxo of T o in M o
c Mxo such that for any ~' E M'
X O
for any y ~ Uxo either of pinch-type
V
and for any zj, j = l,...,a,
be all the points
A n ~ Ho}.
Statement
neighborhood
dim ~(V) = 4, then
N V = ~, ~w intersects
points and if a / O, then
let xij ,
map.
b = deg ~.
(in ~ p S ) ,
Let Mo = {~ E M - ~ o, ~
o E Mo.
rational
a,b as follows:
If d i m ~ ( V ) a = deg
be the canonical
we have:
X O
(d~')y
is a monomorphism
at y (and last possibility
or ~' is a map
holds only
for a finite
number of points of Uxo). Proof of Statement
I.
Let ~D- = ~p5 Ho.
We can assume that
O
affine
coordinates
given by (Zl,...,zs)
Zl,...,z
5 o f ~5o a r e c h o s e n s u c h t h a t
> (Zl,Z2,Zs) ,
zi(Xo)
'no i s
: O, i = 1,2,''',5,
101
the hyperplane z 3 = o does not contain TXo n ~5o and projection (Zl,...,zs) Let
> (z3,z4) is locally biregular on V N ~5o at x o-
~i = zi VN~oS" i = 1,''',5.
parameters at x o. x O in V n ~
We see that ~3,~ 4 are local
There exists a Zariski open neighborhood U of
such that for any y E U, ~3-~3(y),
parameters at y.
~ 4 - ~ ( y ) are local |
Consider ~6 with coordinates
and let z i = zi+eiz~+eiz5, !
i = 1,2,3,
~i = zi VN~ 5 " | !I o
!
|
For the
Jacobians |
I
13 = D(~3,~ ~)
|
and
I
23 = D(~3,~ 4)
we have: I
+ c3 D(~3,~4 ) - c2 ~
Consider
- ~
U × ~2 and an algebraic variety W in
by the equations: |
t
13(~,~)
= o,
f
23(~,c)
|
(el,62,e3, el,62,e3)
= o
"
U × ~2
defined
102
We can find open neighborhoods U origin in g6 such that in
D( ~3"% )
D
'
b(~3,,~4 ) ~
o,
Ux o
of x
xo
o
in U and g of the
x~
(~l~'~ 2~ ) '
i
D(~ 1 , e 2 )
¢
o,
that is, W N [UxoXE ] is non-singular and of complex dimension six. Let W O be the irreducible component of W passing through the point (Xo,O), that is,
Wo n (~o~) : w e: Wo ____> ~6
((~je))
n
be the map induce~ by projection, ~(Wo) be the
algebraic closure of e(Wo) in ~6. dim~a(Wo) < 6
~ -I(~6_S):
In the case
~, = • n (¢6_ s ). the equations
We have that either
or there exists a proper algebraic subvariety S
of ~6 such that map.
,
J
d i m ~
a-l(~6-S) < 6
> ~6-S
is an unramified
denote by S = ~ .
Let
, , we get that for any (el,~2,E3,Cl,C2,e)
' = O and 13 i
2'
23 = O
0
E ~'
define two complex
.
subvarieties of UXo , say CI~,C23 whzch in a neighborhood of any common point of them are non-singular complex curves intersecting transversally. ~' ~'~ ~ Z' and let f: U Xo Fix any (el,E2,e3, e'1,_2,_3, map defined by projection (Zl,Z2,...,z5)
>~3
>(z~,z~,z~).
be the Take any
y ~ Uxo.
If y ~ Ci3 N C23' then ~df)y" is a monomorphism.
Z E C~3 A
c23.
i = 1,2.
We can write Z i = Oi(u,v), i = 1,2, where ~i(u,v) are
q
I
Let
I
i
Suppose I
u = ~3-~3(y), v = ~4-~4(y), Z i = ~i-~i(y) ,
103
D(zi,u) some power series of u,v, ~i(O,O) i = 1,2, and a linear
= O.
transformation
Using the fact D - - ~ ( y
of type Z
> Z -a u l
assume that
~i(u,v)
Z i = ~.u 2 + 2Biuv l
l
have no terms of degree one.
+ Y.v 2 + t.h.d. l
("t.h.d."
means
we can
l
Write "terms of higher
degree".)
We get from the transversality
of Cl2,Cl}
det 82 Y2
~ O, and in particular
YI,Y2 is not zero.
can assume
?l ~ O.
and using
A(O,O)
parameters
u,v
degree Let
one,
--2
Z1 = v =
--
~u
We have now v
We
some neighborhood
= A(u,v)v 2 + B(u)v
2
+ d(u),
b ~ O.
+ t.h.d.,
one (and ~(0,0)
Let ~2 = Z2-~(u'~I)'
see that u,v
are local parameters
~} = B(U'~l)' at y and in
~2 = uv;
at y.
~ ~ O.
= O) and
of y the map f is given by the formulas:
f is of pinch-type
+ C(u)
= ~i and Z 2 = A ( u , Z 1 ) + B ( U , ~ l ) ~ ,
--2 ~i = v ;
that is,
We
d(u) has no terms of
+ 2~u$ + ~ 2
has no terms of degree
= bu + t,h.d,
~( U,~l).
,l(U,V)
in y that
we see that we can find new local
d(O) = O, and Z 2
A(U,~l)
~(U,~l)
= ?i ~ 0
such that
~i = Zl-d(u)"
where
-u =
--
Now writing
one of
) = O,
Statement
I is proved.
104
Let
Now it is easy to see i=l,"'~ j=l,"'~
X,
,J
.
1,3
l=l,'",b j=l,..-,a
that for any point n of M following:
l,j
sufficiently
for any y E V s m
or n is a map of pinch-type denote it by ~I"
X.
either at y.
close to ~o we have the
(d~)y is a monomorphism at y Choose
such a n in M I
and
Because ~ is compact we can choose an open
n e i g h b o r h o o d M I of ~ in M ° such that for any ~ E M I we have: for any y E V s m either
(d~)y is a m o n o m o r p h i s m
at y or ~ is a
map of pinch-type at y. Let d i m ~ ( V )
= 4 (that is, dim V e = N-I
Duality Theorem)). projective N V
For any x E V
lines of ~p5 w h i c h
and
x
sm
let K
connect x
K (V) be the algebraic •
(Corollary
T,x
5 of the
be the union of all
with other points of
closure in ~p5 of
U xEV
KT, x. sm
From C o r o l l a r y x E V,
6 of the Duality T h e o r e m we get that for generic
T N V x
is either a finite number of points or the union
of a projective
line in •
passing through x and of finite number
X
of points.
In b o t h cases we have for generic x E V, dim~K T
< 1 ,X
and thus dim~K (V) ~ 3. ~. i
n KT(V ) = ¢.
Xl,j,
,..
¢
,~,j
W e can find ~' E M I
Let ~ ,
D ~(X) = [Zl,"
be all the points of V s m w i t h
such that a ~, Z'o = ¢' and Try
!
9 zj (xi, O = ¢).
1,j !
Suppose that for zj, O < j < k < a property:
V
i = 1,2,-..,b,
~'
-1
we already have the following
(~I(x'i
,j))
,
= xi, j
105
For any x E V let K
be the algebraic
closure
in ~p5 of all
X
projective
lines connecting
x with other
dim~K x _< 3 and for x E Vsm ,
~x --c Kx.
points of V. Clearly b Let ~ + i = ~ K , .
|
We have
Zk+ 1 E ~n, n ~ + i "
n" in M I arbitrarily
Because
dim~+l
close to n, such that
It is clear that Z'k+l E ~,, N ;(V).
< }
we can find a
~n,, N ~ + i
Suppose
I
= Zk+l"
for some i = 1,2,---,b
there exists a point y ~ ~"-i"" (x ~ ,k+l ) with y ~ Xi,k+ I. P(~,, "xi, ' k+l ) be the 2-plane
in ~p5 containing
~(X~,k+l,y ) be the projective
line containing
Let
~,, and Xi,k+ 1 and Xi,k+ 1 and y.
,
Clearly
(Xi,k+l,y)
c P(~.,,Xi,k+l)
z = ~,, N ~ (Xi,k+l,y). ' '
~(Xi,k+l,y ) ~ Kx, i ,k+l we have z = Z'k+l. Hence Z'k+l E
that
!
Zk+ 1 E KT(V ).
( Cx ~"-i.~"
sufficiently
9
to x~ .. l,j
Z ..l ,
j
Xi,k+l,
O < j < E, '
,,
~(X~,k+l,y ) c K T
--
such that
Because
~" sufficiently
--
z.. j
,
(Xi,k+l,y)
, c KT(V ) "xi ,k+l
Taking
sbl
define
We see ~"
the points
x" g V i = 1,2,...,b, l,j sm' are close to z
~' is of pinch-type
|
j
and II
. . are close
X "
at all x. 1,j
J.,j
we see that for
close to ~' we will have ~"-l(~"(x~',j))
o<_j <_K, 1 < i i b .
and ~
An, , ~ K (V) = ~.
i = 1,9,'''
close to A ° we can uniquely
]
x. 1,j
Thus
But this contradicts
;,k+l ) =
z'~ g %~,, N ~(V)
a point
Because
L~,, ~ K x, = Zk+ 1 l ,k+l ' and (Xi,k+l,y) c_ • , xi ,k+l and
and there exists
= x "l,j'
106
This argument
shows that we can choose
any y E V s m with the property: ~-lii(~ii(Y)) = y.
~
II
is of pinch-type
hyperplane
for
at y, we have
There exists an open n e i g h b o r h o o d MII of ~II in
M I such that for any ~ ~ MII we have: is of pinch-type
such ~II in M I that
at y then
section of V.
~-l(~(y)) For generic
If y E V = Y.
sm
is such that
Let E be a non-singular
projection
know that p(E) has only o r d i n a r y double points.
p: E
>~P2
It follows
we
from
this that we can find a WIII in MII such that there are only a finite number of points in ~rrr(V) which are not ordinary We can assume also that for any x ~ n]]i(V), ~ ( x ) There
exists a n e i g h b o r h o o d MrFr of ~Trr in MII
singularities.
is a finite set.
such that all ~ ~ MTrr
have the same property as ~I]I" Let H 1 be a h y p e r p l a n e points x of Vsm that ~Ixi(Vsm ) , M--I ~
=
in ~p5 which does not contain all such
~]ii(x) is not an ordinary
[~ E Srrr, ~
Statement and
finish the proof of part (2)
if we prove the following S t a t e m e n t s If.
of
c HI}.
It is easy to verify that we will of our T h e o r e m
singularity
and Corollaries.
Let Xl,X 2 E V s m , x I ~ x2, ~iIi(Xl) = -iii(x2)
~rrr(TXl ) = ~iii(~x2 ) .
Then there exist open neighborhoods
Uxi of xi, i = 1,2 ' in Vsm , a n e i g h b o r h o o d M Xl,X 2 of and an open dense M' c M Xl,X 2 Xl,X 2 we have:
nl]I in M I ~
such that for any ~' E M' Xl,X 2
107
if Yl E UX; Y2 ~ Ux 2 then
are such that
n'(yl) = ~'(y2), yl~Y2,
~'(~Yl ) ~ "' (Ty2)"
Corollary of Statement II. for any pair yl,y 2 E Vsm with niV(TYl) ~ niV(TY2).
There exists a ~IV E M]]I such that "IV(Yl) = ~[Ev(Y2) Yl ~ Y2' we have:
Moreover, we can find an open neighborhood
MIV of ~IV in MIII such that for any T; E MIV we have: yl,y 2 E Vsm are such that
~( "ryl)
~(Yl ) = ~(Y2 )' Yl ~ Y2
if
then
F ~( "Y2 )"
Statement III.
Let Xl,X2,X }
be three different points of 3
Vsm with
~Iv(Xl) = ~Iv(x2) = ~Iv(x3) and
dim~[i=~l~iV(TXi)] = =
1.
Then there exist open neighborhoods Ux. of xi, i = 1,2,3, in Vsm , l a neighborhood Mxl,x2,x} of ~IV in MIV and open dense M' c M Xl,X2,X ~ Xl,X2,X ~
such that for any
if Yi E Uxi, i = 1,2,3, are such that
IT. E M' we have: Xl,X2,X ~ Y~ ~ Y2' Y2 ~ Y3' Yl ~ Y~
and ~'(yl) = ~'(y2) = ~'(y~) then dim~ li~__l~'(Tyl)== O. corollary of Statement II!. if
There exists a wV E MIV such that
yl,Y2,y } ~ Vsm , ~v(Yl) = ~(y2) = ~v(y~5) and Yl ~ Y2" Y2 ~ Y~'
Yl ~ Y~
3
then
neighborhood
dim(]:i/~__l~V(Ty2) = O. %
of
~V
Moreover we can find an open
in MIV such that for any
- , %
we have:
108
if
yl,Y2,y 3 E V
sm
are such that ~(yl) = ~(Y2 3
Yl ~ Y2' Y2 ~ Y3; Yl ~ Y3 Statement IV. V
sm with
then
= ~(Y3 )'
dim~ /~(i=l ~Yi ) = O.
Let Xl,X2,X},X ~ be four different points of
nv(xl) = ~v(X2) = ~v(X3) = ~ ( x 4 ) .
Then there exist open neighborhoods Uxi of xi, i = 1,2,3,4, in Vsm , a neighborhood
MXl,...,X 4 such that for any
M
of
~V in %
and open dense
Xl,''',x 4
MXl,''',X 4
~' E M' we have Xl,'-',x 4
~-~ IT'(Ux,) = ~. i=l
Corollary of Statement IV. There exists a ~VI E % for any x ~ ~vI(V) we have:
-i
~vi(X) has less than four elements.
Note that all Corollaries immediately Statements of
such that
follow from the
(arguments are the same as used for the construction
~I (page 104 )). Proof of Statement II. We can choose affine coordinates in ~51 = ~pS-H 1 such that
ITT
on VC~51
Zi(~T(Xl))
is given by (Zl,...,z5)
>
= Zi(~TII(X2) ) = O, i = 1,2,3,
(Zl,Z2,Z3) Zl V' z2 V
are local
parameters of V at x I and x2, z3 = O on Tx nl 5~5 and on Tx2 N ~5I. we can find open neighborhoods U i of x i in ~i' i = 1,2, such that V N Ui is given in U i by equations: Z 3 = Ai(Zl,Z2) , z 4 = Bi(Zl,Z2),Z 5 = Ci(Zl,Z2) , i = 1,2,
109
where
Ai,Bi,Ci
terms of degree BI(O)-B2(O),
are power one,
series of Zl,Z2, Ai(O ) = O, A.I has no
i = 1,2,3,
and one of two numbers
Cl(O)-C2(O ) is not zero
(we use x I ~ x2).
Consider
!
~6 with
coordinates
(EI,£2,E3,£~,£2,e3)
z[ = z=+e~zh+e~z~, ] J J " J D
j = 1,2,3.
and let
Consider
~2 with coordinates
q.z Taking
smaller U 1 and U 2 we can choose an open neighborhood
U of (o) in ~2x~6
such that there exist holomorphic
in U with the following V N Ui ' i = 1,2.
property:
z 3'
=
functions t
Ai(Zl,Z2,el, ~ ' '-
Let W be a complex-analytic
...,e3)
subvariety
~. l
on
in U
given by the equation:
F
=
, , [l(Zl,Z2,¢l,
. . .
It is easy to verify that ~~F 6~ It follows
and non-singular.
induced by pro~ection.
~: W
Vu
~ 6-S
=
o.
) ' ~-z(o) = Cl(O).-c2(o). 3
= BI(O)-B2(O
>~6
or
This remark
an algebraic
such that
subvariety
~: W
5
5
~IX~IX~
> ~6
6
>
be the map
¢6
it is not
variety ~ and a regular
W c Q, W is open in W and ~ = ~ W" a neighborhood
S in
V u ~ ~-S finishes
Let
Considering
to construct
see that there exists analytic
(o)
% ( z l , z
from this that taking U smaller we can assume that W is
Z-dimensional
difficult
_
' 3
~ and
~
We
of (o) in ~6 and a proper
such that either ~v
map
E ~i(u)
the proof of Statement
~'(u)
=
(d~) v is an epimorphism. II.
110
Proof of Statement III. We can choose affine coordinates V N ~
is given by ( Z l , - . . , z s )
in ~
such that ~IV on
> (ZlJZ2,Z3) ,
zi(~iv(xl) ) : zi(~iv(X2) ) = zi(~iv(X3) ) : O, i = 1,2,3, z4(x3) = Zs(X3) = O, z. = O on l on
Tx3 N ~ ,
z2 V' z 3 V
7x. N ~ , 1
i = 1,2, Zl+Z 2 = O
(corresp. z I V' z 3 V ) are local
parameters of V at x I (corresp. at x 2 and x3). open neighborhoods U i of x i in ~ ,
We can find
i = 1,2,3, such that V N Ui
is given in U i by equations: for i = i: z I = Al(Z2,Z3) , z 4 = Bl(Z2,Z3) , z 5 : Cl(Z2,Z3) , for i = 2: z 2 = A2(Zl,=3) ; z 4 : B2(Zl,Z3) , z 5 = C2(Zl,Z3) ,
for
i
=
3:zl+z2=A3(Zl,Z3) ; z4 = B3(Zl,Z3), z 5 = c3(zZ,z3),
where Ai,Bi,C i are power series of the corresponding variables, Ai(O ) = O, A.I has no terms of degree one, i = 1,2,3, B3(O ) = C3(O ) = O and one of four numbers BI(O).B2(O), BI(O)-C2(O) , B2(O)-CI(O),
CI(O).C2(O)
is not equal to zero
(we use that all Xl,X2~X 3 are different). Consider ~6 with coordinates I =
zj
Zj+ejz4+cjZ5,
!
|
I
(el,e2,e3,el,e2,e3)
and let
j = 1,2,3.
Taking smaller Ui, i = 1,2,3, we can choose a positive number r such that there exist holomorphic
functions
111
Al(z2,z3, ~i, ~2, e3, el, c~, e3) • ~3(Z~Z~,El~...jE~)
defined
A2(Zl,Z3, ¢1, e2, " " ", ¢3), for
Iz~l < r,
IEil < r~
IC~I < r, l
i = 1,2,3, which have the following property: z.1 = Al(Z2'Z3' ~ ' ' ~i' -. "" e~)
Zl+Z2 = ~3(zl'z3'el'''''E3) Let~=
[~i,z~
on
V n Ul,
on
V N U3.
' ~ i' ... 'c3) ' ~ ~3x~6' 'z3'
and W be a complex-analytic
subvariety in U given by the equations:
F 1 : z~ - A~l ( . 2 , z'3 , ~ '
i
Iz~l<=,l%l
~
I, • • •, ~ )
!
:
i
o
|
F 3 = z~ + z 2 - A3(Zl,Z3,el,...,e3) It is easy to verify that the matrix ~F 1
~F 1 ~F 1 ~F 1 ~F 1
~F3
~F
has at the point (O) the following
form
lliBiO O ClO O 41 O
B2(O )
O
C2(O )
O
0
O
O
=
O.
112
It follows
from this that taking r smaller we can assume that W
is 6-dimensional
and non-singular.
Now we have only to repeat
almost word by word the last part of the proof of Statement Proof of Statement
IV.
We can choose affine coordinates V n ~
is given by (Zl,...,z5)
zj(nv(Xi) ) = O, j = 1,2,3, z.] = O
on
Tx.] A ~ ,
zj, v,Zj.. V j = 1,2,3,
>
in ~
z4(x~) = z5(x@) = O,
Zl+Z2+Z 3 = O
z I v,Z2 V)
on
Tx~ A ~ ,
are local parameters
(j',j") = (1,2,3)-(j), Uj of xj in ~ ,
such that ~V on
(Zl,Z2,Z3) ,
i = 1,2,5,4,
j = 1,2,3,
(corresp.
neighborhoods
II.
(corresp.
j = 1,2,3,4,
x~).
of V at xj,
We can find open
such that
V N U.3 is
given in U by equations: for j = 1,2,3:
zj = Aj(zj,,zj.),
z~ = Bj(zj,,zj..),
for j = 4: Zl+Z2+Z 3 = A~(Zl,Z2) , z 4 = B~(Zl,Z2) , where
for j = 1,2,3,4
variables,
Aj,Bj,Cj
z 5 = Cj(zj,, ~..), z 5 = C~(Zl,Z2) ,
are power series of the corresponding
Aj(O) = O, Aj has no terms of degree one, B4(O ) = C4(O) =O
and the rank of the matrix
Bl(O)
I is equal three
o B2(o) o
o
Cl(O)
o
o
B3(o)
o c2(o)
o
o
o o c3(o)
(we use that all Xl,X2,X3,X 4 are different).
Consider ~6 with coordinates
(el,...,e3)I
and let
113
z~3 = zj + Ejz~
+ £jzw,~ ~'
j = 1,2,3.
we can choose a positive functions and
Taking
!
!
__%(zJ .,z'_ j..,El, ... ,e3) , j = 1,2,3,
~4(z~,z~,El,.-.,E~)
for
j = 1,2,3
for
j = 4 e
defined
(j ' ,j") = (1,2,3)-(j),
Iz~l < =, IEil < r, IE:l~ <
for
zI'] = %(z'.3,,zji,,el,...,e~)
o
Let U = [(Zl,Z2,Z3,EI,...,E~) and W be a complex-analytic
subvariety
on
,~{)
iz~l<
E ~3X~6,
Fj = Z~] - ~ . ( z j ' , , ~ j ' . ,
~F 1
~F 1
~F~
~F~
has at the point
o
that the matrix ~F 1
~F~
(O) the following
form:
O Ii Bl(°) o o ci(o) o o c2(o) O o B2(o) o o o c3(o) O O B 3 (o) O
O
O
O
O
V Cl U 4. !
r, l£iI
of U defined by the equations:
e l , . . . , E~) =
~F 1
on
V n U.3
TEil<
r,
j=
~ 4 (Zl, ' F 4 = z'+z'+z' 1 2 3 ' z2,el, ...,e 3 ) = O. It is easy to verify
r,
property:
' ' 3' = ~ 4 ( Z l , ' z2,El, Zl+Z2+Z ... ,
e
smaller,
number r such that there exist holomorphic
i = 1,2,3, which have the following
and
U i, i = 1,2,3
O
1,2,3,
114
and thus it has rank 4. It follows
from this that taking r smaller we can assume
that W is 5-dimensional. projection.
and ~ = ~ W"
We
~: W
(u) = ~.
such that W c W, W is open in
subvariety S in g such that
Vu
E ~-s,
It is easy to see that from the c o n d i t i o n
(page Z2)
IV.
I. of
From
we get that K(V) has
Let G be the G r a s s m a n i a n
5.
g of (O) in
the proof of the Statement
it follows that V is not a cone.
Severi's Theorem dimension
~ > 6
(2) of the T h e o r e m
(D. Mumford).
the T h e o r e m
be the map induced by
to construct an algebraic variety
This remark finishes
W e proved part
~6
see that there exists a n e i g h b o r h o o d
~6 and a proper analytic
(3)
~: W
It is not difficult
and a regular map
-i
Let
(complex)
of projective
lines in ~p5
and 2
I~ 1 G
be the canonical canonical
diagram c o r r e s p o n d i n g
line bundle).
f: V X V - ~ line b u n d l e
~G
Constructing
(4 is the diagonal).
induced by ~i under
f and
to G (~I: ~
~ G is the
chords we get a regular map Let
!
~i: ~'
>VXV-~
be the
115
~,
be the induced diagram. in ~p5 of
-2F(~').
Now considering
F
Evidently K(V) is the algebraic closure
Because
~2F: ~,
dim~K(V) = 5, we have K(V) = ~pS. > ~p5
and using Bertini's Theorem
we get that for generic line ~ c ~p5 (n2F)-l(~) |
algebraic curve in Z'
Let Pl: VXV-~
by projection on the first factor.
where ~ :
V
Theorem.
be the map induced
It is easy to verify that
~ ~P~ is the projection with center Z.
the algebraic closure of p ~ ( ~ 2 F ) - l ( ~ ) is such that
>V
is an irreducible
~
in V.
Let ~ be
We can assume that
satisfies to the parts (i) and (2) of the
Then we get from (*) that ~ = ~ I ( s ~ , ( V ) ) o
Thus
"~I(s n (V) and S~ (V) are irreducible algebraic curves.
We proved
part (9) of Theorem i. (~) (D. Mumford) From Corollary 5 and ~ of the Duality Theorem we get that
dim~(V)
= 4.
Let
~: V
~ ~ ~P~
be a projection which
already satisfies the parts (i) and (2) of the Theorem and Z~ C ~pN be the center of ~.
Because codim~Z n = 4 we have that Z~N ~(V) ~ ~.
But that means that ~(V) has pinch points.
Q.E.D.
PART II ELLIPTIC §i.
Deformations fibers.
of elliptic
Definition.
Let
of a (non-singular) complex
surfaces w i t h
f: V
complex
~ A
elliptic
in the fibers of f. analytic
has singular "multiple
8.
f: V
multiple
fiber which
f-l(x)
Kodaira we call
~ A
fibers
curves
is a
("singular
(being reduced)
exists a commutative
f: V
diagram
~ 6
or an elliptic
be an elliptic
curves an
fibration.
fibration which
multiple
fiber" means
is a singular
of proper
map
compact
curve and there are no exceptional
Following
Let
x ~ A,
singular
holomorphic
surface V to a non-singular
fiber space of elliptic
Theorem
"non-stable"
be a proper
curve A such that for generic
(non-singular)
there
SURFACES
curve").
surjective
Then
holomorphic
maps of complex manifolds h / W
where
~ : {T @ ~IITI
coincides
with
F h_l(7):
h-i T/t\
singular
f: V
multiple
> p
~m
! I}, such that > A, for any
-l(T)
Flh_l(o):
h-l(o)
>
p-l(o)
T E ~-(O),
is an elliptic
fibration without
fibers and h }]as no critical
values.
Moreover,
117
the number r(7) of multiple fibers of F h_l(7) and the set of corresponding multiplicities [ml(~),m2(~),.-.,mr(~)(~)~ do not depend on • Proof.
< • ~
O)
The Theorem follows from the following
Lemma 4.
Let
f': V'
~ ~
be a proper surjective map of
complex manifolds with a single critical value (O) E ~ for any
~ E~
-(o),
such that
f.-l(~) is a non-singular elliptic curve and
f'-l(o) is a fiber of type
mXb , m h 2, b h 1
l)
e' > O, O < e < i, and a commutative
There exists e,e' E ~ ,
(see [13]).
Then
diagram of surjective holomorphic maps of complex manifolds W'
, < where
p,
~cx~,
"~6' = [ ~ e ~ I I~1 < ~ ' ] , ~
P': ~eX'~)E' such that
> ~C'
is the projection,
F'lh,_l(o): h'-l(o)
f' f,-l(~):
f'-l(~ e)
F'lh,-l(~): h'-l(~)
= [~E¢ I ~ 1 < c ] ( 9 c c D = ~ 1 )
'
~ De ,
b p'-l(o) for any
F I is a proper map coincides with
T E D e ,~ O 3
>p'-l(T) has as generic fiber a non-singular
elliptic curve, F'-I(o,T) is the unique multiple fiber of F,ih,_l(T) : h,-l(T)
> p.-l(T) whic h is of type mIo
non-singular) and h' has no critical values;
(that is,
118
2) there exist Cl,~ 2 ~ i~ O < E2 < eI < c
and a commutative
diagram of holomorphic maps
iv
D~~
(2)
'
p
where
~
K : [q~l£2
~
*D~
iV and iK are biholomorphic embeddings,
and p~ are evident projections, V~ = f'-l(K), I
F~ : (f' v~)X(ident~ty)'
iK KXO: KXO
coincides with the natural embedding
, V~XO: VK×O embedding
V~
> F ,-l(Djo
~ D £ ×O K
>~£
and
) (: V , ) coincides with the natural
~ V'.
Suppose for a moment that Lemma 4 is already proved.
Then
the proof of Theorem 8 proceeds as follows: Let al,-..,a r
be all the points of A such that
V j = 1,2,.-.,r,
f-l(ai) is a singular multiple fiber, ~ e, j be a small open neighborhood of aj in & such that and V t
~ ~e,j-aj,
~E,j
is isomorphic to D e
f-l(t) are regular fibers of f: V
Vj' = f - l ( D j ), f'j = f ~. : V!]
~ ~E,j"
Using Lemma ~
~ A, and taking
119
e sufficiently small, we construct for all fj: Vj! diagrams analogous to the diagrams (1) and (2).
>
D E,j
the
We shall use the
same notations as in Lemma 4 for the new objects which we obtain adding only index j (that is, we have
W'j,F'j,hj.'.. and so on).
Without loss of generality we can assume that
~¢,,j
depend on j and we shall use
~£,,j.
a local parameter in identification of
D e , instead of
does not Let ~'3 be
~ ¢,j which gives the above-mentioned
De,j
and D
I~j
e (that i s , D e , 2
~ j = [ : j e ~, I : j l £ e 2 , ] ] ' u = ~ - jU__IDj,
Wl u = u x D e "
W u = f-l(u)xD£,,
F U = (f f-l(u))X(identity):
and
e'' PU: WIU
hu: WU " ~
~e'
w U ----->WIu
are evident projections.
We have evident embeddings !
K xD e J
> uxD e (:WlU) !
!
(Kj = [~j ~ ~I
h~j ; ~nd
~ vI~
~24<
×D ]
vK xP~. ].
•
>f-1(u)x~e,(:w)
l~J I < el,J ' el,J < e, 62, j > O])
, P~. = Pu IKjxDe
Taking
e', el, j smaller
E2, ~J greater we can suppose that
an embedding
i--Kj: K j x D e '
(where K'3 = {~J E ~IE2 --< I@j
~"
iK. could be extended to 3 Dc,jxD c , E1 ]
and that
Now (Dc,jxDe,) 3
and
- i~.(KjXDe,) ]
120
has two connected components.
Denote by U' one of them which J Let WI, j be the interior part of
does not contain (0)×26,.
(Dc,jxDc)
,
- u'. j , w (j)
Fj = F'Iw(j): j w (j)
=
Fj,-i (Wl, j), pj
=
p j,
Wl,j
,
- > w I ,J, h J = h'J W( j ) . We can consider
iK. (corresp. iv, j ) as biholomorphic embeddings of K j X ~ E , J V' , (corresp. K X ~ E ) into WI, j (corresp. w(J)). Using the J above-mentioned K j x D 6 , > Wiu (corresp. V'K.×DE, >Wu) J we form new complex manifold W 1 (corresp. W) which is the union
r
wlu
~
r
(j=l t*
r
Wlj
)
K X D , = ~ iK (KjXD ,) j=l j c j=l J
(corresp.
(~ w(J))).
WU r
r
xD j=l
NOW we identify
J
j=l
=
xD j=i ~v,j
WIU , WII,-.',WIr
)
j
(corresp. Wu,W(1),...,w(r) )
with the corresponding open subsets of W 1 (corresp. W). F: W
>~i' p: W1
> D£,,
h: W
> ~E'
as follows:
We define
121
F(x) = ~ Fu(X)
if
x E WU
Fj(E)
if
x E W (j)
p(y)
=
I pu((y) if pj(y)
y
if
Wiu
y E Wl, j ,
hu(X), x E WU, h(x)
=
hi(x), x E W (j)
Using commutativity
of (1) and (2) of Lemma ~ we can
verify that F, p, and h are well defined holomorphic maps and that the diagram: W
D E ,<
W1
P
is commutative. It is easy to see that this diagram satisfies all demands of Theorem 8.
Proof of Lemma 4.
Q.E.D.
Let
~ (w) be the modular function (the V
"absolute invariant"). exist
Kodaira remarks in ([13], §7) that there
N O > O and a convergent power series
~(z)
such that
122
(w) = e -2~iw + ~ Let E > O and
(e2,iw)
Me = ~(~'') ~ ~2
for
Im ~ > N O .
i~ I < E, l~I < ¢].
For £
s u f f i c i e n t l y small we can define the following holomorphic function in M £
~
n
•
I+( qnl-~nl ) ~ ( ~nl-~ nl )
where the positive integer
nI
Let X be a complex variety in
will be defined later. M ×~p2 6
defined by the equation
ZoZ22 B ~z31 ( ~ -' 3ZlZ2o T ) z- 3 =
where ~:
(Zo:Zl:Z2) are homogeneous coordinates in ~ p 2
X
M X~P 2 £
> M E be defined by restriction of projection n I n_ ~ M . Let z = ~ -T £. Evidently we can write E
B(~,T) = B(z) = l-ZBl(Z), series and Bl(O ) ~ O.
Let
where we fix a branch of M ~ = [(@l' ~i ) E ~2
• i~
o
where
Bl(Z ) is a convergent power
~[ n~ m ~ ~ " ~i = @ V B l ( = x-T *)' "i = T ~ l (
~n I n l i -
V B I ( Z ) j and
i~ll < Z,
ITlI < ~],
~
is a sufficiently
small positive number. Taking c smaller we can write the equation of X as follows: 2 31 2o nl _nl~ 3 ZoZ 2 = 4z - 3ZlZ + (-1 + ~i - ~i )Zo
123
For any t E M
-l(t) is a cubic curve in ~p2 which has
E
no singularities on the line z = O. ~ - l ( t ) is singular if and O n1 only if (Ql(t)) nl - (~l(t)) = O (or equivalently (a(t))nl-(T(t)) nl = O).
I Let M E' = {(~,T) E ME ~ntTDi~ O},
Zl x = --,z O
z2 Y - Zo
Taking E smaller we can assume that there exists ~ > O such nI n1 that the equatiQn (on x) 4x3-Sx - i ~ i -~i = (9 has exactly two roots Xl(@,T),x2(=,T) with L e t D ( - i ~,r) = Ix ~ • I I~x
Ixi(Q,T) + I < r}
and
< r when (~,~) E M E . y(~,T) be a segment of
the straight (real) line in D(- ~,r) 1 connecting
Xl(~,T ) and x2(Q,~),
81(Q,T ) = < (x,y) E ~2 Ix E y(@,T), y2 = 4xJ-3x - l ~ lnl -~lnl
>'
!
(~,~) ~ M e • cycle in
~l(~,T) is a homologically non-trivial 1-dimensional
~-l(@,T).
Fixing an orientation of 51(@o,To) for some
(~o,To) E M'
c we can check that for any close ]~th M in M'E with
origin (#o,To),
51(@,T ) has continuous changing along ~ and
returns to 81(#o,T ~ with the same orientation.
Now we can speak
about a continuous family of oriented 1-dimensional cycles
Let ~(~,~) be a holomorphic differential on
~-l(~,T),
(~,T) E M~, which in the part of ~p2 with z ° ~ O is given by
124
~(=,~)
~ --~ ,
-
WI(¢,T ) =
~(~,~).
Wl(¢,T ) is a single-valued
~IV¢'T) holomorphic
function in M'E without zeroes.
for some (G' ,~') ~ M' e
I
~(~, ,T')
= O
we would have
(~ ~ 51
and
(If WI(¢',~' ) = O
51(~',7' )
~(¢',T') = O, ',7')
is homologically
zero).
61 ¢',~') Let
~be
a small open neighborhood
~52(~,, ) ~
~-l(~,T>,
(~,T)E ~ ]
of (@o, To) in M'£J
be a continuous
oriented 1-cycles such that for any (G,v) ~ , generate
denote
HI(~-I(~,T)j~)
w(~,r) =
and
of
m(¢,~)
path
along
~
I ~) ~/~l(~'~) 62( ,
that it is a multi-valued for any close
Im[
7 gives
If t = (¢,~) E M e following equation:
~(@,T)/~l(Q ,T] > O.
~(¢o,~o)+N
~-l(t)
Zl -- , z o
y=
z2 --: z o
analytic
is given in ~p2 by the
ZoZ~ = ~z~-3ZlZ~-B(¢'v)z3 o
y
2
= 4x3
-
3x
We get that absolute invariant of
-
continuation
from W(ao,To) , where N ~ ~.
homogeneous coordinates, x=
We
function in M'£ such that
Y ~ (~o'~o)
then
61(@,T),82(@,T )
and get by standard arguments
holomorphic
y , i n M'e"
family of
B(~,~).
~-l(t)
or in non-
125
(t)(=
g•2 >
27
~)
g2-27g~
27_27~2(~ ~)
=
nl 1 ~
n1
Fix ~ E ~w,
~
We see that
~(t)
+ ~
_ l ~ n l - ~ n l ) ~ (~nl-~ll
1
@nl _ Tnl
I-B2(~, ~)
(~nl-~nl) •
~ I, and denote
J~
results of Kodaira desingularization
has in (O,O) a pole of order n I.
it then follows that if Z of Z
then Z
From
is the minimal
has over (O,O) a singular
fiber
of type I n 1 and (~(~,,,1 = 12~i nll°g T + f(,) , 0 where e is sufficiently in O !
J~J < ~.
Im W(~T,T) > N O
< I'I <
T,
small and f(~) is a holomorphic
function
Taking ~ smaller we can assume that (for O <
l~J < ~)
and thus
J(~T,T) = e-2~iw(~T,~) + ~ (e2~iw(~T 'T)) (for O < i~l < ~). 2~iw(~,~)
Denote
q(@,T) = e
q(~,~)
= Tn~e2~if(T)(O <
holomorphic and
~ : D~
, p(~,T) = @
- • nl.
We have
ITI < ~), that is, q(e~,~)
function in O _< >~2
nl
J~J
< ~.
Let D~=£
is a
[~E~O<J~J<~]
be a holomorphic map defined by
126
> (q(~7,9),p(~7,7).
=
+ ~
we have that
Since
(q(C~T,T))=
]
=
(C~',~) = ~
~e(I~E ) is contained in the subset
where we suppose that E and ~l are sufficiently ~I sufficiently
small
C
is a non-singular
+
C~ of ~2 given by
small•
But for
analytic curve and
because it evidently contains the set = C(P,q) ~ ~2 Ip = we have C
= A.
q,
IpJ < ~e l,
We see that p ( ~ , ~ )
lql < ~i} = q(~T,T)
for 0 < ITl < ~.
Hence q(Q,T) I~u = (~nl-Tnl) l%l
Let z
=
{(~,,)
~ oe
Gnl-~ nl
and
q(~,T) = ~nl-,nl
= o], z~ = z e
be a cyclic group of analytic automorphisms by the transformation
(~,7,w)
> (~,T,(
in
De•
-(O,0),
of M' = M-S generated
nl • n l w ) ).
Using
Kodaira's remark in ([13], p. 597) we construct from M a n d r e new complex manifold which we call Kodaira factor-space and denote by M / ~
(holomorphic map ~: M
> M/~
is given by:
~(z') = ~(z"), z',z" E M, if and only if either z',z" E M' and
127
z' = g'(z") for some g' 6 ~
or z' = z").
(It is easy to
verify that the condition for existence of M / ~
indicated by
Kodaira in [13] is satisfied in our case). Let P 0 (@'T) ~ ~ -l((~'T)) be given by Zo(P o
(=,~)) = o,
Zl(P °
X(o) = [(~,T,Zo,Zl,Z2)
(~,T)) = o,
E XI($,T ) E M"c
z2(P o
and
(~,~)) = i,
) is (Zo'Zl'Z2
a
f
non-singular
point of ~ - I ( @ ' T ) }, ~t = ~ - l ( t ) ~X(o) ' t = (q ,T)~M"e"
Evidently we can consider ~(~,T) as a holomorphic ~t for any t E M;. 81 c (~',~') in
Take (~',T') , T e.
~(~',9')
differential
We can define a 1-cycle
such that 61(Q',T' ) could be connected
with the previously defined 81(~o,9o) by continuous 1-cycles in
~-l(t),
on
family of
t E F, where y is a path in M~
(~o,~o) and (~',T') and Int(Y) c M e - ~ .
connecting
A direct verification
then shows that
t
61 ~',T')
~(~',~')
/
o.
We see that wI(~,T ) is actually defined in (@,~) E Me,
Wl(~,T ) # O.
(single-valued)
(~,~) Let e
(This proves also that wI(@,T ) is a function in M e and Wl(~,T) ~ O,
E 11c).
~I(~,T) = Wl(1,~)~(@,T),((@,T)
2~Ti~(~,,)_-- ~ nl
¥o: X(o)
holomorphic
M e and for each
>M/~
~ nl
~ Me).
Using
we define a holomorphic map
by Y o ( a ) =
[@(a),,(a);
exp(2wi
~
~l(@(a)~ ~(a)),
p o (@'(a),'r(a))
128
where
is taken along a path in
C(~(a),~(a) )
~o(~(a),~(a)) (~(a) = ~ ( ~ ( a ) )
J
T(a) = T ( ~ ( a ) )
onto and a 1-1 map.
Hence
?
•
It is clear
that
is an i s o m o r p h i s m
YO
is
of complex
O
manifolds.
We shall identify X(o ) and M / ~
construction
of logarithmic
that a certain residue fiber of type a topological
transform
class
k mod m
is defined
uniqueness
prime to m.
construction
of a n e i g h b o r h o o d
shows
for a multiple
for such a k could be given.
will be important only that K o d a i r a ' s analytical
(see [!4], p. ??O)
mIb , b ~ i, k is relatively definition
Kodaira's
(Actually For us it
shows the
of a multiple
fiber
mIb , b ~ I, when k is fixed). Take the corresponding
k = k(f')
given in our Lemma and let k' E ~ Let d = g.c.d.
n = mb, n I = mb I.
of nl).
Denote
Pl = exp(2Wi kl). For j = l,...,nl-1 , let n1 yj(~,T) be two holomorphic functions in S~ defined as
follows:
x (~,~) : ~l([~";w~ (~'~)])
>D
be such that kk' ~ 1 (mod m).
(k',b), k' = kid , b = bld ,
(That is the definition
xj(@,T),
for the family f': V'
6
129
where L=I
w0(°") I~j~i nI [ (~-p ") if for some ~' E (1,2,-
,j),
L'
~-Pl ~ = O
We use the following remarks:
~°(P°(a"')) = [:,~;l]
X(o ) n (Zo=O) = (=,)~ M" P°(='~)" and for ~ <
c
1
n 1
(that is, for (~,~
and for (~,T) ~ ~E--~" nl
wj(~,T) ~ I, Zo([Q,T;w j (=,~)]) ~ o,
E ~',
~
an equality ~nl Tnl) N
7-[(~-pl ~) = (
-
, N~=,
~=j +i gives first N > 0 and then
1 =
(~nl-Tnl)N
~=j +i Contradiction).
1
ion n i O: i<
i.
130
Since
~6-M6
= (O,O) we see that
xj(~,T),yj(~,T)
are
holomorphic functions defined in For (~',T') ~ ZE' ~ - I ( ~ . , T . )
is the same rational plane
curve, say C, given in ~p2 by the equation 2
3
2
ZoZ2 = 4Zl-3ZlZo -
z3
o
Hence we can assume that BI(~',T' ) chosen above for (~',~') E Z
is the same 1-cycle on all ~-I(~',T'),
(~',T') E Z"
E
£
(that is, images of 51(~',T' ) by
Me
X ~p2
> ~p2 coincide with
some 1-cycle DIC on the non-singular part ~ of C). zI Xc = -~o C '
z2 Yc = -~o C "
~C =
dXc YC '
~iC =
Now define by
~C" DIC
IC
=
1 ~C ' wlC
Po = (Zo=O'Zl=O'z2 =I) E C,
P Wc(P) = e2~i ~ ~{C Po where p ~ ~
and
Yc
T = ~ Xc+
P ~ is taken along some path in 5, Po T2
(that is, x C = ~--+i, YC
= T(2+6)
~
).
A direct
calculation shows that (changing if necessary the orientation of 81C ) we have
131
Wc+l (£)
-3Wc
1 ~c - 4(Wc-1)2 - ~'
T = iF~Wc_l , Take
A~
Z' E (l,2,.-.,nl)
= [(=,~) ~ M ~
xj L' = x j ( ~ , T )
A~ "
•
-3i~6wc(wc+l) Yc
and let
- Pl ~ =
^~, = ^~,-(o,0),
o],
YJ'~' = YJ(~'T)
A~ ' wj~, = w j ( Q , T ) I
,
Since wj~.
>O
~(Wc_l)~
=
,
or oo when (~,~) --->(O,O)((Q,T)
~ A~,)
A~,
we see
from (i) that lim x
1
lim
~ ~)--,X::p)J, ~' = = ~' ~,~)~A~.
yj
~,~).->(o,~ ~,~)~%.
"~'
= O
1 This shows that xj(O,O) = - 3' yj(O,O) = O. Now let m(x) be a (single-valued)holomorphic in some neighborhood Let
y' = y + ~(x)(x+
1 of x = - ~ by , x
,(x) = 2~x-~-l and
= y-~(x)(x+
,(-~) = iy~.
(y,x' are some local
parameters at the point (x =
-
x (O,O)j = - ~'
we see that (taking E smaller if
necessary)
yj(O,O)_ = O
31'
function defined
y
the following holomorphic
=
O) of ~2).
From
functions in
~£
are well
defined: , yj(~,~) = yj(~,~) + , ( x j ( ~ , ~ ) ) ( ~ j ( ~ , , )
1 + ~),
Xj(Q,T) = yj(Q,~)
+ ~).
- ~(Xj(~,T))(Xj(~,T)
1
132
Because
y~(~,,) : ~(o,,)
3xj(o,,>
-
B(~,,)
-
we have y](~,7)Xj(~,T) where e(G,T)
= (~
nI
-7
n1
)e(.,T)
B(~lT)-i
=
Qnl_Tnl that is,
e(Q,~)
is holomorphic
A,,,
Define on
~E
in
Tj,~,(~,T)
and e(O,O) ¢ O.
=
From (i) we
Xj,~,(~,T)~ " see that on A£.
wj,~,(~,~)+l
Tj,~,(=',~) For
£' E (l,2,''',j),
wj,z,(o,,)
= i]/g
we have n1
=
wJ, ~,(~,~)-i
~
~-~).
~=j+l Thus lim
A~,~(a,,)->(o,o)
w
j,~,(o,
7)
=
O
and
lim
^~,~(~,~)-->~,o)
T
J'~'
(~,~)=-i~.
2 Since
and
T23,~.(~,~) = [~(xj,£.(~,7)] 2 (we use
1 = i~ ,(- ~)
=
y_
we have
T2
YC
(Xc~)2 = 4(Xc-l))
~(xj,~. (g,7)) = -T j ,~,(Q,~)
+
1
O.
We obtain
and
133
J y](q,~) = T~(~-p.T))e.'(@,T)
in M E Now let
where
#' E (j+l,--.,nl). J
wj,~.(c,,~)
= l--[(=-~l$)
%
We have
~,~,(~,$)
x.(~,,) I J
:
A%,
.
and
= Tj,~,(Q,T)
is holomorphic
We have -i
lim w. (q,T) = co ^~,~(',W)-->(O,O) J'Z'
e'(~,T)
Thus
lim T (~,$) = ~ . A~,~(a,W)-->(Op) j,L'
and
yj,,, - Tj j ~ ! (xj,~ s +~I
We obtain n1 J where e'~(~,~) j
%=j +I is holomorphic
in
M
,
(~nl-,nl)e(Q,,)
Hence
= yj(",')X;(q,$)=
e(Q,~) --- e ' . ( g , T ) e ~ ( ~ , ~ ) and J X has o n l y one s i n g u l a r p o i n t
e'
Now
nl
~T(@-91"))e;(~,')e~(@,'). %=i ,i
e'.(O,O) ~ O, ej(O,O) ~ O. J
134
We can find an open neighborhood U of ~ in M e X ~ P
2
such that
(~,T,x',y') are local coordinates in U (x
,
1
= y-,(x)(x~),
y
,
1
Zl
= y~(x)(x~),
z2
x = i--' y = ~ ) " on o is given in U by the equation x'y' = (~nl-T l)e(~,T).
~
= X n U
Now we take nl-i copies of ~pl, say ~P~, i = 1,2,...,nl-l, with homogeneous coordinates (~Oi: ~li~ i = 1,...,nl-1 , and nl-i 1 consider in U × ~ - - ~ P i a subvariety YU defined by the i=l following system of equations: Y'~o1 : ~ll(~-Pl T)e(~'~) =
~l~Oi+l
.
.
,~
i+l~
~Oi%li+l ~ -Pl
),
i = 1,.. ,nl-2 ,
X'~lnl_ 1 = ~Onl_l(~-7 ). It is easy to see that
Pru(Yu) = ~ .
Let
the map induced by projection, XU, T = [a ' ~ ,
Yu,, =
s-l(~jT).
s: YU ---->~ be 7(~(a)) = ,],
Standard verification shows that YU is
non-singular, T.s
s s-l(x_~): s-I(Xu--~) >Xu~ ~ is an isomorphism, U is a holomorphic function on YuWithout critical values, and
SlYu,o: YU,O
>~,0
is a minimal resolution of singularities
of XU, 0 (that is, there are no exceptional curves of first kind of YU,O in s - l ( o ) ) .
135
Now identifying
s-l(x - ~) with X - ~ we get from X-@ and YU U U complex manifold Y with a holomorphic map ~: Y > X such
a new that
(i)
s ;~(X-@):
(ii) T.s
~-I(x-')
is a holomorphic
and (iii) if
>X-@
function on Y without critical values
X T = [a E X, 7(~(a)) = ~], YT = ~-I(xT)'
~Iyo: Yo -->Xo is a minimal resolution Let
' = ~
°;' ~ T ' '
De,~, = [(@,T) E M e ~'-l(a) of
=~'
singular
~
, : YT'
t
~T'
fiber of
of singularities > DE,W,
of X o.
where
For any a 6 De,T,-De,7,
n ze , fiber
,' ~ O, % = O,1,''.,nl-i , are
which have type I 1 and ~' 0 : Y 0
> De
Denote by C (O) the closure of o j = l,...,m-l,
then
elliptic curve (and a regular
T',7'),
fibers of
is a singular
YT
T = T'].
is a non-singular T.),
is an isomorphism,
,0
~-i((o,o))
which is of type Inl -
~-i(~-i(o,o)-@)
in Y.
For
let C (j) be an algebraic curve in YU defined by o
the following system of equations: ~li = O, i = l,..-,j-l,
~o~ = O, i' = j+l,---,nl-l,
x' = y' = Q = T = O.
It is clear that irreducible
C(°)'c'l)o (0 '''''Co( hI-l) are all the
components of
~'-i(o,o).
We assume C (nl)-- C (O) and o
(i) = c(i)N c(i+l) let qo o o '
F(i)
o
_(i) (i) _(i-l) i = O,1,.-.,nl-i , = Co -qo -qo '
136
=
U
i=O Take a point a E ~(J)'o j E (l,2,...,nl-l).
We have
~lj(a) ~ O• ~oj(a) ~ O.
Let U a be an open neighborhood of a in
YU (c y) such that
E Ua,
Va'
have a holomorphic
~lj(a') ~ O, ~oj(a') ~ O.
function in Ua,
uj = ( % )
YU"
We
Our
~0] differential ~(~ T) = d/~ which is defined on each ~t
t ~ (O,O)
can be rewritten in U a by the following way: First we have y = ~' 2+x' ' y' -x' = 2 ~ ( x ) ( x ~ ) . ~(_i) ~ O
there exists a holomorphic
some neighborhood
~(z) defined in
of z = O such that ~(O) ~ O and
1 1 x + ~ = ~(y'-x')~(y'-x'). dx Y
function
Since..
Thus we obtain
[*(Y'-x')+(~'-x')*'(Y'-x')](dy,_~,). y'+x'
Because differentiation
here is along the fibers of
~',
that is,
when Q and T are fixed, and on YU n
n
y'x' = (a I_~ l)e(~,~ ) we have
x'dy'
+ y'dx' = O.
Hence
= L,ky, r~t_ X,)+ky, \ I_ X,)w,ky, ,~ f_ X,)j ,i dy' y, But in Ua,
y
,=
uj
~J
(q-p~)e(~,~)
and we get dyy,'= dUu1 3
and ~ = [~(y'-x')+(y'.x')~'(y'-x')] du~ U.
3
137
This formula shows that we can extend ~(o,7) also to C. ~ du. J where it is equal to ~(O) u.) J Let 5~)'-lube a 1-cycle on ~tJ)o' " defined by lujl = 1. verification then shows that we can include
D )~ "
family of 1-cycles {5[J)(~,~),
8[J)(O,~)E
such that for (~,~) ~ O,
(~,~)E ~],
in a continuous ~(~,,),
~[J)(~,~) is homologically equivalent
in C(a,7 ) to the above constructed
51(a,T ).
) we see that we can extend
Because
du.
o oo) !ioo0)
, 01 0u
lujl=l
J
~1~(~,7)(G,7) ~ O) constructed above
also to the non-singular part of
~'-i(o,o)
holomorphic family {~i(~,~),
(G,7) E ~ e} ,
holomorphic differential on
~(~,T)
Let
A direct
~(J), j = l,''',nl-i ,
such that we get a where
~l(q,~) is a
(~(O,O) = ~O )"
be an analytic surface in X
given by the system: zI
- ZoXj(~,~)
=
0
z 2 - ZoYj(q,7 ) = O. Denote by P(J) the closure in Y of ~-I(~(j)_s). in YU by the following system of equations:
P(J)
is given
138
J
~
,
y' = ']--[" (0'- Pl'r ) ej (0', T ) , .f,=l X' =
(O'-Pl'r)ej (0', 'r) , ~=j +i
j +i ~ll = ~ ( ' - P ~ ) Z=2
e"-l~ol ' 3
~
7-]-
~oj+l
~=j +1
nl
~lj-1 =
T/'[( ~-plT)e~-l~oj_l, ~j
,,
(~-Pl~)ej ~lj +i ,
~
,,
~Onl_l : -F--~ ~-Pl~)ej~inl_l ~=j+l
~lj = e':-l~oj ]
It is clear that P(J) is a cross-section of and that
u~': Y
> ME
p(J) n ~ '-l(o,o) E ~(J) 0
Let ~(o) = ~a E X Zo(a ) = 0], p(O) = ~_i(~(o)) and p(i)(@,T) = p(i) n ~'-I(@,T),
(~,T) E ME,
y(i) = ~_iCX(o~) U ~(i)o , It is clear that {Y(°),...,Y (nl-l)]
i = O,l,...,nl-l. form an open covering of
~(~,,)
Y(o) = ~-l(x(o)) u ~ = ~J
Let M 1 = ~e×~*, S 1 = ~£X ~*, ~ !
be a cyclic group of
analytic automorphisms of M 1 = MI-S 1 generated by the transformation
139
(~,T,w)
>
(G,~,(~
nI
-~
n1
)w).
It is easy to verify that the
Kodaira condition for existence of Kodaira factor-space M1/ ~ is satisfied here (we gave references and little more explanation on page 126). Using e 2ni~(~,T) = ~nlTnl , we define for i = O,. -- Jnl-1 i(i): y(i)
holomorphic maps
¥(i)(a) = [(~(a),T(a);
where
M1/~bY a exp(2~i q ~l(g(al,T(a)))] (i~ ~), T(a)
~ is taken along a path in ~(a),~(a)) N y ( i ) P(iI~(a),• (a) It is clear that each ¥(i) is onto and 1-1 map.
~(i) is an isomorphism of complex manifolds. a Denote w(J)(a)= expC ~ ~l(~(a),T(a))), pTJ)(~(a ),T(a) a
Hence
a E Y(J)
is taken along a path in ~(,(a),T(a)) ~ Y(~)' P(J)(~(a) ,T(a)) j = O,...,nl-1. b Let P2 = Pl" hlj: Y(J)
We define analytic isomorphisms: > y(j+b)(c y)
where the index j is considered as an element of the cyclic group ~i
= ~Z/(nl) of order nl, that is, Y
(nl) y(O) y(nl+l) y(1) etc = , = , .,
140 by
~2)
O'(hlj(a))
=
p20'(a)
"r(hlj(a))
= "r(a)
w(J+b)(hlj(a)) = w(J)(a)pJ~, where = exp
,Tik' "b
0"- I )
Let a E y(]) n Y(]') j ~ j' (that is, (~(a),T(a) ~ (O,O)) We shall prove that
hlj(a ) = hlj,(a).
Let aj = hlj(a),
aj, = hlj,(a ). We have
w(]+b) ( a' J+ )b j) = w( (a j. )
expI2wi( f~x~
a] ~ ~I(P~~(a)'~(a))-
~'P~'J~'(p2o (a), 'r (a))
,
aj , ~ al( P2°(a)'~(a))~ 1
p(j +b)( p20.(a), ,r (a)) p(j ' +b) ( p2.(Q.(,a),r(a))
a]
= exp(2~i ~ ~l(P2~(a),T(a)) .exp(2~i J~ ~l(~Q(a),T(a)) a j, p(O)(p2~(a) 'T(a))
p(j+b) (p2(G(a),T(a))
(o)~
~l(02~(al''r(a) )
P (P2O(a),T(a)) a
.
~3 = exp(2~i
aj,
~I(P2 (Q(a)'~(a))
where we assume w
o
= w
nI
= i.
w~'+b(P2 ~(a)''(a)) w +h(p2~(a),T(a)) j
'
141
From (2) we get w(J+b)(aj) w(J'+b)(a~)
_j-j' wCJ)(a) = p2j-j' w ,(O(a),T(a)) #J'~a) w J (Q(a),,(a))
=
~2
nl
))~~0
Without loss of generality we can assume that (~(a)) -(7(a
(that is, we have to prove only that the maps hlj and hlj , coincide on an open dense subset of Y(J) N y(J'~.
We have
j+b wj +b( p2q(a), ~(a) ) = T ~ ( ~2Q(a)-Pl7(a) )-l(mod((~(a))nl-(7(a) )nl)) L=I
(where congruence is considered in the group ~ ) , J
wj(~(a),~(a)) = ~T(~(a)-Pl~(a ))-l(mOd((Q(a)) %=1
nl_(~(a))nl).
Because j+b 4=1
%=1 J+b T ~ ( ~ _ o ~
=
P2
"~I-b
)
J " q--[(~-DlT) ~= 1
we obtain that Wj+b(P2~(a),~(a))
wj(~(a),~(a))
o
_-- p2 j-b
and (by the same reasons)
n
n
]~T (~(a)-P~'(a))(rood((~(a)) l-('(a)) I)) ~=l-b
142
Wj+b(P2~(a),~(a),=--p~j '-b wj,(~(a),~(a))
0 n ))nl) ,.~(o(a)_~l~,r(a))(mod((o'(a)) I_( 'r(a ~=l-b
Hence
wj. +b(p2~'(a) , 'r(a) wj +b(P2~(a) , 'r(a)
9~_j. w~.(~(a),~(a)) w (~(a),T(a))(mod((~(a)) n 1 -(~(a)) nl) ]
a
and
J
exp(2??i ~ ~l(P2(~(a),~(a)) ) _= l(mad((~(a))nl-(~(a))nl). aj' We see that
aj = a'j, that is, hlj(a) = hlj .(a). |
Now we can define an analytic isomorphism hl: Y(o) by
bY(o)
hi(a ) = hlj(a ) where a ' Y(J). For ~,~) ~ (0,O) let q'(Q,~) be the singular point of C(~,T )-
. We extend h I. to .hl: .Y(o) by
.>Y~)
hl(q'(~,')) = q'(p2~,'),
- , where ~o)-¥(o)U ((~L 2T ) ~ M ~ ~,(a,~))
hl
= h I.
In order to prove that
Y(o) il
h I is analytic, it is enough to prove that h I is continuous. [a (r), r = 1,2,...], lira a (r) = q'(~,,). r-~0o
be an infinite sequence with a (r) ~ Y(o)' We have to prove that
Suppose that it is not true.
~hi(a~))=q'(p2Q
' a (rp)) , p = 1,2,...,} such that
~moohl(
5( = a' ~
a(rp)= p-~colim
, ,).
Then there exists a subsequence
[a (rp)
We obtain
Let
hi-l(a') ~ ~(a,T)"
Contradiction.
p2~,~).
143 Now we extend h I to hl: Y --~Y ,~qo(Iml)=
J = o'l'''''nl-l'
by hl(q~J) ) : q
q~O)' qo(nl+l)
_(i) ' = qo
etc.)
'
As before, we have to prove that h I is continuous. it is not true at some q(oj), j E (O,l,...,nl-l). infinite
sequence
relatively
and
(o
: r = q j)
U r' of
Ur+l
~' r
U' r
that if
[ar, r = 1,2,--.]
ur r L ( ~ ] q
CUr,
that
Take an
)
%=O
(J) = qo
for any r = 1,2,.-..
hl(Ur-q(oJ)).
=
closure
mr E ~(J) o ' ~ ol'm o
Suppose
of q(oj ) in Y, nl-i (o~)
"
Ur-q(oJ) is connected Denote
hl Yio) = hl" "
[Ur, r = 1,2,.-.} where each U r is a
compact open neighborhood
o0 ~ r=l
lj+b),
It is easy to see that the
is compact.
A direct calculation
is an infinite
a r = q (oj) then
r l ~"
Hence
shows
sequence with
hl(ar) = q(j o +b) "
co r= 1 nl-i
~D
Suppose there is a point a' ' ~ ~' r=l r
such that a' ,
~ _ 0 q(o~),
that is,
h~-l(a integer r
') ~
o such that
{a'r' r = 1,2,...] with
nl-I ~ q (o~1 . ~=0 --Uro ~
Then there exists a positive
h~'-l(a ' ) .
Take a sequence
a'r E Uro, r ~ m o o a'r = a ' . '
We have
144
hl -l(a')
,
= r ~
-
hl-l(ar ) ' Uro
Contradiction.
,,-i
(because
hI
')ch"-l(u ' ~u -q(J)
(a r
I
ro
ro
o
Hence nl-i r=l
~=-O
But each U' is connected. r "''D U--r = ~'~+l ~''" eD ~'r
qo
Thus
"
each
and the compactness
is connected.
We see that
r= 1 from this that h I is continuous hl: Y
>Y
is an analytic
covering
is isomorphic
at q(oj ) .
Let
~i
we get that It follows
Thus we proved that
of Y.
equivalent
to 2Z.
of Y corresponding
of
and from
_ = qo _(j+5) " ~oo Ur r= 1
automorphism
Since Y is homotopically that ~l(Y)
~' is connected r
to
~'-l(o,o)
8: Z ~
we have
Y be an unramified
to the subgroup
d~
of 2Z ,
~" = ~ . ,, Z(')= [a' ZlT(~'(a))~---T} • =
z(~): z(~)
~ pc, ~.
It is easy to see that all fibers of that
• is a holomorphic
Hence
each
Z(~)
~ O,
curve
f u n c ~ on on Z without
the fiber
(and a regular
~"-l(a)
fiber of
~ = O,l,''-,nl-i , are singular
have type I
and
are connected
(I~I ~ ¢) is nonsingular,
a E DE, ~ - De, ~ N Z e elliptic
~"
~"-l((~O))
critical
and values.
for any is a nonsingular
~7), ~T l~.(pl~ ~,~)), fibers of ~"7 which .
is a singular
.
.
.
.
fiber of
145
,i
~o:
Z(O) --->De, O which
is of type Inl d = Imbld = Imb = I n .
We can assume that circle
"
c o of C(o,a ~ . , , which
order of their
_(o) qo
"
"
" 'qo(nl-1)
are contained
is a generator
_~n-~) q(o°)' • " "'~o
in some
of nl(Y ) , and that the
in c o coincides
w i t h the order of
indexes. Denote
c'= o
B-l(co)
d-i 8-1(q(oJ)) = U q(rnl+J)" r=O
and let
We can assume that the points q(O),q(1),-o-,q(nl-l),q(nl),-.-,q(2nl-1),-.-,q((d-1)nl), have the same order on c o' as their indexes. has only one subgroup unique analytic
Wl(Y) ~ Z
of index n I we obtain that there exists
automorphism h
Z
Because
"'°,q(dnl-1)
i
h: Z -->Z
such that the diagram
> Z
JB
Y
> Y h1
is commutative
and
h(q(O))
Let G 1 (corresp. automorphisms
= q(b).
G) be a cyclic group of analytic
of Y (corresp.
Z) generated
by h I (corresp.
A direct v e r i f i c a t i o n
shows that G 1 is isomorphic
and that any Gl-orbit
has exactly
isomorphic
to ~ / m Z
m points.
and that any G-orbit
h).
to ~ m Z
We get that G is
has exactly m points.
146
Let
Me
= [(~,~) E ~2
be given by
I~I < 6m,
~(~,T) = (~m,v),
I~l < el,
~ = Z/G,
~: ~ c
~: Z
> ME
> Z/G
be
the canonical map. It is easy to see that we have a commutative diagram of holomorphic maps Z
Let
ff(~) g
= {a E z ~ ( ~ B ( a ) )
=
~},
= h k " go = g Z(o): Z(o)
(~)
~Z(o).
=
ff(~):
ff(~)--->D
,~,
We can assume 0 < k < m.
Evidently we have go(q(o)) = g(q(o)) = q(kb) and for any a E Z(o) $(go(a)) = G(g(a)) = p.f(a) where with the Kodaira construction
> D
£jo
that we can identify If
769)
we obtain that
has unique non-regular
this fiber is of type
mIb
fiber at ~ = O and
with the desired invariant k.
f': V' - - > D 6 with
• ~ 0 the only multiple
corresponds to G = O.
Now comparing
of logarithmic transform and using
Kodaira's arguments in ([14], p. ~(o): ~ ( o )
r2ui~ p = exp[-~--].
fiber of
~(o): ~
We see
~(o) ----->DE,O.
(v): ~(v) ----~D
Hence it is non-singular.
E,T
147 Take positive integers e. < E, i = 1,2,3, such that ¢1 > c2" l Let
A = {(~,T) E ~
YA : ~ ' - l ( A ) -
O).
y
2
e2
PA(O) = p(O) A'
YA is given in AX~ 2 by the equation
= 4x 3 - 3x - B(O',T).
Because A is a Stein manifold we have that Ax~ 2 YA c AX~ 2
are Stein manifolds.
and
It follows from this that
~-l(y~) is a Stein manifold. m-i Let Z A = ~ - ~ hJ(8-1(YA)). Z A is also a Stein manifold, j=O because it is isomorphic to a closed analytic subvariety in !
J(8-1(YA)).
It is clear that
h(ZA) = ZA, ~ " ( Z ~ )
= A and
j=O ~'-I(A)-Z~
is a proper analytic subvariety of
We obtain that
Z~ = B(Z~)
is a Stein manifold and
~"-I(A).
(which is isomorphic to ZA/G )
~(~"-I(A))-~ A is a proper analytic
subset in ~(~"-I(A)). Let ~ = [(~,~) ~
Im
m
£ e2 < I~l < 61,
I~I <
£3],
From the Kodaira construction of logarithmic transform ([14], p. Z?O) it follows that there exists a cross-section of
~ ~,O1
contained in
~,O nz~ - ZA.
> ~o Since
such that
~o
dim~ o = 1
is not we have that
o
148
dim
~o N ( Z ~ Z A ) = O.
Thus (changing if necessary 6i, i = 1,2,3)
we can assume that ~o c Z A. smaller we can prove that
Let
~ T~
(X)
Taking e}
Z~,o"
Z~ is homeomorphic to
~' 'o)%Dc3 ~(D e3 ~ {~ E ~I I~I < E ~ X
A,O = ZA
and
Z~°
~ De3
coincides with the projection
We see that the canonical map
H2(~,=)
(corresponding to the embedding
given by
Z~,oXD£3
> De3.
> H2(~,O,=)
~' c ~i) is an isomorphism. A,O
Since ZA, O and Zl are Stein manifolds we get that the canonical map P i c ( ~ )
~Pic(~,O)
is an isomorphism and there exists a
complex line bundle [~] on Z~
such that [~] Z~,O = [7o] where
[~o] is a line bundle on ~'A,O corresponding to the curve •
!
~O C Z~, O (d~m~ZA, O = 2). with zero-locus ~o"
Let ~o be a cross-section of [~o]
From the exact sequence
J
jO
~ ~ H°(ZA~',o[e]
we have that there exists a cross-section such that
~ Z~,O = ~o"
Let
~ = Ix E Z~ ~(X~ = O].
)
It is
clear that N ~. = ~ A,O o
and
dim~
= 2.
Changing if necessary £i' i = 1,2,3, we obtain that 7 is closed in ~A~ and
~ (~) = ~.
Because
(7. ~ -i(~,~)) = l,
where (~,~) ~ ~, we have that 7 is a cross-section of
~.F~
>~.
149 In order to finish the proof of Lemma ~ we shall use the following Lemma 5.
Let A = [(~,T) E ~21e 2 <
ei > O, i = 1,2,3, and
~:
Z -->A
curves.
Let
x E Z
~(Q,T)
any (~,T) E A Suppose that
I~l < e3} ,
be a proper surjective
holomorphic map of complex manifolds epimorphism for any
I~J < el,
such that ( d ~ ) x
and all fibers of ~
be a holomorphic
is an
are elliptic
function in ~, which for
is equal to the absolute invariant of ~(Q,T)IT= 0 is not a constant
~-l(~,~).
(as function of ~)
and that there exists a holomorphic cross-section ~ of ~ :
Z --~A.
Then there exist positive numbers e~l' i = 1,2,3, such that !
|
!
!
e3--< ¢3' e2 -< 62 < el -< 61' and if Z
=
~-
o)'
~o = ~
Z~: Z'o
!
Ao = [(~,T)6A T=O, e2
>A'o, De~ = [7E~
I~I<
there exists a commutative diagram of holomorphic maps iz Z '×D o E'
> Z
Xid
Dc.<
}
P
A'×D
O
,
e3 ~
"
~
A
De3 where iz and iA are biholomorphic
embeddings,
, then
150
D£3 = (~ E ~II~I < £3} , iD(~ ) = T, p'(@,~) = ~, canonical projections,
iA A~×O: A~XO
with the natural embedding
p,q are
>p'-l(o)
A'
>p'-l(o)
coincides
and
O
izlz~o: z'×Oo > ~-1(p'-1(°11 coincides embedding
Z °'
---'--->
~
with the natural
-1 ( p 0-1 (O)
Proof of Lemma 5N
Changing,
if necessary,
~ has no zeroes in A i
Ei, i = 1,2,3, differential
el, i = 1,2,3, we may assume that
There exist positive numbers i
|
e% < e3, E 2 < e 2 < e I < e I
such that the
equation
d_~ _- _ ~(,,T) d~ ~(~,~)
has unique holomorphic
solution ~ = ~(T~qo) with i
'(O;~o) = ~o, !
{go I
!
IT] < e3, e2 < !
!
< e I and e2 <
I'I < e3, e2 < I@oi < eI •
for
pefine
,
~:Ao ~x.~ ~ q(=o,,r) = ('(';~o),') iAIAo×O: A'o × O ----~p'-l(o) ,,
~,
p'-l(o)
and
,
(Aot:(m ")Ee21.~=o, e2
It is easy to see that iA is a biholomorphic
A'o
I,(,;~o) l < e 1
!
embedding,
coincides w i t h the natural embedding
p'i A = IDP"
151
Let Z' = Z ×A (A;×DE)) (corresponding to ~ i': Z' --->Z, ~': Z' ---->A' × D' o c5
and iA) ,
be canonical projections,
2'" i17' ~'"
~7 Hence
~b-~ +b~-
+
J'(~o,T) does not depend on T.
be a canonical projection,
j'-
b~
J" =J
= o.
Let p": A' X D ~ o E
A;xO"
>A' O
We see that
Changing, if necessary, £;, i = 1,2,3, we can assume that there exists an open covering [Ui,i E g] of V i = UiXDe; , i E ~
A'o such that if
then there exists a continuous family
[81i(~,T),62i(~,~),(~,~) ~ Vi],
where
51i(~,,),521~,T) E Hl(~,-i(~,,,=),
Dli(~,,),~2(~,~) generate
HI(~'-I(~,~),~), and a holomorphic family [~i(~,~),(~,,) E Vi} where ~i(~,~) is a non-zero holomorphic differential on ~,-I(Q,~). We can assume also that if (Q,~) ~ V i N Vj, i,j E ~,
then
81j(~,') = aijSli(~,~) + bij~2i(~,'), ~2j(=,~) = cijSli(=,~ ) + dij~2i(0,~) where
aij,bij,cij,dij ~ ZZ
depend only of i and j, and that
152
Im
~C 'r) ~ i ( ~ ' ' )
>
0
,
V i ~ ~.
52i Q,
Let
%i(~"0= ~z" (='~') ~.(=,~), "
=2i(=,,- ) = 52"! (="~) cq(c,,,¢),
~li(~, ~ )
®i(~,,) = w2i(~, )
1 '
Note that mi(~,T ) does not depend on T. we would get that for some
If it would not be so,
~o E A',o ~(~o,~) is a non-constant
holomorphic function, that is, it has a continuum set of values. Then
~' ~=~o
We can write
is not a constant.
Contradiction
(with ~
wi(~,¢ ) = wi(o ).
Let (Q,~) ~ V i N Vj, i,j ~ J.
I This shows that
~
We have
) = cij%(~) + d.
cijwi(~)+dij ~ O.
Since =
B2jI~,,l~J (~',) - cijwi~)+dij we obtain that
~j(Q,T) =
:~i(~''))
1 ~i(~,~). cijwi(~)+dij
O
= O).
153
Let G. be a grQup of analytic automorphisms l of U.X~ which have the following form: l (@,~)
>
(~, ~+nlwi(@)+n2) "
nl,n 2 E Z .
Define a holomorphic map: ¥i
:~
'-i V
( i)
> ui~/c i x
by
~i(x) : [°(~'(~))' 7'I}'(~))~(~,(x))] x is taken along a path in
where
~'-l(~'(x)
~'('(x)) Let
~oi
_-
: ~,-l( ~il~,-l(ui×o) ui~°) -->ui~/Gi"
It is clear that define
~oi is an isomorphism.
v.l : ~'-l(vi)
Let x E ~'-l(vi) Mi(X)'-Mj(X).
Let
> Z'XD o 63. by N ~'-I(vj),
Denote T.l = y-lol,and
Mi(x)=
i,jEJ.
(Ti$i(x),,(~(x)).
We shall prove that
x i = Ti~i(x), xj = TjYj(x).
We have
@(~'(x)) = ~(~'~xi)) = ~(~'(xj)), that is, ~'(xi) and for
~ = i,j x~
7'
= ~'(xj),
(~'(x~)~ ~(~'(x~)) =
x 7'I~'(x))~(~'(x))
(mod(l,w~(~(~'(x))).
154
Now
X
X
~(~(x)) = ((x)) ]
l cijWi(~'(x))+di7
D' I~ ' ( x ) )
~i(~'(x))
and
X.l
xi
.~, (
~(~I(xj ))= cijWi(~'(x))+dij
(xj)) j
~'
I~'(Xi) )[i(~'(xi))=
X
1
eij~i(~'(x))+dij
[~
$i(~,(x) +
'(x~
niW i(~(~'(x) )+n~] =
xj
x
,(x))] (~' (x)) +
~ % ( ~(X))5 niSi(~(x))+
'(xj))(rood(I, mj~(~' (x)))) ~'(
(xj) ~j(~
+n~B2(~'~)) (nl,n ~ ~ ZZ).
We see that x. = x.. Thus Vi(x) = 9j(x) and we can define a 3 holomorphic map 9: Z' > Z'o × De3, with ~ ~,-I(v i) = ~i" It is easy to verify that v is an isomorphism.
Taking
iz = i'*(V -I)
we finish the proof of Lemma 5.
Q.E.D.
This also finishes the proof of Lemma ~ (and of Theorem 8). Q .E .D.
155 Theorem 8a. multiple
Let
f: V
'~ A
fibers only of type mI
be an elliptic
.
fibration with
Then there exists a commutative
O
diagram of proper
surjective
holomorphic
maps of complex manifolds
W
D ~
W
1
where D = [7 ~ ~1171 < l} such that coincides with E D~,
f: V --->4,
F h_l(T):
which has singular Proof.
FIh-l(o):
h-l(o)
> p-l(o)
h has no critical values and for ~ y
h-l(~)
>P-I(~)
is an elliptic
fibration
fibers only of types mI o and I 1.
Using Lemma 5 and the same arguments which we used
for deduction
of Theorem 8 from Lemma
4 (see pp. 118-121) we see
that Theorem 8a will be proved if we prove the following Lemma 6.
Let
f': V'
>D
be a proper surjective
map of
complex manifolds with a single critical value o ~ D
such that
for any
curve and
f'-l(o)
~ ~ D-o,
f,-l(Q)
is a non-multiple
positive numbers surjective
£,el,
holomorphic
is a non-singular
fiber of f': V' ---~D.
£ < i, and a commutative maps of c ~ p l e x
manifolds
|
Del<
elliptic
p'
DeXDe I
Then there exist diagram of
150 where D eI
= [" ~ ~ I I'I < el],
p': D XD e e1 such that a)
De =
[~ ~ ~I '~' < 6), D c C D,
> D e l is the projection, F' is a proper map,
F' h ,_i(o): h'-l(o)
, : f,-l( f If,-l(De) De)
> p'-l(o)
coincides with
> De;
b)
h' has no critical values;
c)
For any • 6 DEI-O,
F' h.-l(~): h'-l(~)
> p,-l(,)
has as generic fiber a non-singular elliptic curve and all singular fibers of d) F': W'
F' h,-l(7)h'-l(7)
are of type Ii;
There exists a holomorphic cross-section ~ of > D e x Dci.
Proof of Lemma 6. that
> p,-l(7)
Taking 6 sufficiently small we can assume
f 'I f,_l(De): f,-l( De)
> De
is obtained by minimal
resolution of singularities from the family of elliptic curves which is given in D X~P 2 by the following equation: 6 2 =
where (Zo:Zl:Z2) are homogeneous coordinates in ~ p 2
p(=),q(~)
are holomorphic functions of ~, G E DE, and d(~) = 4(p(~)) 3 + 27(q(=)) 2 = = O
is equal to zero (in DE) only at
(see [15], Ch. VII, ~6).
157
Suppose for a moment that (p,)3q + 2(q,)3 ~ O ( p )'2 p + 3 ( q , ) 2
~ O (where p' = dp dQ' q, = ~ ) "
and
Then
}(p')~q - 2(p')2pq ' ~ O
and because p' ~ 0 (if not, q' ~ 0
and p ~ const, q ~ const,
A ~ const) we have
and q2 = Cp~ C ~ ~.
Now q =
V~
p~/2,
q, = ~
~p'q-2q'p ~ O ~pl/2p,,
(q,)2 = C ~p(p,)2 and (p')2p + 3C~p(p,)2 ~ O, C = - 2-~ (we used p ~ 0 (because p' ~ 0)).
We see that A(~) ~ O.
Contradiction. We obtain that almost for all ao E ~, (p')3q+2(q')}-aoP'[(p')2p+5(q')2] function of
Q(E DE).
demand also that
is a non-zero holomorphic
Choose a ° with such a property and
a O / O, q(~)-aoP(~ ) ~ 0 and (ao)3+p(O)ao-q(o)
Now let X be a complex analytic
subvariety in
~ O.
DeXD¢,X~p2
defined by the following equation: 2 =
Suppose that
~ = (~,;;zL:;l:~2)
E X
is either a singular
point of X or a critical point of the function T considered as a holomorphic -
function on X.
-
)
Z2 + ~z I +
Then we have + 3Zo(q(F)+ao~ ) = 0
(p(~)'+V) = o;
2z2z ° = 0 p,(~)~iz'~2o + q ' ( ~ ) ~
: O.
158
We
see t h a t ~o ~ O, z~2 = O a n d
4[p(~)+?] 3 + e Z [ q ( ~ ) ~
° 7] e = o
2q' (~) [p(~) +V] -3P' (~)[q(F) + a o ? ] = O.
Suppose
p'(~) = O, q'(~) = O.
assume
that ~ = O (because
Taking
E' s m a l l e r ,
4[p(o)+T]3 namely
Then
t h a t the e q u a t i o n
+ e T [ q ( o ) + a o T ]2 = O has o n l y one z e r o • w i t h
T = O.
~ O, w e o b t a i n
t h a t the h o l o m o r p h i c
one z e r o in D
6
a n d at the p o i n t ~ = O
h a s at m o s t o n e z e r o
Consider q(~)
T with
E smaller,
Hence we can
q ( q ) - a o P ( Q ) h a s at m o s t (if this z e r o e x i s t s ) .
that the e q u a t i n n
p(o)+T = O
ITI < e' a n d t h i s is • = O (if t h i s
t h a t is, if p(o) = O). n o w the c a s e p'(~) ~ O. + a ~ = 2 o ~
p ( ~ ) + ~ = O.
Taking
function
E' s m a l l e r w e can a s s u m e
zero exists,
I~I < E'
T h u s w e g e t ~ = O.
q ( ~ ) + a O ~ = O a n d q ( ~ ) - a o p(~) = O.
Taking
e smaller we can
it c a n n o t be p'(@) ~ O, q'(@) ~ O).
w e can a s s u m e
If p'(~) = O, q'(~)
assume
taking
Again
w e g e t ~ = O.
Then
q'(__~_Ir
p ' ( ~ ) t p(7) +7]
and
~(p(~)+;)3
+ 2T.~ (q(F)) 2 9 ( P ' ( ~ ) ) 2 ( P ( ~ ) + 7 ) 2 = O.
If p ( ~ ) + ~ = O, w e o b t a i n as a b o v e Then
t h a t ~ = O.
Suppose
p(~)+~
O.
159
p(~÷;= -) (q,(~))2 (p,(~))2'
q ( ~ ) ~ ; : -2 (q,(~))3 o (p,(~))3
and
q(~)-'aoP(~)
= -2 (q,(~))3
(p,(~))22 ' (p,(~))3 + ~ao(q'(~l)
(p,(~))3q(~)+2[q,(~)]3
Taking
e smaller,
(p,)3q
has at m o s t exists).
_ ao p , ( ~ ) [ ( p , ( ~ ) ) 2 p ( ~ ) + 3 ( q , ( ~ ) ) 2 ]
w e can a s s u m e
+ 2(q,)3
one z e r o in D
As above, we
E
see
that the h o l o m o r p h i c
= O.
function
_ aoP,[(p,)2p+3(q,)2]
a n d at t h e p o i n t ~ = 0 (if t h i s z e r o from the c u b i c
equation
4[p(o)+,] 3 + 27[q(O)~o~] 2 = 0 that ~ = O. We proved that the
t h a t X has no s i n g u l a r
function
T X
points with
has no c r i t i c a l
Let S be a complex
subspace
of D
6'
values XD
£
• = O and
in D£,-o.
defined by
~(~,~) = 4(p(~)+~) 3 + 2 7 ( q ( ~ ) ~ 0 ~ ) 2 = o. Suppose For any But
z' =
t h a t S has a m u l t i p l e (Q',T')
~ S1
we have
component ~(~'~')
S 1 with S 1 ~
(O,O).
= O, ~--~(~',T') = O.
160 ~---6~;(q,,) = 12(p(o)+,) 2 + 54ao(q(=)+ao,) and w e h a v e q(~') + ao7'
= - ~o(p(G')+7)2
,
4(p(~')+~') 3 + ~ Z 4-!--(p(~')+T') 4
=
O.
81a 2 o Suppose
p(a')+T'
A s above, p(Q')+T'
= O.
w e get a. = O and = -3ao,
q(~')+a O
q(~')
Since
q ( G ' ) + a O 7' = O 7' = O.
=
a n d q ( Q ' ) - a o p(~') = O.
let p(~')+T'
q(Q')
- aoP(@')
= a3o"
Now taking
E and
Contradiction. restriction
These
~ E D . E
we obtain
~(q,~o)
Let Qo b e one of t h e s e roots. = O, q ( @ o ) - % p ( @ o )
taking This
e smaller, contradicts
see that S 1 does not exist.
E' smaller,
- o the e q u a t i o n
Then
a 3o
- q(O) ~ O for any We
~ O.
and
a ~ + a o P ( O ) - q ( o ) ~ O w e can assume,
a 3 + a p(~) o o
q(Qo)+a~o
Now
= -2a3o
- aop(.')
that
T o E De,
Then
= O has o n l y
Suppose
= O and,
arguments
to X of p r o j e c t i o n
that
that
D e ,×D E X~P 2
simple
roots.
p(@o)+7o
as above,
show that
for any
if
we g: X
>DE,
= O.
see t h a t ~ De,
Then 7o = O. is the
and
l
X T o = CX 6 X l g ( x ) = 70] , f: X to X of p r o j e c t i o n
De. XDE x~P2
>
DE'XDE >D~7o
is the r e s t r i c t i o n = f IX 7 0 $ xT o --->T°XDe'
161
then for T
~ O the family
fT : XTo
O
fibers of type I 1 (see [15], Ch. VII, It is well known rational
~s
only singular
§6).
that X o has as singularities
double-point.
singularities
> ~o×D E
O
Taking
for the family
simultaneous
only a single
resolution
of
{X~, • E DE, ] (see [6],[Z]),
we get
a diagram W
Cl ~
with desired
properties
d) is evident comes
p'
e
eI
a), b), c).
(the corresponding
from the cross-section
f: X --->De. XDe).
¢
holomorphic
cross-section
(z O = O, z I = O, z 2 = i~
of Q.E.D.
§2.
Lefshetz
fibrations of 2-toruses.
Definition 3.
Let f: M --->S be a differential map of connected
compact oriented differential manifolds dim M = 4, dim S = 2.
(which may have boundaries),
We say that f: M --~ S is a Lefshetz
fibration if the following is true:
a) ~M = f-l(~s); b) there is a finite set of points al,... , a
flf-l(s_i=lai)~ :
is a differential c)
f-l(s - i__~lai)
ell (df)x is an epimorphism there
exist
fibers;
such that
for any x E f-l(ai)-ci,
neighborhoods
and complex coordinates
~a.i=ll
H 2 ( f - l ( a i ) , = ) = = and there
exists a single point c. E f-l(ai) l
c2)
~ S -
fiber bundle with connected
for any i E (1,2,.-.,~)
E S-~S such that
Bi
of
a. in 1
S,
U. of 1
c. in 1
X.x in Bi and Zil,Zi2 in Ui, which define
in B. and U. the same orientations as global orientations l l
of S
and M restricted to B i a n d
and
f Ui: U.l
>B.l
correspondinglyj
f(Ui)
is given by the following
formula:
Xi
M
Ui
2
2
= Zil
+ zi2
We shall call a l , a 2 , - . - , a
•
critical values of f.
= Bi
163
Remark
1.
If a ~ S - U a i , i=l
on a.
If this genus is equal to one, we shall call f: M
"Lefshetz Remark
fibration
2.
the genus of f-l(a)
of 2-toruses
does not depend ~ S
over S"
If
ao E S - ~ a i, j ~ (1,2,-..,~) and Y is a smooth i=l path in S connecting a ° and aj and such that a i ~ Y,
i = 1,2,--.,j-l,j+l,...,~, Lefshetz
theory
we define by usual arguments
(see [16]),
so-called
5. ~ H l ( f - l ( a o ) , = ) corresponding ] Definition
4.
Let
T 2 be a 2-torus, preserving
orientation,
f: M -->S,
identification M
a,~
=
M
which on
M
_
of 2-toruses,
be all critical
Identify
Let N a = f-l(~Da) '
canonically f-l(Da)
f-l(Da)
values
of T 2
map
of
- U a i, D a be a small closed 2-disk i=l
in
~D a w i t h S 1 and
~: N a
b ~ ( D a X T 2 ) be
(and by the
defined by
= DaXT2 ) .
f-l(Da) ~ ( D a X T 2
-
fibration
> ~ ( T 2) a differential
- i__~lai with the center a.
f-l(Da) with DaXT2.
cycle"
to y.. 3
be a Lefshetz
~: S 1
al, • ..,a ~
a E int(S)
a diffeomorphism
vanishing
n(T 2) be the group of all diffeomorphisms
(S 1 is a circle),
int(S)
f: M---~S
"Lefshetz
of
DenOte
) and let
f
a,~
: M
is equal to f and on D a
a,~
~2
S
be a map
is equal to
164
projection D a XT 2 --~ D a .
canonical Lefshetz
fibration of 2-toruses
s-twisting
at the point a.
Definition
5.
Let
We call fa,~ : M a,s
obtained
fl: M1 ---~Sl'
f2:M2
---->S2
We say that fl and f2 are isomorphic
fibrations)
if there exist o r i e n t a t i o n - p r e s e r v i n g
>M2,
R e m a r k ~.
~: S 1 ----->S2 such that
are Lefshetz (as Lefshetz diffeomorphisms
~fl = f2 ~"
It is easy to prove that the s - t w i s t i n g
f
ajs
can always be defined by some ~ o : S 1 ---> ~o(T 2) where the component of the identity element depends only on the class of s o in assume that in the notations of
a
from f: M --->S b y
fibrations.
@: M 1
> S
: M
aj~
--->S
~o(T 2 ) is
in ~(T 2) and that it
Wl(~o(T2),Id).
fa,s,Ma,~
Hence we can
symbol s means an element
,l(~o(T2),Id).
Lemma ?.
Let
f a,s : M a,s
f: M ~S
> S be a L e f s h e t z
be some s - t w i s t i n g
fibration of 2-toruses and
of f: M --~S.
S u p p o s e that
the canonical h o m o m o r p h i s m
~I(S - i=~lai , a) is an epimorphism. ~:
f ~ =
f
a,~
> Au~(Hl(f-l(a),
=))
Then there exists an i s o m o r p h i s m
of L e f s h e t z
[~: S --->S, ~: M --~M
fibrations
a,fx
]
f: M---> S and f
such that
a,~
: M
a,~
(i) ~ = identity;
--->S,
165
(ii) if
Tl: f-l(Da) ~
f-l(Da) and
~2: f-I a,Q ( Da) > Da
of
~ ~ Da X T 2
is the trivialization
of
which we obtain in the construction of Q-twisting,
~If_l(Da ) = T~lo~ 1.
-)
Proof. i: T 2
is a trivialization
> D a which we used in the c o n s t r u c t i o n of Q-twisting
f-1 ( D ) a,~ a then
Da X T 2
It is well known that the natural embedding
>Qo(T 2) (i(y)(x) = X+y) is a homotopy equivalence
(see [l?]).
Hence we can identify
~l(~o(T2),Id) Let
and consider Q as an element of Hl(f-l(a),~Z ). Y:
[0,I]
> s - t/a
i=l be a path in S - ~ a i i=l
with
Yw: Hl(f-l(a),Z)
be the canonical automorphism an isotopy
~t: S ~ > S ,
with Hl(T2,Z~ )
l
Y(O) = Y(1) = a
and
....> Hl(f-l(a),=)
correspQnding
to Y.
There exists
t ~ [O,i], ~o = i d e n t i t ~ such that for
some open U
"t U = identity
C
S
-
De, U D
~ ai, i=l
(for any t E [O,i]), ,t(a) = y(t) and ,l(Da) = Da.
We see that there exists an isomorphism of Lefshetz
fibrations,
~':
fa3 Q
~' = [~': S --->S, ~': Me, Q
> fa,¥~(Q) ~ M a , y ~ ( Q )]
~)The proof of Lemma Z which we give here is based on an idea of D. Mumford.
166
such that ~' = ~l
a,o.
where
Jc~: f'l(S-Da) and
> f-1 a ,~ (S-D) a
Jy.x.(O,): f-I(s-D a)
are canonical identifications. 9t: M a,Y.(e) f
a,ye(~) "Vt
Denote
>
~
~l_t. ,11
,y
a
Now we can find an isotopy
o
t ~ [O,1]
)( <)identity.
fa,y~(~) and
~_
We obtain an isomorphism of Lefshetz
%Y
~,Y:
fa,y~(i,
Ma,y~(~)' ~o = identity, such that V
~" = vlO~'.
fibrations
+
fa,~
> fa,Y~(~)'
{id: S --> S,
,¥: Ma, e
>
(~)},
It is clear that
a,o,
Let el,e 2 be any free basis of Hl(f-l(a)~X ).
Since
fa,e = (fa,~l)a,~2 for ~ = ~i+~2 we have to prove our Lemma only for the case ~ = e I. with 0(el) = el+e 2 and
Let 0 E Aut(Hl(f-l(a),=)) ~: [O,i] --->S - i=~lai
be an automorphism be a closed path
167
in S - ~ a i with ~(0) = a and ~ i=l .... el,?
{id: S -->S,
Take
: O.
~. . . . el,~: Ma,e I
>M a
,el+e 2
}
constructed above. Now let ~' E A u t)( H l (~f - l ( a ) , ~ ~'(e2) = -e I and
be an automorphism with
y': [O,1] --~S - ~ a i i=l
S - U ai with y'(O) = a i=l
and
y~ = ~'.
be a closed path in
We have the following
chain of isomorphisms N
f a,e I
f a,el+e 2
(fa,el)a,e 2 -- (fa,el)a,Y~(e2)
(fa,e I )a,_el -- fa,o
Denote by ~:
f
> fa
' ~ = ~:
=
f-
S --~S,
~: M --->Ma,el ]
,e 1
the isomorphism which we obtained.
It is clear that ~ = identity
and that there exists an open subset U c S-Da, U D ~V1ai,
~If-l(u) 5: [0,i]
=
" I . 3el 1f-l(U) ~ S - ~a i i=l
such that
Now take b E U - U ai, a' ~ ~D and let i=l a be a smooth path in
S - ~a i i=l
with
5(o) = b, 5(1) = a' Using a trivialization ~: f---> fa,el such that
of f: M
~S
over 5 we can improve
~ f_l(a. ) = 3el[f_l(a. ).
Now from
168
I
f-l(a, ) =
je I f_l( a
')
it easily follows that we can finally
improve % such that it will satisfy all demands of our Lemma. Q.E .D.
Remark to Lemma ?. Lemma ?.
Let f: M
> S
be the same as in
Suppose in addition that dS ~ ~ and that there exists
a connected component s of ~S such that the fiber bundle m
f £: f-l(£) --->~
is trivial.
some trivialization diffeomorphism
of f £.
f-l(£)
Let Let
~:
f-l(£) - - ~
~
T2
be
~ ~ ~l(nO( T 2 ),Id) and
~
(over £) corresponding
to ~.
~ f-l(~)
Then there exists a diffeomorphism ~: S --~M
be a
such that f~ =
and ~If-l(~) = ~. Proof.
Let ~ be a closed 2-disk with 8~ = ~,
M = M U~ (DXT2), S = S Uidentity --
by
--
--
D, o n
~
.
.
.
~I M = f, ~IDXT 2 = canonical projection ~xT 2
the center of D.
f: M --->S.
Let a be
> D.
Using ~ and the given trivialization
m
f-l(D) = D~T 2
be defined
f: M • > S .
we define an ~-twisting
Let
~: f
¥=
f
-~_,c~
: M
-a,~_
>S
--
of
> f
{id: s--~_s, ~: M
>M
]
be an isomorphism which exists by Lemma Z and has the property Denote
~ = (~
)-l
< ~_(s) _
Using
f-l(s) = M
and the
(ii)
169
identification consider
of
f-i (S) w i t h M as in D e f i n i t i o n
8 as an a u t o d i f f e o m o r p h i s m
that f~ = ~
and
Lemma 7a.
of M.
Now it is easy to see
~ f-l(~) = ~.
Let
fl: M1
Q.E.D.
> S, f 2 : M 2
> S
fibrations with the same set of critical values, U w h i c h is not empty. Let a E S - ~a.. i=l 1 S u p p o s e that
,,l(s-
ta L/ai,a) i=l
Qj a i , a ) i=l
say [al,.--,a
> Nu~(FIl( f l l ( a ) , ~..) )
fl
>f2'
of L e f s h e t z
= [~: S --~S,
canonical
and
fibrations
~: M 1
such that ~(a) = a, ~ induces identity on
~I(S
>M2} , ai - ~ ,a) and i=l
~Ifil(a ) = identity. Proof.
],
coincide and are epimorphisms.
Then there exists an isomorphism
~:
be two L e f s h e t z
f?l(a) = f~l(a) and the corresponding U
~I(S-
homomorphisms
4 we can
It follows from Lemma 7 that it is sufficient to
prove the following Statement.
Suppose that ~S is non-empty and connected,
S = S' U D, where D is a closed 2-disk, ~D, D contains
exactly one of al,...,a
D ~ ~S
is a segment in
, say el, S' is a
2 - m a n i f o l d with b o u n d a r y and S' N D = ~S' n ~D = ~D - ~D n ~S.
170
Denote by
M'j = f-l(s'),3 f'j = fj M~: M'j
>S', j = 1,2.
i
I
that there exists an isomorphism ~': fl ~' = ~,': S # --~S~ %': M1 - ~ M
,
> f2'
such that
induces identity on "l(S'- i=2~a''a)l and ~'(S'nD) = S'nD. ~: fl such that ~I(S-
Suppose
,'(a) = a,,'
%' f[l(a) = id,
Then there exists an isomorphism
> f2'
~ = [~: s --~s, ~: M 1 and
~ S' = ~'' $ S' = $'
~
> M 2}
induces identity on
Wail. i=l
Proof of the Statement. Let cj, i = 1,2, be the singular point of
f;l(al).
From the definition of a Lefshetz fibration
we have that for each j = 1,2 there exist neighborhoods B of a 1 in D, U. of c. in f~l(D) and complex coordinates ~. in B, J J J J Zjl,Zj2 in Uj which define the same orientations as the global orientations of S and M, restricted to B andUj correspondingly, and are such that
f(Uj) = B and fluj:2 Uj
following formula:
2 kj = Zjl+Zj2.
De, j = {b E B kj(b) < e},
a[ E D
- -
.
a I be a point in
J
.
>B
is given
For a small ¢ > O let e,j
be defined by kj(a~) = ~, ii
int(~S'N~D), a 2 = ~'(al) , ¥i: [O,1]
be a smooth path with Yl(O) = el, " be a (smooth) embedding with
by the
yl (l) = el, '
rl: IXI
>D-DE, 1 --->D-De, 1
171
1 ~l(OXl) c int(~S'n~D), ~l(lXl) = ~Dc,l, rl(lX~) = YI" Then there exists an embedding
r2: i×i
> D - D ,2
with
~2(OXI) ~ int(~S'0~D), ~2(IxI ) C De, 2 and a diffeomorphism ~: s' u rl(ZXZ) u D c,1
such that
~IS. = ~',
b E D£,l,
~2(~(b)) = ~l(b).
~S' U ~2 (xxx) U n c,2
~°~i = ~2'
~(Dc,1) = De, 2
and for any
Define the Lefshetz vanishing cycle
8j ' Hl(f~l(a~) ) as the homology class containing the circle ~j: (Re Zjl)2 + (Re zj2)2 = e,
Im Zjl = Im zj2 = O,
where orientation on ~
is taken according to the order 3 (Re Zjl , Re zj2 ) of the corresponding real coordinates. Let Y2 = ~(¥i )' 5"= j
(jy-1)~(Sj), j = 1,2.
From the classical
Pickard-Lefshetz formula and from our assumptions about the representations of
~I(S - U ai,a ) in i=l
A u ~ H l(fll(a),~) = A u ~ H l(f~l(a),z)
I
corresponding to fl and f2 it follows that (*' fil(el) )~(B1) = _+B2.
In the case when we have z21,z22.
ii
-B 2
"
we change the numeration of
Then for the new 62 we will have (t
fil( al ))~(B )
=
B2 .
172 For j = 1,2 and a small e' > O denote by
Rj,e. = {x E uj]lZjl(X)
l 2 + tzj2(x) l 2
Taking E << e' we can assume that for each j = 1,2 and for any b ~ De,j,
f;l(b) is transversal to ~Rj,e..
Define a
diffeomorphism m
T: f i l ( D E , l ) n RI, e'
> f21(Dc,2)
A R2,e,
by
~2l(,(.)) It is clear that
= Zn(X), m
f2 T = ~fl
z22(.(.)) and
is a tubular neighborhood of ~] in
= ~12(.).
~
~(81) = ~2"
--1
!
Since fj (aj)NRj,E.
fj-l(aj), and
(~' Ifll(a {)).(51) = D2" we can construct a diffeomorphism ~": f~l(s'u rl(IXI))--~ f~l(s'U ~2(IxI)) such that ~" S' = ~''
f2 "~'' = (~ S'L~l(I×I))
° (fl fil(s'U~l(l×I) )
and
~" Ifl i(rl (l×I)) n RI,E' : T Ifll(rl(l~i))n~,~ Without loss of generality we can assume that rl(lXl ) = {b E De, 1 Ikl(b) I : c, - 7 Then
<- arg kl(b ) ~ ~].
173 r 2 ( l x I ) = [b 6 De, 2 I i 2 ( b ) " For t 6 [- ~, ~]
and
= c, - ~ <_ arg k2(b) <_ [ } .
define ejt : ~b E D ¢,j IIm kj(b)-Im eit},
~ . ~ ejt , j = 1,2. Qj = t6[-~,~]
Now choose a Riemannian metric,
gl' on f[I(D ) such that for any b 6 De,l, orthogonal to ~RI,c.
%:
f[l(b) will be
and construct a family of trajectories
[O,l]
> fil(%)
- fil(%)
n ~i,~,'
where
y E f[l(~l(1Xl) flqy([O,l]) for any s ~ [O,i],
) _ f~l(~l(lXl)
: e. 3 arg k l ( f l ( y ) ) '
~([O,1])
the point qy(S) and
) n RI, E,,
is orthogonal to f~l(fl(qy(s))) at
fl(qy(S)) depends only on fl(y ).
It is clear
that for y E fll(~l(iXI)) N ~RI,6. we have qy([O~l])Cfll(Ql)N~Rl~£.. Now define a family of trajectories q,~.: [0,i] --~ f~l(Q2) A ~R2,e. ,
~' ~ f ~ l ( ~ 2 ( l × I ) )
N ~R2,£, as f o l l o w s :
q'7' : ~ ' q~-l(~,) We can choose a Riemannian metric g2 on f~l(D) such that for any b ~ D£,2,
f21(b) will be orthogonal to ~R2.e. and for any
174
s E [0,i] •
q '~,([0~]) will be orthogonal to f21(f2( --' q ~, (S))) at
the point q'~,(s).
Now using g2 we extend the family {q ! ~,} to
the family of trajectories
q':y, [O~l]
> fjl(Q2)-fjl(Q2)~ R2,e,
with the properties analogous to the properties of the family [qy}.
Using the families
[qy} and [qy',} and the diffeomorohism~
we extend ~" to a diffeomorphism ~": fll(S' U ~l(I×I)
U QI)~>fjl(s'
U ~2(I×I)
U Q2 )
such that ~
s,url(Z×i)
" , ,
f2"~ '' = (~ S'U~"I(I×I)UQI)
° (flIfiI(s'u~I(I×I)UQI#
Our statement easily follows from the existence of ~" with these properties.
Definition 6. 2-toruses f: M (2-dimensional
Q.E.D.
We shall say that a Lefshetz fibration of
> S is regular if S is diffeomorphic
to S 2
sphere), and the set of critical values of f is not
empty. Definition 7. fibrations,
Let fi: Mol
> S i, i = 1,2
be two Lefshetz
~S 1 = ~S 2 = ~, such that the non-singular
and f2 have the same genus.
Let
fibers of fl
a (i) E Si, i = 1,2, be some
non-critical values of fi" Dali~) be a closed 2-disk in S.l with
175
the center a (i) which does not contain critical values of fi' and ~: ~Da(1)
Identify
> ~Da(2) be some orientation reversing diffeomorphism.
f~l(Da(i) ) with Da(i)XC , where C is the typical non-singular
fiber of fl and f2" L e t by
~ = ~ × (Identity)
~: f~l(~Da(1) )
~ f~l(~Da(2) ) be defined
and
M = M 1 ~ M 2 = (M 1 - f~l(Da(1) ) D~ (M 2 - f~l(Da(2))),
S = S 1 ~ S2 = (S 1 - D e ( l )
f: M --->S be defined by clear that f: M ---~S
f Mi-f~l(Da( ~ ) = fi Mi-f~l(Da(ii ). It is
is a Lefshetz
"direct sum of fl and f2" and write Theorem 9. of 2-toruses,
Let
f : M o
a Lefshetz
) U8 (S2-Da(2)),
fibration and we call it f = fl ~ f2"
f: M --~S be a regular Lefshetz ~ S o be a Lefshetz
fibration
fibration obtained from
o
pencil of cubic curves in ~p2 by blowing up of (nine)
base points of this Lefshetz teristic of M.
pencil.
Let e(M) be the Euler charac-
Then e(M) > O, e(M) ~ O (mod 12) and f: M --->S
is isomorphic to the direct sum of e(M) copies of f : M --->So. 12 o o Proof.
Let al,...,a ~ be all critical values of f: M
a O E S - i=l~a,l Co = f-l(ao)'
YI'''''Y~
paths which connect a o with al,...,a ~
be disjoint smooth respectively,
> S,
176
I
|
YI,...,Y~
~I,...,Yu, to
be generators of ~I(S@i,...,@
¥i' .. "' y'U'
be automorphisms
51,...,5 ~ E HI(Co,Z )
'
corresponding
~/ai,ao) i=l
to
YI,...,yU
with (el.e2) = I.
of
which correspond to HI(Co,Z ) corresponding
be Lefshetz vanishing cycles
and el,e 2 be generators of HI(Co,~ )
Let Dj, j = O,i,...,~, be small closed disjoint
2-disks in S with the center aj, j = 0,i,...,~. the sets Yi N Dj, i E (1,2,-..,~), empty and that each of the sets has only one point.
j E (1,2,--.,~),
i ~ j, are
Yi N ~Di, Yi N ~Do, i = 1,2,-'-,~,
Let a io = ¥i N 8Do, a'i = ¥i N ~D i and ~i be
the part of ¥i from a o till a~l" Y'= i
We can assume that
We can assume that
~ i ' ~ D i ' ~ 1 (~Di is oriented as the boundary of D ~ a n d
the order of points alo,...,a o on ~Do
that
where ~Do is oriented as
the boundary of D o coincides with the order of their indexes. Now classical Picard-Lefshetz @i(z) = z-(z.Bi)Bi,
formula tells us that
z ~ HI(Co,~),
i = 1,2~.--,~.
It is clear
also that @I @2 "'" @~ = Id (we write the multiplication automorphisms
of
of HI(Co,~ ) here and further from the left to the
right (as the multiplication
of corresponding
matrixes:)
and use
the notation z@ = @(z)). Let ~ (corresp. ~) be the automorphism of Hl(Co,~ ) defined by z~ = z-(z.el)e I (corresp. z~ = z-(z.e2)e2) , z ~ Hl(Co,= ). It is well known that each 6.1 is a primitive element of HI(Co,Z ). Hence there exists an automorphism A i of HI(Co,Z ) preserving
177
intersection form and such that elA i = Di"
For any z G HI(Co,Z )
we have zA'-l~A'i 1 = (zA~l-(zA~l'el)el)Ai =
z-(zAi'el)elAi
z-(zAilAi.elAi)elA i = z - (z.~i)5 i.
Thus
O. = A.-1 ~ A. 1 1 1
and
~(A~ i=l
For i = 1,2,...,~-i
let
qi
1
x Ai) = Id.
: ~I(S - i=l~ai'a°) --->~l(S - i=~la'z'aO)
be an automorphism of ~I(S - ~ a . , a ) defined by i=l i o
q i ( Y i ) = yl, !
!
qi(Yi+l) =
I
,qi(Yi_l) = y i-l' qi(Yi) = y i+l' ,-iy
.
.
.
.
.
y, °
Yi+l iYi+l" qi(Yi+2 ) = Yi+2' " " q i ( ~ ) |
It is easy to see that
qi(yi),...,qi(y~) correspond to
some new choice of disjoint smooth paths
YI,...,Y~
connecting
a O with al,...,a ~ and the same is true for q~l(Yi),...,q~l(Y~) (see Figs. 8 and 9). qi(Y1),--.,qi(Y~)
We denote these new paths by
(corresp. q~l(Yl),---,q[l(Y~).
transformations qi and
q[lelementary
We call
transformations of the
paths on the base. NOW if G is a group and (Xl,...,x)
is a ~-tuple of elements
of G, we call the transformations (Xl,''',x ~)
-i
>(Xl,''',Xi_l,Xi+l,Xi+ixixi+l,Xi+2,''',x ~)
178 Fig. 8 (For qi )
qi ai
Fig. 9 (For q~l)
~o
179
and (Xl,''',X ~)
> (Xl,''',Xi_l,XiXi+ 1 xZl,xi,xi+2,''',X ~)
elementary transformations of ~-tuples in G. Using el,e 2 as a basis we identify H l ( C o , ~ ) with ~
and
the group of all automorphisms of HI(Co,~ ) preserving intersection form with the group SL(2,~).
It is clear that an elementary
transformation of the paths on the base correspond to some elementary transformation of the ~-tuple (~l,...,@)
= SL(2,~)
and vice-versa. We see that if ( ~ l , - - - , ~ ) (~i,...,~)
c SL(2,2Z) is obtained from
by some finite sequence of elementary transformations
then there exists a set of disjoint smooth paths ~I,...,~ ~ on S connecting a ° with al,...,a ~ and such that ~ i ' ' ' ' ' ~ to ll,...,Y~
by the same way as ~ i , . . . , @
correspond
correspond to Yl,...,Y .
Now we shall use the following Lemma 8.
Let
x =
iO
II 1 1 II' ~ =
~-iII I and
I
AI,...,A ~ E SL(2,Z)
be such that if ~i = A~ 1 ~ Ai' i = 1,2,-..,~, then •
" "
.
"
:
II °iII .
Then , ~ O (mod 2) and there exists a
finite sequence of elementary transformations starting with some elementary transformation of ( ~ i , . . . , ~ ) is the resulting ~-tuple in SL(2,~) then
such that if ( ~ i , . . . , ~ )
180
Proof.
We use the following T h e o r e m
proof see A p p e n d i x
Let G be a g r o u p with two g e n e r a t o r s
and b and relations
then
a 3 = b2 = 1 (that isj G is isomorphic
[(~/~)~
and Q I , Q 2 , . . . , Q ~
E G
(=,~=)]).
resulting
such that if ( Z I , . . . ~ U )
is equal to
to the T h e o r e m of R. Livne be the same as above and yl,.'.,y~
be such that each of Yi' i = 1,2,-.-,~, So,Sl,S 2
is the
So,Sl,S2.
Let G , a , b , S o , S l , S 2
elements
i = 1,2,...,~,
some elementary
subset of G then e a c h ~ i , i = 1,2,.--,~j
Complement
to the
for i = O,1,2,
Yi = Q[iSoQi'
starting with
of (yl,...,y~)
one of the elements
a
Then there exists a finite sequence of
transformations
transformation
Let s. = a 2 - ~ a i 1
be such that if
yl. Y 2 " . . . ' y ~ = i.
elementary
(For the
II, page 223.)
T h e o r e m of R. Livne:
free-product
of R. Livne.
and
E G
is equal to one of the
yl.....y ~ = i.
Then
~ ~ O (mod 2) and
there exists a finite sequence of e l e m e n t a r y
transformations
starting with some e l e m e n t a r y
of (yl,Y2,.-.,y~)
such that if ( ~ i , . . . , ~ ) U any j = 1,2,-..,~,
transformation
is the r e s u l t i n g
[2j-i = Sl' ~2j = s2"
~-tuple in G then for
181 Proof.
Note that a cyclic
shift in a tuple
be obtained by a finite sequence (see A p p e n d i x
Let us call the statement
the set
•
> O, ~ ~ O, and define < [~,~]
of the Complement
m
[~,~]
following
of all pairs
that
with
yj = s o .
to the Theorem
"Statement
[~,k]"
order in ~:
We say
or ~i = ~ ' ~i < ~ "
consisting
of
[~,~], ~ g ~, ~ E =,
the following
is not true.
[~,k] be a minimal Suppose
are given
if either ~l < ~
be the subset of "Statement
transformations
of yj in yl,...,y ~
R. Livne in the case w h e n ~,k
[~i]
can
If).
Denote by k the number
Now consider
of elementary
(Xl,...,x)
of all Suppose
[~,~]
Let ,'
for which the
that ~' ~ ~ and let
elemelit of m' k ~ O.
Then we can write
yl'....y ~
in the
form mI m2 m~, yl.....y ~ = So XlS ° X 2 .... So X~,
where N
(i)
] k. ~. X. = ( I Sl l'j s J i=O
then for all i = 1,2,-..,Nj, ~, (~ =
~,
k '
> O, o,j --
~N J ,j
> 0 and if N. ~ 0 J
ki, j > O, ~i-l,j > 0
Nj
m + (ki,j+Li,j) j= 1 j j 1 i=O
~, X = j
m ). J
182 We can write also Yl
m2 "'" Y~ = X l S o X2
Suppose that k
= O.
o,j
-i s 2 SoS 2 = s I we ~an reduce situation
to the "Statement
contradicts
[~,~] ~ m'.
m1 ...
X
Is O
Then because
SoS 2 = s2(s21SoS2)
(by an e l e m e n t a r y
transformation)
[~,k-l]" w h i c h is true.
The same arguments
and our
This > O
show that ~ N J,J
(we use
SlS O = (SlSoSll)s I and SlSoSl I = s 2).
U s i n g elementary factors of Xj,
transformations
j = 1,2,.--,~',
set of
we can get different expressions
for Xj,
j
Sl,S 2.
A m o n g all these expressions we can find a maximal
each Xj,
1,2,--.,
on the (ordered)
• as a positive w o r d w r i t t e n
j = 1,2,-..,~')
according
in the letters
to l e x i c o g r a p h i c a l
(s I is the first letter and s 2 is the second one). that we can assume that we already have y l , . . . , y ~ Xj,
j = 1,2,...,~,
formations
(for
order
This shows such that each
cannot be t r a n s f o r m e d by elementary
trans-
of its factors into a w o r d w h i c h in the form (i)
(that is, when w e put c o r r e s p o n d i n g is greater a c c o r d i n g
s I and s 2 instead of yi)
to l e x i c o g r a p h i c a l
Take j ~ (1,2,.-.,~')
with N
~ O.
order than X
]
.
Now suppose that some
J kij = i, i ~ (I,2,...,Nj).
We have that in the e x p r e s s i o n
Xj there is a subproduct w h i c h has the following Using
SlS2S 1 = S2SlS2,
form:
(1) of
S2SlS 2.
that is , s I = s 2-i s -i I S2SlS2, we get
183
S2SlS 2 = Sl(S -i 1 S2Sl)S 2 = SlS2[s~l ( sllS2sl)s2]
It follows replace
from this that using elementary
S2SlS 2 by SlS2S 1.
in the l e x i c o g r a p h i c a l ki, j > i,
we can
the m a x i m a l i t y
of X. 3
Thus we obtain
k in S2SlS2 °'j.
k X. = Sl°'Js2s I . . . . . . . J k o,j > S2SlS2 we can transform s I s2s 1
Lo, j = i.
substitutions
SlS2S 1
ordering.
transformations
i = 1,2,---,Nj.
Suppose that Using
This contradicts
= SlS2Sl"
We have
SlS2S 1
As we showed above,
any substitution
> S2SlS2 is a sequence of elementary
Thus we come to the situation w h e r e leads to the c o n t r a d i c t i o n
of the form
transformations.
Xj = s 2 ......
as we explained above.
, but this Thus we obtain
~o,j > i. Now suppose that for some i ~ (2,3,...,Nj-I), We have that in the expression
(i) of X
~i-l,j = ~i,j = i.
there is a subproduct w h i c h 3
has the following
form:
SlS2 s~ i ' J s 2 s 1 .
Using
substitutions ki,j
SlS2S 1 ---~ S2SlS2 and k i j ~ 2 k, .-2 SlS2SlS2SlS2S2 l']
.
we t r a n s f o r m
Hence performing
SlS2S 1
elementary
s2s I i n
transformations,
we can come to the situation w h e n a part of X. has the form 3 SlS2SlS2SlS 2. cyclic
But
SlS2SlS2SlS 2 = I.
This
shows that using a
shift we can reduce our situation to the S t a t e m e n t
[~-6,~] w h i c h is true.
This
184
contradicts
[~,k ] ~ i'
$i-l,j = 1
then
~. > I. l,]
i ~ (2,3,...,Nj), Suppose
We see that if for some i ~ (2,3,...,Nj-I) Let us prove now that if for some
~i-l,j = 1
ki_l, j = 2.
then
ki_l, j _> 3
and
Then in the expression 2 2 S2SlS2S 1.
k.~,j >_ 3.
(I) for X
have a part of the following
form:
transformations
this part to SlS2SlS2SlS 2.
contradicts
we transform
[~,k]
~ m'
As above,
by elementary
Yi,j
with
[~,~]
(corresp.
second)
We proved that YI,j
kind if
which also
For We shall say that ~i-l,j > 1 (corresp.
is of the first kind and that if
is of the second kind then Yi-l,j
is of the
first kind.
follows from this that we can find a set Zl,j,...,Ztj,j elements either
of G such that
to some Yi,j
'
(i) each Z~,j,
i E (1,2,-..,Nj)
first kind or to some product
ko, j
J Z~,j s~N,J
~--[
~ = l,-..,tj, where
Yi_l,j-Yi,j,
where Y. . is of the second kind, 1,3 t sI
This
in SlS2SlS2SlS 2.
E m'
~i-l,j ki,j , we denote Yi,j = s2 sI .
Yi,j" is of the first ~i-l,j = 1).
2 2 SlS2SlS 2
transformations
we get a contradiction
i = 1,2,..-,Nj
elementary
In this case we have in the
(1) for X. a part of the form J
can be transformed
we
(as we saw above).
Now suppose that k. = 2. l,j expression
Using
J
and
= X.
Y.1 , j
of
is equal
is of the
i ~ (2,3,--.,Nj)
It
185 For any element c E G we define a reduced form of c by writing c as a positive word in the letters a,b (generators of G) and performing all possible cancellations
(using a 3 = b 2 = ~.
It
is clear that c has unique reduced form. Let ZL, j be such that Z~,j is equal to some Y'l,3" where Yi,j" i C (l,...,Nj),is of the first kind. verify that the reduced form of Z~ with some R~,j ~ G
can be written as bRm
(we use s2X-l'Jsll'3
~i-l,j ~ 2, ki, j ~ 2 and
Then it is easy to
= ~2
s2 = ba 2, s I = aba).
which is equal to some Yi_l,jYi,j, is of the second kind and Yi-l,j
U~l
.ha '
Consider Z~,j
i E (2,3,'..,Nj) where Yi,j
is of the first kind.
We have
~i-2,j h 2, ~i-l,j = 1 and as we proved above, ki_1, j h ~, ki, j h 3 Now we can write Z~,j = Yi_l,jYi,j = s2i-2'Js ki_l,3s2s~i,j I
~i_2, j -2 ki_l, j-3 ki, j-} = s2 b sI ab s I aba . It is easy to see from that formula that the reduced form of Z~,j can be written as (bR~,jba) with some R~,j E G. We define
~
J
=
I
1 ~
if
N
3
= O
I~"j(bR~ jba). 1~=1
,
186
It is clear that if Nj ~ O, then the reduced is equal to some R
j
V~b~=l R~,jba)
E G.
We have
X
= s
l
~s ~
SN-,j
2 J
If N •
Wj
~ i, we cannot have cancellations
s 1 and
~j
and between
~j
and could be written as bR baj with
ko,j
j
form of
~j
and s 2.
in
Sl°'J , ,j
j
~ O, that is,
~Nj, j s
between
If N.j = O, that is, ~--~i = I,
we cannot have cancellations in sko,j I s 2~o,j between s I and s 2. k -i ~ N 2 o,j j Now s o X j = a babas I ~--[j s 2 3' xjs o = s~ °' j ~ j
s2N3' j-lba2.a2b =
k ~N -i = Sl°'J~-~, 'J s 2 J'J bah.
These formulas show that we have no possibilities cancellations
in SoX j
and
This contradicts
for further
Xjs o.
the equality
~Yi i=l
= i.
Now let us consider the case ~ = O, that is, all yl,...,y ~ are equal either to s I or to s 2. in the following
form:
(21
N k. ~. x = l q - sils2 ~ i=O
where
N
i=O
('ki+~i)
=
U.
We can write X = 7 ~ y i ( =
i)
187
Using that k
a cyclic
> O,
l
N > O.
~
B y the
can be written is e q u a l
> 0
i
of indexes
for a l l
in the
HEnce
for
as a b o v e
ko Sl ~ s
form
~N 2
we
~N
s I bRbas 2
cancellations
we
can a s s u m e
It is clear can
where
ko
X =
further
(1,2,...,~)
i = O,1,...,N.
same arguments
to bRba.
possibilities
shift
prove
here
the r e d u c e d
and we have in X.
This
return
center
of S L ( 2 , = ) ,
be
canonical
the
i = 1,2,...,~, is w e l l
known
corresponding b 2 = i.
get
of
no contradicts Q.E.D.
L e t us
we
that X
form
X = i.
apply
that
resulting
El
PSL(2,~)
of R.
=
1
(mod 2) a n d
of
x, ~2
generated
of P S L ( 2 , ~ )
y'
=y.
"'"
8.
L e t Z b e the
,: S L ( 2 , = ) y = ,(~), x = aba,
---~PSL(2,Z) ~ i = ~ ( ~ i )'
y = ba 2.
by a and b and
b y the r e l a t i o n s
~i = ~(Ai)-ix~(Ai)
and the Complement there
starting
(~i'''''~)
:
and
Livne,
transformations
Clearly
is g e n e r a t e d
are
~I-....~U
subset
=
x = ~(~),
2 a = yx, b = y x. that
of L e m m a
SL(2,=~Z,
homomorphis,~,
~ ~ O
transformation
proof
PSL(2,=)=
the T h e o r e m
elementary
to the
relations
Since
that
now
such
exists with that
some if
that
all
a3 = 1 and we
can
to it.
a finite
It
Thus
sequence
of
elementary
(~l,''',~/)
is the
then
~2~-1
:
x,
~
:
y,
...,
~,-i
:
x,
188
It is c l e a r t h a t w e c a n lift our e l e m e n t a r y g r o u p SL(2,2Z). elementary
T h u s w e get t h a t t h e r e
transformations
transformation resulting
exists a finite
starting with
of ( ~ i , . . - , O )
transformations
such that
sequence
of
some e l e m e n t a r y if ( ~ I , . . . , ~ U }
is the
s u b s e t of SL(2,2Z) t h e n
®(~_l ) = x, ~(~_2) = y , ..,,(~_2~_1 ) = x, ~(~-2~ ,(~_~_i ) -- x, , ( ~ ) From the definition t h a t e a c h --l e.,
to ~).
obtain that
of elementary
transformation
is c o n j u g a t e
"'"
is c o n j u g a t e
to x
s h o w s t h a t if ~) is an (corresp.
,(~)) = y) t h e n 8 = x (corresp.
for any j = 1 , 2 , . . . ,
~,
it f o l l o w s
to x (and t o ~, b e c a u s e
A direct verification
o f SL(2,2Z) w h i c h
,(~)) = x (corresp.
) -- Y "
= y.
i = 1,2,..-,~,
is c o n j u g a t e element
to the
to ~) a n d
~ = ~).
We
--~2j-I = x, ~)2j = ~" Q .E .D.
N o w w e r e t u r n to the p r o o f of T h e o r e m g e t t h a t w e can a s s u m e t h a t for a n y j = 1 , 2 , . . . , ~ , verification ~l ~ ~'
paths
t h a t ~l"
yl,...,¥
B 2 j _ l = el,
"''~12 = 1.
We
1 ~ 1.
From Lemma are chosen
B2j = e 2.
s h o w s t h a t if ~i is a p o s i t i v e
~i < 12, t h e n ~ i . . . . . ~
9.
integer with
It is e a s y to v e r i f y
see t h a t ~ ~ O (mod 12) "
f : M o o
fibration which
> S
o
a n d let
is t h e d i r e c t
f: M
so t h a t
A direct
•
Consider
8 we
} S
sum of k c o p i e s
Let k - ~ 12"
be a Lefshetz of
fo: M o
> So"
189
Let [al,...,a ] be the set of critical values of T, La
E S - ~ai. Applying to ~: M >~ our above considerations i=l we construct a system of disjoint smooth paths ~I'''''V~ connecting a O with al,..
,a~
such that corresponding Pickard-
Lefshetz transformations are given by the matrices ~ i ' ' ' ' ' ~ with
~2j-i = ~" 9--2j = ~" j = 1,2,...,~, in some free basis
el,e 2
of Hl(~-l(~o),~). Now we get from Lemma ?a that ~ is isomorphic to f. Q.E.D.
Corollary i.
Let f: M ---~S be a Lefshetz fibration of
2-toruses, ~S = ~. f: M
Then M is simply-connected if and only if
> S is regular. Proof.
a) Suppose f: M ----~S is regular.
be the set of critical values of f: M
Let (al,...,a)
}S,
S' = S-Uai,__ M' = f-l(s'), f' = f M': M' --~S', a ~ S' i=l o C O = f-l(ao) .
From Theorem 9 it follows that there exists a set of smooth disjoint paths ¥i,...,¥~ in S connecting a ° with
al,...,a ~
such that the corresponding Lefshetz vanishing cycles 51,...,5 ~ E H I ( C o , ~ ) generate HI(Co,Z ).
That means that the
190
image of ~l(Co)
in
~l(M)
is trivial.
fibre bundle we have the following
(3)
~l(co) Let Di,
i = 1,2,'-.,~,
f-l(Di)
with a point collection
From the definition
) generate
= ~
l
in S with
of Lefshetz
.
~l(M',Xo)
~I(M')
~I(M)
~ ~ Oo and
S is diffeomorphic
of f: M --->S is empty.
~2(s)
o
its typical
> ~l(co)
is an
sequence
~ ~ ~I(M)
Suppose Then
~I''''' ~ ~ in
(3) that
~I(M) = O.
Contradiction.
to 2-sphere.
~. l
~I(S) ~ O we take any
Then there exists a
~l(M) ~ O.
and denoting by C
Thus If
each
such that
and images of
is an epimorphism. that ~I(M) = O.
Connecting
~, of l
path in M' we get a
of
~l(S,ao)
Clearly
1
Hence we see from the exact
--->~l(M)
~ ~I(S),
Let ~
x ° ~ C o by some smooth
b) Suppose
values
--->D i.
are trivial.
epimorphism.
f~(~) = ~
be a small closed 2-disk
C~I,..-,~ ~] of elements
~(al),-..,8(~
~l(Co)
sequence:
we get that there exists a local cross-section
f f-l(ni):
~l(M,Xo)
exact
f': M' ---> S' is a
> ~l(M') --F-->'TI(S').
the center ai, s i = ~D i. fibration
Since
Thus
such that ~I(S) = O
and
that the set of critical
f: M --~S
is a fiber bundle
fiber we have an exact
> ~I(M) ---~l(S).
sequence:
191
But ~l(S) = O, ~2(S) = Z, ~l(Co) = = O ~ .
Hence
~l(M) ~ O.
Contradiction.
corollar~ 2.
Q .E .D.
Let
fl: M1
regular Lefshetz fibrations.
> Sl' f 2 : M 2
> S 2 be two
Then fl and f2 are isomorphic
(as Lefshetz fibrations) if and only if the corresponding two-dimensional Betti numbers b2(Ml) and Proof.
b2(M2) are equal.
Immediately follows from Theorem 9.
Q.E.D.
The next Corollary gives another approach to a result of A. Kas (see [i~]). Corollary 3 (A. Kas).
Let V 1 and V 2 be elliptic surfaces
over ~p2 with no multiple fibers, with at least one singular fibre and with no exceptional curve contained in a fiber.
Then
V 1 and V 2 are diffeomorphic if and only if b2(V1) = b2(V2). Proof.
It follows from Theorem 8a that we can assume that
V 1 and V 2 have only singular fibers of type I I. corresponding maps
fl: V1
>~pl,
Lefshetz fibrations of 2-toruses.
f2:V2
Then the
> ~pl
are regular
Now Corollary 3 follows from
Corollary 2. Theorem iO. ~)
Let f: V
>
6 be an analytic fibration of
elliptic curves, ~(f) be the number of multiple fibers of
~)This Theorem generalizes results of Kodaira's work on homotopy K3 surfaces (see [19]. The case ~(f) = O was proved by A. Kas (see [18]).
192
~.
f: V
Then V is simply-connected if and only if the
following conditions are satisfied: (i) (ii)
A is isomorphic to ~pl, there exists a fiber of f: V - - ~
which (when reduced)
is a singular curve; (iii)
O ~ ~(f) ~ 2 and in the case ~(f) = 2 the multiplicities ml,m 2 of multiple fibers are relatively prime numbers.
Proof.
a) Suppose that ~I(V) = O.
Then (i) is evident.
Proof of (iii) is contained in the proof of Proposition 2 of [19]. Consider (ii).
Suppose that all fibers of f: V
singular curves.
> ~
are non-
Let a,b E ~ be such that for any c ~ ~-a-b,
f-l(c) is not a multiple fiber of f: V - - ~ ,
D a and ~
be small
closed 2-disks with the centers in a and b respectively, sa = ~Da, sb = ~D b.
Since f-l(a) is non-singular, that is, a
2-torus, and (f-l(a).f-l(a))v = O we have that the differential normal bundle of f-l(~) in V is trivial.
Hence f-l(sa)
(corresp. f-l(Da) ) is diffeomorphic to s a XT 2 (corresp. D a XT 2) (f-l(sa) is the boundary of a regular neighborhood f-l(Da) of f-l(a) in V). that
sa
We obtain dim~Hl(f-l(Sa),~ ) = 3-
> A-De- ~
is a homotopy equivalence.
It is clear Hence f-l(Sa)
is homotopy equivalent to f-l(A-Da-Db) and dimoHl(f-l(~-Da-Db),~) get that
= 3.
From f-l(Da) ~ Da MT2
we easily
193
dim~H2(f-l(Da) , f-l(sa);~ ) = i. By the same reasons f-l(sb);@ ) = i. It is clear that H 2(v,f-l(A-~Da-Db ),@) = H2 (f-l(D a).f-l(sa);@)~H2 (f-l(Db) ,f-I(%9;@).
Hence
dim~H2(V,f-l(A--~a-Db);@ ) = 2
H2 (V, f-i (A---~Ta-Db);, ) we see that HI(V,~ ) ~ O.
and from the exact sequence
> Hl( f-l(A-Da-Db) ,~ )
Thus ~l(V) ~ O.
> HI(V,~)
Contradiction.
(ii) is proved. b) Now suppose that conditions (i),(ii),(iii) are satisfied and prove
,I(V) = O.
From T h e ~ e m s
8 and 8a it follows that we can assume that
all singular fibers of V are of type I 1 or m oI. be such that for any c ~ a-a-b, of f: V
> A, al,...,a u ~ A
Let
a,b ~
f-l(c) is not a multiple fibre
be such that
f-l(ai) , i = I,''',~,
are all the singular fibers of f: V --~A which have type I I. From Kodaira's theory of logarithmic transform it follows that there exists an analytic fibration of elliptic curves 7: ~ - - > A such that
~ ~-~A-a-b):
~-l(A-a-b)
> A-a-b is isomorphic to
194
flf-l(A_a_b):
f-l(A-a-b)
> A-a-b and ~-l(a),~-~b)
are regular
fibers of ~: ~ --->A. Take a O ~ A-a-b - ~ a i. From Theorem 9 we i=l obtain that there exists a set of disjoint smooth paths yl,...,Y~ in A-a-b connecting a O with al,...,a ~ such that the corresponding Lefshetz vanishing cycles 51,...,5 ~ ~ Hl(f-l(ao),Z ) generate Hl(f-l(ao),Z ).
We see that
the image of ~l(f-l(ao )) in WI(V) is equal to zero. Let D a (correSpo Db) be a small closed disk in A-b-i=~la i (correspo in A - a - U a i ) with the center a i=l s a = ~Da (corresp. sb = ~Db) , S~a (corresp. of
~ ~-l(~a): ~-l(sa) ~ > s
a
(corresp. b), S~b) be a cross-section
(corresp. ~ T-l(
: ~-l(sb)--->Sb). sb )
Let ~a
(corresp.
(corresp.
S--b) be a cross-section
flf_l(Sb):
f-l(sb)
~ sb)
of flf_l(sa):
which corresponds to
~a (corresp. S~b) (recall that ~ ~ ( A - a - b ) @If-l(A_a_b) ) . _
are equal to one.)
f-l(b)).
In [19], p. 68 (Proof of Lemma 6) Kodaira
= O we infer that ~ a
(corresp. ~brob) is
equivalent, to zero in V.
Let A' = A-a-b-~ai,-i=l C O = f-l(ao) .
--ma Sb) such that the loop sa
is homotopic on V to some loop in f-l(ao). From
Im[~l(f-l(ao )) --~l(V)] homotopically
of the
(Possibly both or one of ma,m b
shows that we can choose s a (correspo (corresp. ~ )
is isomorphic to
Let m a (corresp. m b be the multiplicity
fiber f-l(a) (corresp.
f-l(sa)~ s a
f-l(A') = M'
2
~' = fl,S l : M' ---> A' •
--
195
Since f': M'
> A' is a fiber bundle, we have the following
exact sequence: (~)
~I(Co)
>r~l(M')
~ nl(A' ). a . with j=l 3
Let Di, i = 1,2,.--,~, be a small closed circle in A-a-b-
jdi
the center ai, s i = ~D i. flf-l(si):
There exists a cross-section si
f-!(si) --~ s i.
elements
GI'''''Q~'Qa '~b
generate
~I(A'),
Using in
Sl,...,su,Sa,Sb
~I(M') that 8(~i),.-.,~(~
images of
~i''''" ~ ~ in
qa (corresp. ~b) is a conjugate to the loop is clear that the canonical homomorphism epimorphism.
we construct such
S(~alS(~bl ) is in a subgroup of
by 8(~i) ,---,8(~),
of
),~(~a),8(~b)
~I(M') generated
~I(V) are trivial and ~a
(corresp.
~: ~I(M')
s--b). It
) ~I(V) is an
We see (from (~)I that any z g ~l(M') can be written
in the form z = z'.z", where z' E 8.(Ul(Co)), z" is in the subgroup of
nl(M') generated by
~i,..-,~
of
~I(V)
~(Qa) and
generated by
,me.
Then ~(z) is in the subgroup
~(~a) generates
~I(V)
(because
~l(V) = ~(~l(M')). We have also that by
~i,''', ~
and
q ~-i is in the subgroup of ab
~'(el),~'(e2)
where
~I(M') generated
el,e 2 are ~enerators~ of
~l(Co).
We see that ~(QaOb I) = i, that is, ~(~a) = ~(~b). As we _m e _m b mentioned above, sa , sb are homotopically equivalent to some loops in C O .
ma ~mb That means that qa ' b
8'(~i(Co) ).
are conjugate to some elements of
From ~(8'(~i(Co) ) = O and ~(@a) = ~(~b) we obtain that
[~(~a)] ma = [~(~b) ]
= O.
we have ~(~a) = O.
Hence
Because m a and mb are relatively prime ~I(V) = O.
Q.E.D.
§3.
Kodaira
fibrations of 2-toruses.
W e introduce now the following notations: Let k , m ~ ~
be such that m > 1 a~d k is r e l a t i v e l y prime to m.
For 6 > O denote D a u t o m o r p h i s m s of
= [m ~ ~I IGI < 6}.
£
Let G be a g r o u p of
I
D 1 × ~
consisting of t r a n s f o r m a t i o n s
Em
(Q,~) ---> (q,~+nli+n2) , nl,n 2 ~ ZZ( ~ E D 1 , ~ E ~)
and let F(D i) = D 1 × ~/G. E~
Denote by
c o r r e s p o n d i n g to (G,~) ~ D 1 X ~. a n a l y t i c a u t o m o r p h i s m s of
g:
and
F
m,k
[~,~]
= F(D ! ) / ~ £m
> [p~,
•
Let
~
be a cyclic g r o u p of
F(D ~) g e n e r a t e d by 6~ k ~ + ~]
Denote by
c o r r e s p o n d i n g to [~,~] ~ F(D !) £m given by
[g,~] the point on F(D¢!)m
E~
[~,~]~ and b y
2wi p = e m ,
where
the point on F
fmk:
F
> D m,k
m,k £
the map
fm,k([G,~] ~) = ~m.
D e f i n i t i o n ~.
Let
f: M ---~S
be a d i f f e r e n t i a l map of c o m p a c t
o r i e n t e d d i f f e r e n t i a l manifolds,
dim M = 4, dim S = 2, ~S = ~.
say that f: M --->S
fibration w i t h V m u l t i p l e fibers
where
~ ~ ~ , ~ ~ O
is a K o d a i r a
if the following is true:
We
197
a) if
9 = O,
b) if
~ > O
closed disjoint
f: M --~S
fibration
then there exist ~ points, 2-disks
Cl,...,c ~ respectively, where
is a Lefshetz
El
say Cl,...,c 9 ~ S,
...,E ~ c S with the centers
9 pairs of integers
for any j = 1,2,...,v,
m
> 1 and k J
m., and v commutative J
of 2-toruses,
in
(ml,kl),...,(m~,~), is relatively
prime to
J
diagrams
f-l(Ej)
~ Fmj,kj
f I
]fmj,kj
E.
>
D
j = 1,2,...,~,
,
such that
(i)
f M_&
f_l(~j )
:
M-
~ 7 f-l(Ej ) j=l
>S
- k/E j=l j
j=l is a Lefshetz (ii)
fibration
of 2-toruses;
for any j = 1,2,...,~,
~
and ~ 3
diffeomorphisms complex
(where orientations
are orientation
preserving
J of D
e and F m j , k j
are defined by
structure).
Let T(f)
(c S) be the set of critical
= O and the set of critical
values
of
values
of f: M
~ S if
198
f
-i M-j=~if
if
:
V > O, T'(f) = ~ if
We call T(f) degenerate)
(corresp.
~ = O T'(f)
>S
- ~" J E . j=l j
and T'(f) = {Cl,...,c 9] if 9 > O.
set of non-degenerate
(corresp.
critical values of f: M --->S.
Definition v multiple
M - ~ f -~I ( E j ) " j=l
(Ej)
8~. Let
f: M
> S be a Kodaira
fibration with
fibers and in the case ~ > O let [Cl,..-,c ~} be the
set of degenerate
critical values
of
f: M
> S, and
{EI,...,Eg} , [ ( m l , k l ) , . . . , ( m ~ , ~ ) ] , C ( ~ l , ) l ) , . . . , ( ~ , ) ~ ) } same as in Definition
8.
F'(D£) = [[@,~] Following
Let
D'6 = [G ~ ~ 0 <
Kodaira we define a map
Am,k([~,¢]~)
=
[ m
~F'(D) k
~ _ 2~i
Let ~ > O and F(De) j (corresp. F(DE)
I~I < £],
• k = f m,K -l(D). 6 ~ F(D6) ~ ~ O], Fm,
Am,k: F~,k by
be the
(corresp.
log =] De,j) , j = 1 , 2 , ' - . , ~ ,
b e ~ copies
of
De).
Define a new ~-manifold ~ as union of M - ~
f-l(cj)
and
j=l
F(De)I,.-.,F(De) 9 where we identify x ~ f-l(Ej-cj), with
Amj,kj~j(x ) E F'(De)j.
j = 1,2,.-.,v,
199
Define the 2 - m a n i f o l d ~ as union of S - ~ ]c j=l ] De,I,..-,De, v where we identify a E E -c.,] J ,j(a)
E D£,j.
~(x)
and
j = 1,2,...,~,
Let
~: ~
>~
~
f(x)
if
X ~ M - ~ ~ C
~j(x)
if
x ~ F(D6) j where
be a map defined by
j=l J
=
~ ([~,~])j
In the case M = O we take ~ = M, ~ = S, ~ = f. that
f: M --->~
is a L e f s h e t z
set of n o n - d e g e n e r a t e
fibration
-
and that the coincides
f: M ---~S (by evident embedding
Uoj=l J
We shall call to Kodaira
fibration
D e f i n i t i o n 9. call
f: M
..> ~
Let f~ M --~S
f: M --->S regular
e(M) 12
genus of M.
and
fibration
be a Kodaira
if S is diffeomorphic
Let e(M) be the Euler =
the Lefshetz
corresponding
f: M --~S.
the set of n o n - d e g e n e r a t e
~(S)
f: M --~ S
= Q.
It is clear
of 2-toruses
critical values of
w i t h the set of critical values of s
with
fibration.
We
to a 2-sphere and
critical values of f is not empty. characteristic
(following F. Hirzebruch)
of M.
W e define
call it a r i t h m e t i c a l
200
Lemma multiple
9.
Let
fibers
f: M --->S be a Kodaira
f: M
~
fibration with
be the corresponding
Lefshetz
fibration. Then
i)
e(M) = e(~); if
ii) e(M) --O
is regular,
(mod 12), that is, Z ( M )
Proof. values
f: M --~S
of
i) Let
~Cl,...,c 9]
f: M --~S.
e(F(DE) j - F'(DE)j)
e(M) = e(M,
is a positive
integer.
be the set of degenerate
It is clear
that e(f-l(cj))
= O, j = 1,2,...,9.
9 ~f-l(cj)), j=l
then e(M) > O and
critical
~O~
We have
9 e(~) = e(M, ~ - l ( c j ) ) . j=l
e(M) = e(~).
Hence ii)
Immediately
is a regular
follows
Lefshetz
from i) and Theorem
fibration
9. (7: ~ - - ~
if f: M --~ S is regular). Q .E .D.
Lemma
iO.
multiple
fibers.
diffeomorphic Proof. Consider fibration
Let
f: M --~S be a regular Suppose
that ~ ~ 1 and
Kodaira
fibration
~(M) = i.
with
Then M is
to P ~ 9Q. If ~ = O then Lemma
the case
9 = i.
corresponding
Let
i0 follows 7: ~ - - > ~
to f: M --~S.
from Theorem
9.
be the Lefshetz
By Lemma
9
Z(~) = ~(M) = 1
201
and by Theorem 9 Identify
7: ~ - - ~
f: M - - ~
with
is isomorphic to fo: Mo fo: Mo
> So .
> So"
Let c I be the
degenerate critical value of f: M---~S and El, (ml,kl),(~l,~l) be the same as in Definition ~.
Identify M-f-l(cl),F(De) 1
(corresp. S-Cl,De,l) with their images in M (corresp. ~). ~i be the center of De, I.
Let
Without loss of generality we can
assume that DE, 1 (considered in So) is contained in some coordim te neighborhood U of ~i in So (= ~pl) with complex coordinate T and that T(~I) = 0 and De, 1 is given in U by I~I < E.
Let ela,e2a , a C De, 1 ,be a basis of Hl(~-l(a),= )
corresponding to the vectors i,l E ~ by our identification ~'I(DE,I) = F(De) 1 = D6j I x ~/G.
Using the family
{ela,e2a;a ~ DE,I~ we can identify the complex manifold f~I(DE,I) with De,IX ~/G W
where
G
is the group consisting of analytic
automorphisms (T,~)
> (v, ~+nl.J(T)+n2) , nl,n 2 ~ 77.. ('I" ~ DE,I, ~ ~ ~)
and (u(T) is a holomorphic function in D6,1 with Im W(T) > O. 1 Let E* = eml, D£. = {@ E ~ I~I < e*], G*W be the group of
I
analytic automorphisms of D£.X~ consisting of transformations (,,T) ---> (Q, ~+nlw(~ml)+n2) , n.,n~A n ~ = (Q ~ D and F(De.,w ) = D £ . ~ / G ~ .
~ E ~)
As above we denote by [~,~] the
202
point on F(DE.,W ) corresponding to (~,~) E D6. X(~). Let ~
be
the cyclic group of analytic automorphisms of F(De.,w) generated by the transformation [a,~]
and
~[p~,
Fml,kl, W = F(De.,W)/~W.
kl ~ + N-~I],
= e
p
ml
Denote by [~,~]~ the point on
Fml,kl,Wl corresponding to [~,~] E F(DE.,W).
Let
D6, 1 = [~ E D6, 1 T / 0},
Fml,kl, m = [[~,~]~ E Fml,kl,~l ~ ~ 0}. Define a holomorphic map
aml,l,l:Fml,ki, mI
f l(D ,I) k1
^ml,xl,~l ([q'~]~) = [~ ' ~-2-~Y log ~]Mo,~ where we denote by [T,~]Mo, ~ the point on f~l(De,l) corresponding to (~,~) ~ D6,1X~ by our identification
fol(DE,l) = D6,1X~/G W.
Define a new complex manifold M o as union of
Fml,kl, ~
and
" Mo-f~l(~l) where we identify x E Fml,kl, w (C Fml,k 1 'W ) with Aml,kl,W(x) ~ f~l(De,l)(C Mo-fol(~l)).
203 A
It is clear logarithmic fo: Mo to
that M ° is obtained
transform
> so
at ~i (see
[~], p. Z 6 ~
be the holomorphic
fo: Mo
from M o by Kodaira's >.
Let
map canonically
> So"
Let KMo (corresp.
K~o) be the canonical
Using
Kodaira
for canonical
class
for elliptic
(see [ ~ ] ,
7Z2 )
we have
formula fibers
KMo = -[f~l(a)],
bundle
(corresp. ~o)"
multiple
corresponding
p.
KAM0 =-[fol(a)]
Evidently
[f~l(a)]
a ~ So-~ 1
of M O and the
surface with
+ (ml-l)[fol(Cl)]. : m [f~l(~l)
].
: -[f~l(~l)
Hence K~
]
0 and all puri-genuses
of M O vanish. Since /k iO) we have that M is a rational o
~l(Mo)
(Theorem
surface.
A e(Mo) = e(Mo) = 12 we have b2(Mo) from some minimal minimal
rational
or to S2XS 2
rational surface
we s e e
that
= 10.
surface by :-processes.
is diffeomorphic M° i s
a diffeomorphism
either
diffeomorphic
Vl(T ) = Im ~(T) and
defined by > Re
~ + Ul(~)Im
Since
AM O could be obtained
to
~ + ivl(~)Im
Because
any
to P or to P ~ Q P ~ ~Q.
We shall prove now that M is diffeomorphic Let Ul(~ ) = R e ~(~),
= 0
to M oA
: ~
>~
be
204
(A T is a non-degenerate
real-linear orientation-preserving
transformation because Vl(~ ) = Im w(~) > 0). for the points of
f-l(El)
As above, we use
(corresp. ~:I(D6,1) ) the notation Define a map
o
~ l ( D l) by
A: f-1(E z)
[~,~]~
> [~, Aamz(~)]~o •
It is easy to verify that A is a diffeomorphism
(fo A ~:I(D,I))-A Let
£ ' = ~,
M' = f-l(s,),
o
"o = fo ( o ), i%:
M'o
(aN),
~H.
s' = s-:~', s' = s -~, o o
= fo](so ). b M'o
to our constructions
> So' ~:- f: l(s) '-
is equal to
>
f-1(~
Let iM: M' --->M~,
/k A fo: MO
)
Y = {" ~ D~, l I'I _< ~'J , ~:,[1(y) , _s:
= A f-z(~):
corresponding
= f f-l( E l
and that
> f:l(s)_
of
be isomorphisms 7: M
~
and
be a diffeomorphism which
205
Let ~ be an element in Q (T 2) corresponding to ~ trivialization of f;l(~) D6,1X~/G.
>s
and the
given by f;l(~) c f;l(D£,l) =
Using our choice of {el,a'e2,a;a E DE,I ] for the
identification E ~o(T2).
f~l(D¢,l) = De,I×~/G w
Now by Remark to Lemma ?
diffeomorphism
f~l(£) = _ ~"
f-l(~) = A f-l(~),
A
I
Define
it is easy to verify that (see p. 168) we obtain a ~: M
~ M O by
f-l(S') = (i~o)-i ° ~ " iM"
We see
^
that M is diffeomorphic to M . O
Thus M is diffeomorphic to
p ~ 9Q
QED
Definition iO. fibrations.
Let fi: Mi --> Si' i = 1,2, be two }(odaira
We define direct sum of Kodaira fibrations
fl~f2: MI~M 2
> S 1 ~ S2
by the same way that in Definition ?
we defined direct sum of Lefshetz fibrations (see Def. Z, P. I?~). Lemma Ii. fibrations, to
Let fi: Mi --->Si' i = 1,2, be two Kodaira
f: M --~S be a Kodaira fibration which is isomorphic
fl~f2: MI~JM2
> S I ~ S 2.
Suppose that nl(Ml) = O = ~I(M) = O
and the intersection form of M 1 is of odd type. Let U be an open 2-disk in S2 which does not contain critical values of f2'
b,c E U,
Ybc be a smooth path in U
connecting b with c, sb c fjl(b), Sc c fjl(c) be two smooth circles such that
Ybc Sb, s c
generate Hl(f-l(c),~ ) where
206
>Hl(f-l(c),~ ) is the canonical isomorphism
Ybc*: Hl(f-l(b)'Z) corresponding
i)
to Ybc"
M # P ~Q ~-manifold
ii)
Then
is diffeomorphic to obtained
if fl: M1
M 1 ~ M2, where M 2 is a
f r o m M2 b y s u r g e r i e s
>S 1 is isomorphic to fo : M o
along
sb and Scl
~=> S o
(see the formulation of Theorem 9) then M ~ P is di f feomorphic to
P ~ M Proof.
2
Let a (i) E Si, i = 1,2
be some non-critical value
of fi' C(i) = fl l(a(i))' ~(i): TC(i) neighborhood of C(i ) in M..l Since
~ C(i )
be a tubular
(C~i))M. = O, i = 1,2, there l
exists an isomorphism ~ of fiber bundles ~(i) I~TC(1): ~TC(I ) and
~2) ~TC(2): ~TC(2)
Let
B: C(I )
~.
~ C(2 ) which reverses orientation of fibers.
~C(2 ) be an isomorphism of bases corresponding to
Using the definition of direct sum of Kodaira fibrations, we see
that we can identify M with [MI-TC(1)] x I ~ C(1), E 1 -i
~[)
b C(I )
=
[M2-TC(2)].
Let
be a small open 2-disk on C(I ) with the center Xl ,
~l)(C(1)_El) ' TCi2 )
s = ~(~(1)(Xl))
~
=
(2)(C(2)-8EI). ~-i
as a subset in M.
obtained from M by surgery along s.
Let M
Oonsider
be a 4-manifold
Now using the same arguments
as on pages 45-47 we see that there exists an orientation reversing
207
diffeomorphism diffeomorphic
8": ~TCil )
>S . o
Q-processes say
fl: M1
b S1
We know that M
o
is isomorphic to ~p2 with nine
~
that
xI
~i: M1
fl: M1 ~(1) N C(1)
=
~il) = QICil)"
and
,-I (1)(Xl)
exceptional curves, f : M o o > So
=
contraction
= Q(TC(~).
TCil ) with T~il)
MI-~I(Z(1)) . that
T~il)
of
> S 1 with fo: Mo
> ~i be a canonical
identify
is isomorphic to
such that the corresponding
Identifying
such that M ~ is
to
Now suppose that f : M o o
>~TCi2 )
~(i)
we can assume
n Tc(1). Let
of
Z (I) to a point,
It is evident that we can
(by identification
MI-~(1) with
Using the same arguments as on page
M ~ P is diffeomorphic
> So .
~8
we see
to
[MI-TC~)] U~ [M2-TCi2)]. NOW applying a result of R. Mandelbaum mentioned and used on page to M 1 ~ M 2 Lemma). f : M o o
(where M 2
55
we get that M ~ is diffeomorphic
is defined as in the formulation of the
In the case when fl: M1 >S o
(see [5] ) which we
we obtain that
M ~
M 1 ~ M2, that is, to P # 8Q ~ M~.
> S 1 is isomorphic to P is diffeomorphic
to
Since the i n t e r s e c t i o n form of
208
M 1 is of odd type we have that the i n t e r s e c t i o n of odd type.
U s i n g -l(M) = O and results of W ~ I I
obtain that M-- is d i f f e o m o r p h i c
M # P ~ Q ~ M 1 ~ M ~2 Lemma multiple
12.
f: M - ~ S be a Kodaira
Thus Q .E .D.
fibration w i t h
Then M is s i m p l y - c o n n e c t e d
following conditions
if and only if the
are satisfied:
f: M --~S is regular~
(ii)
0 ~ 9 ~ 2
multiplicities Proof.
and in the case ~ = 2 the c o r r e s p o n d i n g
m I and m 2 are r e l a t i v e l y
prime.
That is almost w o r d - b y - w o r d
of T h e o r e m
i0.
Lemma
13.
to a
to M ~ P ~ Q.
(see [8]) we
and in case ii) M ~ P ~ P ~ 8Q ~ M ~
Let
fibers.
(i)
form of M ~ is also
~ense space
repetition
of the proof Q.E.D.
Let L be a 3 - d i m e n s i o n a l
m a n i f o l d diffeomorphic
(see [20]), M = LXS 1 , C 1 = pxS 1 where
p E L,
C 2 be a smooth circle in M with C 2 N Cl = ~, N be a 4-manifold obtained
from M by surgeries along C 1 and C 2.
WI(N) = O.
T h e n N is diffeomorphic
Suppose that
to an S 2 - b u n d l e
over S 2
(that is, to $2×S 2 or to P ~ Q)o Proof.
i=
Let
~"= 1
[~i ~ ~I }~i '
%-- c,q ,, lql <_
= i], s i =
{~i E e ll,il = 1},
D1 = {~i E ~ i~ll <_ i],
20g = ~I~2, U
Z
Z = DIXS2,
where
~ = ~x~3, Y = ZXS 3.
~: ~
> ~Z
We can identify L with
is defined by
m
,(TI,T2) = ( ~ can assume that ,(Lj~j~3)
~,~
), a,b,c,d ' Z, detll ac
M = ~ U~ Y, ~: ~ - - ~ S Y
= ~(~j~)×Lj
II = i.
Now• we
is defined by
and p ~ Z, p = (O,1)~ Cl = pxS 3.
Denote by I o = Ix ~ $2, - [ ~ arg Te(x ) ~ ~} ,
I'o = S2-I ° ,
D(T3) = DIXIoXY , y ~ $3, T3(y ) = T3, Te I = ~ D(,3)(union in Y). Let ~: TC1 --->C1 be defined by
X(Vl,~2,~ 3) = (O,l,V3). We can consider
~: TC I --->C~ as a tubular neighborhood of C 1 in M.
Now we have two possibilities
for a surgery of M along C 1 which
correspond to two non-equivalent
trivializations
We can assume that these two trivializations, fo: TCI
>D(1)XS3'
fo(Tl'V2'~3)
fl: TC1
of k: TC 1
~ C1,
say
> D(1)XS3, are defined as follows:
: ((~l''2'l)'v3)'
fl(Tl'~2'~3 ) = ((Vl "~l''e'l)''3)"
Define an autodiffeomorphism u: Y --->Y by ~(TI,~2,T3) = (TIT3,T2,T3). ~. = ~ - l
~: ~
to verify that
> ~.
Let
~'(~1,72,~3)
Let ~' = ~ BY: BY II a' c ' b' d' II = = (~',72~
~Y
II ac bd II-1 " 3 ,T3).
and It is easy
Now define an
210
,- ~ , - ~ ' , ~ )
autodiffeomorphism ~: ~ - - > ~ We obtain an isomorphism easy to see that
by ~(~I,~2,T3) = t~ 1 3,T2 3
8: M ---~M with 8
= ~,
= ~.
8(C1) = CI, 8(TCI) = TC1, XB = &.
It is
Since
(8-i D(1)xid)'fl'8(Tl'T2'T 3) = (8-1 D(1)Xid)°fl(Vl~ 3'T2'T3 ) = = (8-1 D(1) X id)((T1,T2,1),T3) (~-l D(1)
X
id).flo 8 = fo"
= ((TI,~2,1),~3)
Hence
~-i
we have that
transforms
fl
in
fo
and we have to consider only the case when our surgery corresponds to the trivialization from M b y s u r g e r y
fo"
Denote by M[CI] a b-manifold obtained
a l o n g C1 c o r r e s p o n d i n g
M[cz] as follows: Let D 3 = [T3 E ~I IT31 Uf, (~D(1)XD3) , where M[cz] = ~
to fo"
~ 1].
We can construct Then
o
f'= o fo ~T(C1): ~T(C1)
> ~D(1)XS3"
NOW for any x 6 D 1 denote by
s2(~) = (xxI~xs 3) u
(xx~I~ 3)
where union is taken in
M[c1]
M-7 T, (xX~:CoXD3)
E 3D(1)XD3) o
Let Xo E DI, ~l(Xo ) = O, TS2(Xo ) =
k': TS2(Xo )
~ / S2(x) (union in M[Cl] ) and x~D l
> S2(Xo )
211
be defined as follows:
If
z (S2(x), x ( D~, z = xXy,
where
~E ((~,oXS 3) u(~i~ ×D3) (boundaries are identified by evident way), then
~'(z) = XoXY.
Identify S2(Xo) with (Io×S3> U (~I~KD3) and let D 3 = I~XD 3 be a 3-disk with boundary (I~×S3) U (~I~xD3) = S2(Xo ). identify
l': TS2(Xo )
~ S2(Xo ) with
pr: DI×S2(Xo)
Now > S2(Xo )
and let X be a t-manifold obtained from M[Cl] by surgery along S2(Xo ) corresponding to the given trivialization.
We have
x = M[c l] - Ts2(Xo ) u (s i x z'o x 03)
where the boundaries are identified by evident way. Using
M-~cI : M-(DlXloXS3) = (ff~31U(olXIoXS31 and
M[CI]-TS2(Xo) = {[(Z'XS3)U(DIXIoXS3)]U[()D(1)XD3]} -
- {(ozxzoxs3) u (OlX~ZoXO3)} :
(~'X~)U(SlXZoXD 3)
we have
x = [(~'xs~) u (slXZoXD3)] u
[SlXZoXD3].
We see that we can identify X with
(ffx~3lU(SlXS2×D3)(~x~3)u,(~zxo3)
212
which is isomorphic to
(~3)
Uid ( ~ × ~ 3 ) , where
Using
we see that X is isomorphic to
S3XS 1
and we can consider M[Cl]
as a ~-manifold obtained from S3xs I by surgery along a smooth circle, say ~i' embedded in S3XS I. can assume that defined° by ~2"
C 2 c (~-~).
Withou~h loss of generality we
Thus the image of C 2 in X is well
Denote the corresponding smooth circle in $3×S 1 (~ X) Now we can consider N as a 4-manifold obtained from
S3XS 1 by surgeries along ~i and ~2 (~ln~2 = ~)" = xxsl' ~i (rasp. ~2) be homologically (rasp. n2~ ).
Let x ~ S 3,
equivalent to
nj
Since ~I(N) = 0 we have that either n I = O, n 2 ~ 0
or n I { O, n 2 = O
or
n I ~ O, n 2 ~ O, nl,n 2 are relatively prime.
Consider the third case, that is, n I ~ O, n 2 ~ O, nl,n 2 are relatively prime.
We can assume n I > O, O < n 2 ~ n I.
Let
n I = n2q+r , where r,q ~ ~. O ~ r < n2, ~(r) be a smooth circle in S3xS 1 homologically equivalent to
r~
and such that
~(r) ~ ~i = ~' ~(r) n L2 = ~, x(n2 ) be a 4-manifold obtained from X by surgery along ~2 and ~ , ~ ' ( r ) in X(n2).
It is clear that
be the images of ~l,~(r)
~l(X(n2)) = ~/n2= and that ~'l,~'(r)
213
are homologically to homotopically the following
equivalent equivalent
remark
in 4-manifolds
in X(n2)
(that is,
embeddings
of W a l l
surggry
process
number
Thus we could assume (n',O). X(n')
Because
For 1-manifolds
in X(n2)
that is, we obtain N from $3×S 1 by surgeries the pair
Now we use
may be replaced by an isotopy.
see that we obtain N by performing
Hence we can replace
correspond
of S 1 in X(n2~.
(see [8], p. 135):
every homotopy
after finite
~,~'(r)
along
(nl,n2) by (n2,r) .
along
~'(r),
42 and ~(r). Repeating
of steps we come to the pair
from the beginning
that our pair
~I(N) = O we have n' = I.
= X(1) ~ S ~ and N is diffeomorphic
We
this
(n',O).
(nl,n2)
is
But then
to an S2-bundle
over S 2. Q.E.D.
Lemma
i~.
Let
f: M -->S be a Kodaira
that ~I(M) = O and X(M) = 1.
2P
Suppose
is diffeomorphic
to
9Q Proof.
By Lemma Lemma
Let 9 be the number
of multiple
12, v ~ 2 and f: M ---> S is regular.
follows
from Lemma
i0.
From Lemma 9 and Theorem fibration Let
Then M ~ P
Fibration.
corresponding
Consider
If ~ ~ 1 then our
the case ~ = 2.
9 it follows
to f: M ---~S
fibers of f: M --->S.
that the Lefshetz
coincides
with
fo: Mo --> So"
{ci,c2] , CEI,E2~ , [(ml,kl),(m2,k2)] , {(,i,~i),(,2,~2)]
defined
for f: M --->S as in Definition
~.
From Definition
be 8&it
214
follows
2 now that we can consider M ° as union of M - j~=if-l(cj)
and F(D£)I,F(D6) 2 with
where we identify
x ~ f-l(Ej-cj),
A mj ,kj@j(x ) E F'(D6) j (see Definition
also S o with S so that for j = 1,2
8 a) We can identify
f~l(Ej) will be equal to F(D 6
Thus we have an identification where
j = 1,2~
of f~l(Ej)
(i,l) means the group of automorphisms
)j
with Ej×q~/(i,l)>,
of • consisting
of
transformations ~ ~ + nli + n 2, Let elj,e2j corresponding
be the basis
to the vectors
there exists a smooth
to ~
~:
~(ell)
SL(2,=)
x=
i and 1 on ~.
critical
values
Hl(f-l(cl),~ )
has the following
Let
of homologies
II-
of f : M -->S and the o o o
= el2,
~(e21) .
=
Tt is well known ~ and ~.
of the lengths
alphabet
which
correspond
y which
connect
critical
paths
values
corresponding
e22.
denote by d(A) the minimum
of all smooth
c I and c 2 and such
property:
O1 II, ~ =
,y
We shall prove that
~Hl(f-l(c2),~),
is generated by the matrices
x,y,x
of H l ( f ~ l ( c j ) , = )
path ~ on S O connecting
that ~ does not contain isomorphism
nl,n 2 ~ ~ .
of fo: Mo ---~So"
that the group
For any A ~ SL(2,ZZ)
of all words
to A.
Let ~
in the
be the set
c I with c 2 and do not contain
For any y E ~
let
215
Qy: Hl(f-l(cl),~ ) --->Hl(f-l(c2),~ ) be the isomorphism corresponding to Y and Ay be the element of SL(2,~)
corresponding
to ~y and to
the bases ell,e21 of Hl(f-l(el),= ) and e12,e22 of Hl(f-l(c2),= ). Denote by
dy = d(Ay).
of n with d~ = 7. = ~). that
Let
~ =
We claim that
Suppose that
d~ > O.
min dy y6~
and ~
be an element
d~ = 0 (that is, we can take Take a closed 2-disk D in So such
~ c int(D) and D does not contain critical values of
fo: Mo --> So"
From Lemma 8 it follows that we can find two
critical values,
say al~a2, of fo: Mo --~So
¥I,Y2 on S O such that
Y1 N Y2 = Cl' Y1 (corresp. M2) connects c 1
with a I (corresp. a2) , Y1 (corresp. values of
and two smooth paths
fo: Mo --->So
y2) does not contain critical
different from a I Ccorresp, a2) and if
~i (corresp. ~2) denotes the automorphism of H I ( (f-i el),~) corresponding A(~I)
to Y1 (corresp.
y2) then the unimodular 2-matrix
(corresp. A(~2) ) which corresponds to ~i (corresp. G2) and
to the basis ell,e21 of
Hl
It is easy to see that we can assume that each of yi,Y2 intersects ~D only in one point. have
Hence we can change ¥1,¥2 so that we will
~i N ~ = Cl 9 Y2 N ~ = c~.
. . . . 1 ~-i alphabet x,y,x ,y length.
Let
W(A~) be a word in the
which corresponds
to
A T and has the minimal
We can assume that there exists a small circle s on S O
with the center c I and such that each of
¥i,Y2,~
intersects s
216
only in one point.
Let
bl = ¥i N s, b 2 = Y2 N s , b = ~ ~ S.
Let us say that we are in the case I (corresp. (bl,b,b2)
corresponds
to positive
(corresp.
Let ~ be the first letter in W(A~) here the c o m p o s i t i o n as m u l t i p l i c a t i o n (I,~')
(corresp.
from the left to the right,
d~, = d(A~,) < d(A~) = d~. Thus
II) and ~ = ~'
~-i ,y ,
Now for each of
new path ~' as it is shown in Fig.
It is easy to v e r i f y directly
that AT, = ~
.
iO.
Hence
We obtain a contradiction w i t h the
d~ = O and we take ~ = ~.
Let D be a closed 2-disk on S O such that E --
= int(D), 3
j = 1,2, ~ C int(£).
Using
~ ( e l l ) = el2 , ~ ( e 2 1 ) = e22
can construct a t r i v i a l i z a t i o n
fol(D)
of
--
D --
where
(We shall write
~ i (II,e')) w h e r e ~' is equal to one of x,y,x
our eight cases we construct
of d~.
r o t a t i o n of s.
We say that we are in the case
if we are in the case I (corresp.
minimality
negative)
from the left.
of transformations
of matrices:).
II) if the triple
T 2 = ~/(i,l),
f~l(£) _ _ ~ ,
say
> D × T2
> i d
such that for
w i t h our previo~as i d e n t i f i c a t i o n
fo f~l(D):
we
D --
fol(Ej),
j = 1,2,
~ coincides
fol(Ej) = Ej X (~/(i,1)).
217 Fig. I0
~2
Case (I,~)
Case (I,~)
N
% Case (I;~-l)
~2
Case (II,~)
case
w (i,g-~)
7
case (II,7)
/
Case (II,~-1)
Case (II,~ -I)
218 N o w it is easy to v e r i f y direct which
sum of
fo: M o --->So
is c o n s t r u c t e d
coordinates
that
and a K o d a i r a
as follows:
(~o:~i).
f: M ---> S is i s o m o r p h i c
• = ~
We
{x E ~ p l l ~ ( x ) <_ el,
shall use the f o l l o w i n g S1
:
{w ~ ~
G be a c y c l i c
i~l
group
=
E' =
-
-
[x ~ cpl,
D
homogeneous
~l'
notations:
1},
> $2
~o
T' = •
E =
f2:M2
w e take S 2 = ~ p l w i t h
~l
Let
fib=ation
to the
=
Let
< e},
k,m
~II~I
Ca ~
of a u t o m o r p h i s m s
T'(x)
<
£ < i.
E ~ •
6}
of D 1 X S 1
generated
by
2hi ~: L
[~,w] ---> [~6,wpk],
m,k
[O,w] N
mjk
b e the p o i n t of Lm,k, >D e
= f-i (o -o) -~mjK
Let
p = e m
~ ~ D 1 E
w ( S~
= (D 1 X S I ) / G E~ --- '
f--m,k: L m , k
L"
where
E
Co,Coo
b e the m a p d e f i n e d b y
E ~pl
manifold
where we identify
to [~,w]
f~,k([~w]~)
be d e f i n e d
by
U = ~pl_(coUCoo).
L as the u n i o n
x ~ L
T(Co) = O,
of UXS I,
, x = [Q,w] N, w i t h ml,k 1
-k
[~
mI
E D 1 X ~l 6~ = Qm
and
"
and E. = E -Cos E.' = E'-Cco, 3-dimensional
corresponding
, ~ ,
,w~]~T~
1
sl
] ~ E. x_
cu
sl
x_
~'(coo ) = O,
Define
a
L m l , k l ' L m 2 , k 2'
219 and
x' ~ L
m2jk 2
m2
[(~,)
, x' = [~',w'] ~ ~,
-k
with
iii {
, w,(_[~_q.)
be d e f i n e d by
2]
~ E:
X S 1 = U X S J-.
enx
f(x)
N O W we take
M~
certain
when
x E Lml,kl,
f_m2Jk2(X)
when
x g Lm2,k 2
and
f2 = f ° ( P r L ) "
llii ) we have
that M ~ P is d i f f e o m o r p h i c
is a b - m a n i f o l d
disjoint
smooth
f : L ---> ~ p 1
s
x)
M 2 = LXS 1
By L e m m a ~2here
:
Let
obtained
circles
to P ~ 8 ~ M ~
from M 2 by s u r g e r i e s
s b and
s
constructed
along
in L e m m a
ii.
C
This
construction
we can take
(see the f o r m u l a t i o n
s b = pXS I, s
of L e m m a
= ~×q, w h e r e
ii) is such that
p E L, q E S 1 and
~ is a
C
smooth
circle
then ~I(M~) to a l e n s e
= O.
Since
~I(M)
N o w it is easy
s p a c e and b e c a u s e
representative that M 2
in L.
intersection diffeomorphic
= O
of ~I(L).
to an S 2 - b u n d l e
the c i r c l e Using
# P~
Q
(see
[8]).
Lemma
over S 2 "
form of P ~ ~Q is odd, we have to P ~ Q
~I(M #P) = O and
to see that L is d i f f e o m o r p h i c
~I(M2)
for a g e n e r a t o r
is d i f f e o m o r p h i c
= O w e have
~ must be a i}, w e o b t a i n Because
that P ~ 8Q 4 M 2
the is
T h u s M ~ p ~ 2P ~ 9Q. O .E .D.
220
Theorem multiple positive
ll.
Let
fibers
integer
i)
f: M --->S be a Kodaira
such that ~I(M) = O. and the following
if ~(M) = 1 and
ii)
~ ~ 2
fibration
Then
with
X(M) = e(M)12 is a
is true:
then M is diffeomorphic
in all cases M ~ P is diffeomorphic
to P ~ gQ~
to
2×(M)P 4 (IO×(M)-I)Q. Proof.
i) follows
from Lemma
~I(M) = O we have that X(M)
12 and Lemma
is a positive
iO.
Because
integer by Lemma
12 and
Lemma ~. ii)
Using Lemma
the following Statement
that M' ~ P
Suppose
that X(M) > i and that
f': M' --->S' with ~I(M') is diffeomorphic
M # P is diffeomorphic
that the Lefshetz
for any Kodaira
= O and X(M') < X(M) we have
to 2X(M')P ~ (IOX(M')-I)Q.
(e).
fibration
From Lemma of 2-toruses
12 we know that ~ < 2 and ~-: M----> ~
to f: M --->S is regular.
We have also e(M) = e(M)
Since
from Theorem
X(M) > l, we obtain
regular Lefshetz b ~
~2: %
~ 2 "
Then
to 2X(M)P ~ (IOX(M)-I)Qo
Proof of Statement
~: M
to prove
inductive
(~).
fibration
i@ we see that it would be enough
fibration
is isomoDphic
of 2-toruses to the direct
Using Definitions
corresponding
(Lemma 9).
9 that there exists a ~2: %
~ ~2
sum of f : M o o
such that > S
o
and
~, 8 ~ and lO we can construct
a
221 Kodaira fibre tion
f2:M2
---~$2 with v multiple fibers such that
: M2
>~2
is the Lefshetz
f2 : M 2
>S 2
and
of
fibration corresponding
to
f: M ---->S is isomorphic to the direct sum
fo: Mo ----->So and
f2:M2
that M ~ P is diffeomorphic
----->S2"
By Lemma ii ii) we have
to P # ~Q # M ~
where
M2
is
obtained from M 2 by surgeries along two smooth disjoint circles embedded in M 2. Lemma 12 to (we use that
M2
Let
f: M
~l(M) = 0).
S 2 ~ S 2 be either S2×S 2 or P ~ Q. ~S and
~S 2
we obtain
~I(M2) = 0
Then by the results of Wall (see [8]) we get
is diffeomorphic
diffeomorphic
f2:M2
Applying
to M 2 ~ 2(S 2 x S 2 ). ~
to P ~ ~Q ~ M 2 # 2($2XS 2) .
Hence M # P is
Since the intersection
form of P # 8Q ~ M 2 is of odd type we obtain (again referring to [8]) that (P ~ Q ~ M 2 ) ~ 2 ( S 2 × S 2 ) = ( ~ Q 4 ~ ) # 2 ( P ~ ) It is clear that X(M) = X(M2)+X(Mo),
~M2~P~IOQ
that is, ~(M2) = X(M)-l.
Using the supposition of the induction we have that M 2 ~ P ~ 2×(M2)P ~ IO×(M2)-I)Q. Hence
M~P=M 2~P4loQ)
=(M 2~p) # (2P# loQ) =
2X(M2)P ~ (IOX(N2)-L)Q # 2P # IOQ 2×(M)P ~ (lo×(M)-I)Q.
Q.E.D.
).
~4.
T o p o l o g y of s i m p l y - c o n n e c t e d e l l i p t i c surfaces. T h e o r e m 12.
Let V be a s i m p l y - c o n n e c t e d elliptic
surface.
T h e n V is almost c o m p l e t e l y decomposable. Proof.
W i t h o u t loss of g e n e r a l i t y we can assume that V is
a m i n i m a l e l l i p t i c surface, map
f: V
that is, there exists a h o l o m o r p h i c
• ~ ~, w h e r e ~ is a c o m p a c t R i e m a n n surface
for g e n e r i c x E A,
such that
f-l(A) is an e l l i p t i c curve and there are no
e x c e p t i o n a l curves in the fibers of f.
F r o m T h e o r e m s 8 and 8a it
follows that we can a s s u m e that all singular
fibers of f: V
>
are of type m oI or I I. It is easy to see then that f±bration.
f: V
> ~ w i l l be a K o d a i r a
N o w the t h e o r e m follows from T h e o r e m Ii.
Corollary.
Let V I , V 2
Q.E.D.
be s i m p l y - c o n n e c t e d e l l i p t i c surfaces
w i t h b2(Vl) = b2(V2) , ~(Vl) = ~(V2)
(~(V) is the signature of V).
T h e n V 1 ~ P is d i f f e o m o r p h i c to V 2 ~ P.
APPENDIX
R. Livne.
II
A theorem about the modular group.
Let G be a group,
and (al,...,an)
be the set of all n-tuples
(bl,--.,bn)
i)
b l b 2 . . . b n = ala2...an;
2)
there is a permutation conjugate
be an n-tuple in G. in G w h i c h
Let T
satisfy:
~ of l,'--,n such that b. is l
to a~(i).
For 1 < i < n-I define t r a n s f o r m a t i o n s
R. on T:
Rj(bl,...,bj_l,bj,bj+l,bj+2,...
) =
(bl. • .bj_l,bj +l,bj-i+ibjbj +l,bj +2' " " " ) "
It is clear that R. maps T into itself: 3 still satisfied after their application.
the conditions Moreover,
i), 2) are
R. are bijective. 3
In fact, the inverses are given by
R~l(bl,...~j_l,bj,bj+l,bj+2"''')= (bl' "'~j-l'bjbj+lb~l'bj,bj+2"" "")" Both R. and R[ 1 w i l l be called 3 3 T h e analysis
"the elementary
of "global monodromy"
the following algebraic
in complex geometry motivates
problem:
Let C be the group of t r a n s f o r m a t i o n s sequences of elementary
transformations".
of T w h i c h are equal to finite
transformations.
D e s c r i b e the action of C
224
on T.
(For e x a m p l e ,
w h e n it is t r a n s i t i v e ? )
T h e c a s e w h e n G is a free g r o u p a n d generators
of G w a s
considered
b y E. A r t i n
t h a t in t h i s c a s e C a c t s t r a n s i t i v e l y Let al.....a n = A . we get
if it is i) w e
a r e free
in [2~].
He proved
.R2.R 1 to
(bl,-..,bn)
on T.
A p. p l y.i n g . R n _ 1.
(b2,'..,bn,A-~iA).
(especially
{al,..-,an]
N o w if A is in the c e n t e r see t h a t C c o n t a i n s
~ T
of the g r o u p G
any cyclic
s h i f t on
(bl,''',bn). N o w let G b e the m o d u l a r
group ~ ~ 5~Z2,
w h i c h w e p r e s e n t as
I
[ a , b l a 3 = b 2 = i]. a a n d b. where
Any
Each element
e a c h t. is a , a l
2
element
in G c a n be e x p r e s s e d
g E G has a unique or b, a n d s u c c e s s i v e
presentation
We call such a presentation
"reduced"
~(g) = k, t h e l e n g t h of g.
Let
s o = a2b,
a) S 2 S l S o S 2 S l S o
s I = aba,
s 2 = b a 2.
It is c l e a r
that
to e a c h o t h e r
They
g = t I ... t k
clearly
(by p o w e r s
for a n e l e m e n t
satisfy:
g-i is t k
!
-.. t I w h e r e
I b
if
t i = b,
a
if
ti = a ,
a
if
t
= a2 . l
of a).
g in r e d u c e d |
f o r m of t h e i n v e r s e
t'i =
as t l - . . . ' t k ,
= 1.
b) T h e y a r e c o n j u g a t e
reduced
in
t. 's c a n n o t b e t w o b ' s l
or t w o p o w e r s o f a. and define
as a w o r d
form t l . . - ~ ,
the
225
If we conjugate
s o or s 2 by en element
up with one of the s.. l one of them, Q-labaQ
Therefore,
any conjugate
and then we call it "short"
= Q-iSlQ , where Q is expressed
Q-I is w r i t t e n
l, we end
of the s
l
is either
or else it is
in reduced
form tl---tk,
as tk...t I and t I = b.
In this case we say this conjugate Theorem.
t of length
is "long".
Let gl,...,g n be conjugates
of the el, such that
gl....-g n = i. Then,
by successive
the n-typle with
application
(gl,...,gn)
of~lemen~ary
can be transformed
transformations
to an n-tuple
(hl,...,hn)
each h. short. l Proof.
Express
each of the long gi as Q[iSlQi,
L(gl,...,gn)
= ~(Qi).
We carry the proof by induction L(gl,..-,gn) induction,
Assertion. Z(gi.gi+l) Proof.
on L(gl,...,gn).
= O, all the g's are short, we need the
as required.
If For the
following
For some i, 1 ~ i ~ n-l, we have
< max(k,~) Assume
we have ~(gigi+l)
and define
where
k = Z(gi) , ~ = ~(gi+l ).
the contrary,
h ~(gi),~(gi+l )"
that
for all i, 1 ~ i ~ n-l,
W e shall show that the
226
expression
gl.--g n cannot reduce
L e t i be
fixed and k =
gi = t k ' ' ' t l ' j = l,...,k,
~(gi) , ~ = ~ ( g i + l ).
gi+l = ~i'''~" j' = i , . . . , ~ ,
The reduced
to i, a c o n t r a d i c t i o n .
where
Write
~(tj) = ~(~j ,) = l,
(that is,
in r e d u c e d
form).
form of g i g i + 1 is t h e n e i t h e r
tk...tm+irt~m+l...~
or
tk...tm+lr
or
where
~(r) i i. There i)
are two cases:
L(r) = O, t h a t is, r = i.
gigi+ 1 = tL+l...~, Contradiction.
we have
If
g i g i + 1 = t k . . . t m + 1 or
~(gigi+l)
=
l~-kl < m a x ( k , ~ ) .
Hence
g i g i + l = t k . . . t m + I t L + l . . . ~ ~. Suppose of
m > O.
gigi+l,
Since
tk...tm+ItL+l...~ ~
form
o n e o f t h e t m + l , t L + 1 m u s t be e q u a l t o a or a 2 a n d
the s e c o n d m u s t b e e q u a l to b. e q u a l to b a n d b e c a u s e that either
is t h e r e d u c e d
t ~ mm
tm+ 1 = tm = b
B u t t h e n one of tm, L
= 1 we have or
tL+ 1 = <
t
m
= ~
m
= b.
= b.
must be That means
Contradiction.
T h u s m = O. 2)
~(r) = i, t h a t is, r = a or a 2 , t m + 1 = t m + 1 = b.
t. = t. for 1 < j < m - I a n d t = ~ = a 3 3 --m m k,~ ~ ~(gigi+l)
= k+~-2m+l,
gi m u s t b e s o or s2,
or a.
h e n c e k , ~ ~ 2m-l.
since all other conjugates
In t h i s c a s e
We have If k is even, of the s I a r e of
227
odd length.
Therefore
2
k = 2, m = l, t 1
=
a
or
Hence
a.
gi
=
s 2 .
W e h a v e k ~ 2m in this case. Similarly, is odd, w r i t e
if ~ is even,
gi = Q ~ l a b a Q i '
then where
If k
~ ~ 2m a n d g i + l = So" q = ~(Qi) ~ O.
W e have
k = 2q+3. Suppose contradicts
2m-1 = k. tm = a
2
T h e n m = q+2 a n d w e have t
or a.
Therefore
odd, w e h a v e k ~ 2 m + i > 2m. Now
denote
each gi'
l that
is,
ri,i+l,
Similarly,
•
in the r e d u c e d
--
"'"
for any j = 1 , 2 , . . . , k i ,
considered
above
m i , i + I.
Our a b o v e
k > 2m-i and b e c a u s e
for e a c h i = 1 , 2 , - - - , n ,
i = 1,2,..-,n,
k
k is
if ~ is odd,
~ > 2m.
k i = ~(gi)
and w r i t e
form as follows:
'
t(i)_ ~(i) j - ki- j +i"
Denote
r,m
respectively
L e t ro, 1 = r n , n + 1 = l, mol. = m n , n + l = O. shows
the
following:
a) If m. ~ 0 then r = a or a ~,i+l i,i+l
2
and t
(i) ~(i+l) = = b. m.l , i + l +I mi ,i+~{
t h e n r i , i + 1 = i.
b) If k.l is even and m i , i + 1 ~ O, m i - l , i = O, ri_l, i = I. gi = a2b = So
l
(for the pair gi,gi+l)
consideration
If m.l,i+l = O,
= b which
m
and
If
then gi = ha2 = s2 a n d mi_l,i~0(k
i even),
m.l,i+l = O, r i , i + 1 = 1.
m i _ l , i + m i , i + 1 <_ ki-1.
We
then see here that
228
k.-i k.-i c) If k.l is odd, then mi-l,i --< a2 ' mi,i+l -< __12 , that is, mi_l, i + mi,i+ 1 ~ ki-l. that
Now from
mi_l,i+l ~ ki-(mi,i+l+l)+l
~(i)
mi_l,i+mi,i+ 1 ~ ki-i it follows , that is, in the reduced form of gi'
is either on the left side from
mi-i, i +i t (i) mi,i+l+l = tki-(mi,i+l+l ) +i
or
~(i) coincides with mi_l,i+l ..
(~)
gl
=
" gn
Now we can write
t (i) mi,i+l+l" . . .
~(i)
~ (~(i) +i i=l mi-l,i
ki-(mi,i+l+l)
+l)ri,i+l"
Note that if ri,i+ 1 = 1 (i ~ n-l), then one of ~(i) k.-(m . . . . + 1 )+i ' ~(i+l)m l l,l+± i,i+l +I equal to b.
If
is equal to a 2 or a and the second is
ri,i+ 1 ~ I, then ri,i+ 1 = a
~(i) = ~(i+l) = b. ki-(mi,i+l+l)+l mi,i+l+l (~) is the reduced fo~m of gl...g n. Contradiction.
2
or a
and
We see that the right side in
Hence
gl...g n ~ i.
The sssertion is proved.
We return to the proof of the theorem: If ~(gigi+l) < ~(gi) we shall show that application of R.~ or R?II will reduce L if not both g i and g i + l a r e s h o r t ,
~(gi~+l) < ~(gi+l).
and s i m i l a r l y
if
229
Assuming,
as we might,
~(gi) ~ ~(gi+l) we show that
~(gi) > ~(gi+l ) (if %(gi) & L(gi+l) ~(gi) < ~(gi+l) ).
it will
In fact this is clear if gi+l is short.
it is long, then gi = Q?labal Q'l
and
gi+l = Qi~l abe Qi+l"
~(Qi) = ~(Qi+l ), the b's do not cancel, ~(gigi+l) > ~(gi),~(gi+l). -1 ~(gi+lgigi+l)
< ~(gi).
L = L(gl,...,gn). reduce L.
If
Thus
~(gi ) > ~(gi+l ).
~(gi) < Z(gi+l)
application
and all these possibilities
the sequence
u = j'-i .
2 ~ u' ~ u, of elementary So,Sl,S 2.
is (Sl,So) ,
can be transformed
Now if one of gj, j = 1,2,..',n,
can be transformed
transformations
= gj'-i
that for any u', by a finite sequence
to ( YI'" ' "''Yu',) ' where Yu' ' is any of
For u' = 2 it is clear.
Suppose
it is true for some
u' < u. Define a function v: (O,1,2) v(2) = O, v(1) = 2. transform
Now let
is
j'.
Yl = gi" Y2 = gi+l'''''Yu
Let us prove by induction
(yl,...,yu,)
of R?la will
the case where
long, say gj. for j' > i+l, take the s m a l l e ~ s u c h
where
Now we have
In this case, (gi,gi+l)
to each other by R.I and R?I.I
Con~der
If
and
To finish the proof we must consider
or (s2,sl)
If
Applying R i we evidently will reduce
both gi and gi+l are short. (So,S2)
follow that
> (O,1,2)
Yu'+l = sT"
by
v(O) = l,
~ E (O,1,2).
(YI' • .. 'Yu''Yu'+l ~ to (YI' ' ...,yu,,Yu,+l ) '
with
We can
230
yd, = SV(T). (s2,sl)
!
NOW the pair yu,,Yu,+l
is (Sl,So) , (So,S2)
and all these possibilities
other by elementary
can be transformed
transformations.
The inductive
or
to each
statement
is proved. We see that we can assume that gj'-I is any of So,Sl,S 2 . for one such sT, T = O,1,2, The same arguments j' < i.
The theorem
Z(s gj
,
) < ~(gj
work if there exists
is proved.
,
).
But
We now apply R j .
a long gj, with
Bibliography [I]
Wall, C.T.C., "On simply-connected Soc. 39 (1964), 141-1~9.
b-manifolds",
J. London Math.
[2]
Mandelbaum, R., Moishezon, B., "On the topological structure of non-singular algebraic surfaces in ~p3.. Topology, N1 (1976),
23-40. [3]
Bombieri, E., "Canonical models of surfaces of general type", Publ. Math. IHES, No. 42 (1973), 171-219.
[4]
Mandelbaum, R., Moishezon, B., "On the topological structure of simply-connected algebraic surfaces", Bulletin of AMS, Vol. 82, No. 5 (1976), 731-733.
[5]
Mandelbaum, R., "Irrational connected sums and the topology of algebraic surfaces", to appear.
[6]
Tyurina, G., "The resolution of singularities of flat deformations of double rational points", Funktsional Analiz i Prilozhen. (Functional analysis and its applications) 4: 1(1970), 77-83.
[7]
Brieskorn, E., "Die Aufl6sung der rationalen Singularit~ten holomopher Abbildungen", Math. Ann., 178 (1968), 255-270.
[8]
Wall, C.T.C., "Diffeomorphisms Soc. 39 (1964), 131-140.
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Artin, M., "On isolated rational singularities Amer. J. Math., 88 (1966), 129-136.
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[i0] Mandelbaum, ~., Moishezon, B., "On the topology of simply-connected algebraic surfaces", to appear. [ii] Kodaira, K., "Pluricanonical systems on algebraic surfaces of general type", J. Math. Soc. Japan 20 (1968), 170-192. [12] Severi, F., "Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a' suoi punti tripli apparenti", Rendiconti del circolo mat. di Palermo, T. XV (1901), 33-51.
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Subject Index Page
Almost completely decomposable (4-manifold) Arithmetical genus (of a Kodaira fibration)
l 199
Canonical singular locus Completely decomposable (4-manifold)
Direct sum of Kodaira fibrations Lefshetz fibrations
205 174, 175
Dual projective variety
82
Duality Theorem
82
Elementary transformation of Paths M-tuples in a group
177 177, 178, 223
E l l i p t i c fibration
I16
Isomorphism of Lefshetz fibrations
164
Kodaira factor-space
126
Kodaira fibration
196
Lefshetz fibration
162
of 2-toruses corresponding to Kodaira fibration
163 198, 199
234
Page Minimal desingularization
4
Ordinary singular point
Pinch-point Regular Kodaira fibration Lefshetz fibration
Satisfactorily cyclic extension Singular multiple fiber
199 174 69 ll6
Topological normalization
69
Topologically normal (algebraic function field)
69
Triplanar point Twisting (~-) Variety of chords
9 163, 164
72
