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

F. Gherardelli ( E d.)

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

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

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

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

Printed on acid-free paper

Springer.com

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

«COMPLEX ANALYSIS» Coordinatore: Prof. F. Gherardelli

A. ANDREOTTI:

Nine lectures on complex analysis

J. J.

Propagations of singularities for the Cauchy» Riemann equations

177

The mixed case of the direct image theorem and its applications

281

KOHN:

YUM-TONG SID:

Pag.

»

1

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

NINE LECTURES ON COMPLES ANALYSIS

ALDO

Corso

tenuto

a

ANDREOTTI

Bressanone

dal

3 al

12

giugno

1973

- 3-

A . Andreotti

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

September

1973, Aldo Andreotti.

-4 -

A. An d r eot t i

CONTENTS Chapter

I. 1.1. 1 ..2.

1.3. 1.4. Chapter

II..

Chapter III.

Chapter

Chapter

IV.

V.

5.1. 5:2.

5.3. 5.4.

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

1

4 5 12

19 22

27

31 35

40

43 45 47 52

56 59

63 67

71 73

74 82

-5-

.~ .

Chapter

VI.

6.1. 6.2. 6.3.

6.4. Chapter

VII . 7.1. 7.2.

7.3.

Chapter VIII.

8.1. 8.2. 8.3. 8.4. Chapter

IX.

9.1. 9.2.

9.6.

Bibliography.

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

Andreotti

86 92 94

103 111 113 118 119 124

128 13Q

131

138 147

150 154 155

158 160 163

166

- 6-

A . Andreotti

I.

Chapter 1.1 of

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

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

ions

lC

n

II:

I I

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

Zl"",Zn' A f unction

poi nt

Z

o

it

.!l on which

J1.

f

f: .a. - lC

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

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

p ans i on f or every Here a

0),

aa...

.1 I'"

a.

.I '

Z

,'"""n

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

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

A map

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

ae

=

~=l

a! ca)(z -8. )(z -8. )

etp

+6 Because

have

~ ~( a) d-

l)

.

a:

do

fl

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

d.p

and because

'" ~ d.-«. a)

d

~,

~

+

= ta
o( IIz-a

/13)

is r eal-valued, we must

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

In particular the quadratic form

Eugenio Elia Levi,

(3

aci~- k. a)

= Cl-

c{p

4(a) •

-18-

A. Andreotti

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

ata. a acts on

~

J(a)v

~

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

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

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

Then

tL'

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

del.

~Y

takes the following Taylor exp ansion:

~

~

(z)

= 4(0)

+ 2 Re

~

+ L(Oo(z) + 0«1 z 11 3 ) •

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

Then au plane to

=~:z

=u - U

E.a./

4>(z) 0

o (lz 2 J2

sufficiently small, if

Izp+ 112

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

The second one is proved with

a similar argument. c)

Theorem

(1. 4. 2).

(36]) •

(E. E. Levi

set of holomorphY with a smooth boundary.

.Q be an open

~

Then the Levi-form at

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

Assume, if possible, that

~.

oe

= o.

(0)

+

L(

F

-;>

v+v

is ho1omorphic

the map

13)

D:xF-;.F

given by

(A, v)-;:>Av

is ho1omorphic.

Given two ho1omorphic line bundles

X a mOrphism f : F L)

-7

E

(or bundle map)

such that

ff:: W 0

f

(F, IT, X),

(E,,,,-', X)

is a ho10morphic map

over

-28-

A. Andre.atii.

1i) is

for every

x IS X

the induced map

f

a:-linear.

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

ff,

X)

(X

x:

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

is said to be trivial if prX ' X).

~ a:~

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

such that

j

fr0'f

-1 i

(x, y) = x,

we have two trivializations of ~i

.. -1 ( x,

0 ~ j

On F

U

i

() U j

and thus

v ) = ( x,

gij(X)

v)

with

(1)

is holomorphic and never zero

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

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

5

on

-29-

A. Andreotti

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

l

( f, zr , X)

we c an consider

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

r (Xi

F)

=l s

: X ~ F

ho LomorphLc , tro s

5

~

= Ui

of holomorphic functions such th at (cf. b)

Given two holomorphic sections

51

30 ).

= ~ sl1~

and

So = f 80i1

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

given locally by

X,

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

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

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

=~

with

x

~

p, q

6

X we can find a nei ghbor hood

!I(v) ,

q

Ei

)/(V) - D(V).

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

Px

llx

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

[48]).

-30-

A. Andreotti Let

fV(xi)] i€I

Then on

V(X

be a covering of

=~j

qi

i.e.

By the Euclid lemma

Piqj = Pjqi

Pj

must divide

Pi

and

must divide

Pj

i.e.

Pi'

Pj

Pi = gij

z ero.

with

gij

a unit in

gij

qi

Moreover on

V(X

i)

on

i

H(V )

(l

j

= (l

V(x

gij

vex j)

qj

n V(Xk )

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

1 Pi!

coll ection

r qi ~

CI.

1

gij5

line bunJle

holo mor ~h ic

gives

~.

we must have

holo mor phic s e c t i on giv es

Do

is a s e t of transition

F

over

"i

of

m Let

51

F

the collectand the of

with

F

So

as r equired.

So

(F, 7r, X)

on a covering

X,

holomorphic s e c tion

not i de ntically zero on ill1y open set.

c)

H(V )

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

this means th at

So

with such neighborhoods.

i)

Pi

ion

X

be a holo morphic line b un dl,e on

U = ~ u~ i"I

of

X

X

by tr an s it ion functi ons

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

Fit,

(~, 'ill' X)

t gij) 1 '2

•

gi ve n

f

gil>.

-31 -

.A. .Al1dreott i

We can then consi der t h e grad ed ring

rl)

=

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

=

then

st

then

J(X.

£

field of

flex,

reX,

rex, rl),

r~).

F)

If

X

is an integral domain and one can consider the

quotieits

=f..!1 So

F)

r

is connec ted, as we always assume,

rl)

sl' so£ [(X,

- f or some

1,

so';

01.

We have in particul ar Theorem

~ (X, F) c

!!!lit wherever t hi s i s d efi ned . In oth2r words f l ••••• f are analyt i cal ly d ependent i f at any k point whe r e e ac h one o f t hese f unc t i ons i s holomorphic the Jacobian .J

aUp .•.• f k ) 1'\ ( ) o zl.···. zn

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

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

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

I n f act if

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

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

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

k.s...n . fl' ••• ,fk _ l be a

- 3 7-

A. Andreotti

Differenti atin g this

But UX

a

k

i ~e ntity

we get

thus we get a non-trivial line ar r elation

(f) ~ 0,

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

=

Zs

=

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

=

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

Indeed for each

fi

t here exists a holomorphic line bundle

and holomorphic sections t(i) 1

= :tIT t 0

t(i) with 1

t(i) ~ 0

0

Fi

such xhat

- 38 -

lJ. . .An d r eotti

F = Fl

Taking

th en

••• F

k

k

= i1J"l

So

(1) t(i) si = to • • • 1

Moreover

t~k)€

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

r (X,

and

So J! O.

F)

and we have

Y

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

such that

i) ii) 11i)

1v)

Fl p

a

1

~

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

~

i s trivial i

S(15

ai

u p' ~

)

Co

.=:>

Y

Y

at e ach point

a

1

t he f unctions

f l, •• • , f k

are

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

holomorphic and

(i)

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

we may ass ume t h at v)

vi)

GJ Pa f

is trivi al i

is holomorphic at each p oi nt

ai •

-39-

A. Andreotti

As in the lemma

1/~/1

= ek

IIG 1\ = e

we define

(2.3.2)

=F

~

(where

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

x k +l

variabl es of d e gr ee

xk

xl"'"

and of degree

r

s

s; q i So. r

1 $. i

for

••• ,

d

So k

1 k xl ••• x k k' ex k+l

and

0

sk'

0'

l)€

f(X,

~r

over

S

• G

)

k~

x k+l

s, d k+l s, e ,

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

d



c1

~l'

0

k+l

'

=£ c... where

(k times»

• •• F

c,)

Let

t

of

•

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

h

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

a ssociates to

(1(sO, •••• sk' 6

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

0,

0-

1

)

The map which

th e Taylor exp ansion up

P(f l, •••• f k , f)

at each point

N

lm

where

/1(.~

e-

8-

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

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

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

-40-

A.Andreotti

Now the target apace has a dimension

6=N

r

k r Jl + Bwl + l+It) k

= N «(krH + :c..lJ

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

k-I

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

So If we select

a

such that

s+l) Nkltl then, if

r

t

is sufficiently large, we get

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

r.

and there fore

Ker

£,

.;.

o.

Algebraic fields of meromorphic functions.

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

d

= di ma: V.

Q)

-41-

.A• Andreotti

We want to prove the

followin~

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

f

si

i =So

Secondly we find coordinate polycilinders P ai such that

i)

F

restricted to a neighborhood of

U

pI

~

is ~

pI

~

• 1

~

i

~

is trivial

ii) 11i)

~

z»

Y

~

(i)

k

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

Nt

-42-.

A. Andrec tti

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

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

lQ~ l~i~N

.'

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

Gl~

G such that

f

="-1d t1

and satisfying

is trivial.

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

II Gil = e and we choose the centers also

f

is holomorphic,

of

P

but not on

l ~~ ~

in

f.

We also

t

V(~)

so that at

1 ~ i ~ No'

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

A. Andreotti Clearly find

Q

E

1I:(f1"'"

f

k, such th at

5t (X)

fk ,

lI:(fl"'"

f)

r,

C

?( (X).

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

=

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

Q)

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

11:( fl"'"

f k )1 •

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

1.

This means that

h

a:( f

£

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

!N(a:)

l' : X -

point of

X.

IN(a:)

Let

X

Im ')

~

Y

Let

l,

• •• , f

k,

f)

n = d i maf at some

is contained in an i rreducible alge-

of the s ame dimensions than

Y

therefore the

be a pseudoconcave manifold and let

be a holomornhic map of rank

braic variety ~.

we can

h) = lI:(f l,· .. , f k , Q).

Then c(

h ~ '1«X) , .I

Let

X.

be the smallest algebraic subvariety of

containing

(X).

Certainly

Y

exists and is i rreduc-

ible, it is defined by the homogeneous prime ideal

~Y = ?p where

a:[zo"'"

Let

!?. (y) f ~

II: [Zo"'"

~1

r

zN]

pOl =

°S

denotes the gr ad e d ring of homo geneous

!N~~?'

polynomials on element

Eo

be the field

I(Y)

0

f r ational fu nctions on

= ~q

f

pq' - p'q (,

This s hows th at

mero morphic function on

with

q t

eous polynomials

rf\.

Y.

Any

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

X.

~Y' f OJ

If

f

= ~q = £: q

is a well d e f i n e d

we h ave t herefore d e f i ned

necessarily inj ective, ho momorphism

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

then

a,

-44-

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

Now

di mlCY = tr ansende nce d e gre ~ of

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

0

R (y) _

t .r-anecend enc e

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

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

1.

b) Let

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

non-sin g ul "~

by t he

IN(~)

is

al ge brai c s ub-

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

lPn(lC)

of the s ame dimensi on then

B.

(e

r.

(1 51).

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

2.

3.

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

Prove

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

-45-

A Andreotti

Properly discontinuous pseudoconcave groups:

Chapter III.

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

r

connected manifold and

r

We will say th at compact set

Aut(X)

c

X

is a complex

a group of automorphisms of

is properlY discontinuous on

K c X

X

if for every

the set

fr~ flyKnK"~J is a finite set.

K

In particular taking for

have that the isotropy group of

t, is a finite group.

X

=fYcC!yx

o

~

y e tJ(x ) o

I y

€

U(x

a point

f xos we

o = xoS

0

rx

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

II

o

-invariant relatively

for s ome

o)

'(f.

r ,

then rt e

rx

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

v(n,

for

E.

>

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

0

X.

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

o

rl

o

for c. " [. o ' S(£.)::/

a

1e

fx • o

Then

e,

If for any

r , 'f

,J.

~

rx

is f i n it e if

S(E)

wi th

-y

o

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

the eqUivalence relation

phiBms of ['-automorphic line bundles F

is the trivial bundle.

~F,"'; X,~

pyH

in which

- 48 -

A. An d r e otti

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

c)

Given a

we

can consider the sp ace of

r (X,

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

So

element of

p

0r

F,

X

* )t(Z).

r -automorphic

[-invariant sections of it:

and

The proof is straightforward and is thus repeat the

consid~rations

p

*it(Z)

holomorphic line bundle and two sO; 0,

sl

such that

omitted.

sl m = -.

So

We can also

of section 2.2 c) in this more general

case. 3.2. a)

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

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

z

= Zo

(t l, t

h o od

L) ii)

We will say th at fin d a complex

~ an

2)

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

V of

i n E,

Zo

V (151-2, {z

L(q)z

Let now

r

l~

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

v~ •

X

the orbit 0

f

rz

o

x.

i.e. an

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

x ~

th en

contains

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

-49-

.A. Andreotti

=

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

wh ere

Z

= X/f.

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

= k~O

['(X,

~/

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

C!

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

rI

and

r 2'

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

A. And reotti

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

is c ompact.

l'

Su"pose th at G = ['1

th at

condi tions

f

Set

Then

3.3

r2

is pseudoconcave. is pseudoconc ave.

an d

ii)

It is enough to show Let

n c.

specified above in

X

satisfy the

a)

r = r l'

for

1 = ~ G u ••• o ~ G.

n, = ail n.

r=

for

(1

i)

1

u ••• u

ail

n.

veri fies th e same condi t 'ions

G.

Siegel modular group.

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

n

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

be an int eger,

be the s e t of

(r3]).

nAn

n > 0,

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

rt c«,

G

t

0:)

Z = Z,

Im Z >

°.

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

o:tn(n+l).

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

Note t h at

n

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

i~l

lR

>


Z

~

be the normalization of

is also a projective algebraic variety Since

(531) ~

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

Z factors through the normalization

holomorphic map

N*

Since T

is of rank

n

everywhere, and since ",*

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

Let

"

T

the closure

0

being

must be one-to-one and

A is a

We are thus reduced to show that ~e

*

Z

f the graph

T

,.,

of 1"

in

~inite set.

X)I.

lIN( an.'

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

l

(x , t)

X > : T

and set

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

U

(1)

is of r an k

n

)( (X) =

-:;>

*

Z

"'* ., ( a)

be the n atural =

t3d -1 (

a).

Let

and not cont aining any

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

a:

Z.

-71-

A. Andreotti

er·

Now

:z,

must be one-to-one as

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

(3/

-1 ·

mu",t also be one-to-one. Z·

op~ ;bl!)thus a neighborhood in

.A

:.fI..,

-;>

n,

We · cla1ll that each component

-1

(U-a)

must

(U)

and since

=

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

N.

of 7' (a)

B

-1

~

= d. \3-1 : a - r· ( a) ~ U

normal, the holomorphic map A holollorphic map

pI

is normal; so

extends to a

which is not of co-

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

is of codimension 2. 2,

B

then A

must have a non-

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

a

tP

A.·

Now we remark that

(tj)

compact.

and

Thus for each

.-

rv·.

is a proper map because

C(

a '" A T (a)

many compact analytic subsets of

Z

•

is

is a union of finitely

=Y

Z - l (Bc )

of codimension

one and finitely many irreducible componentg of the singular set of

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

tV.

previous ariument it follows that if

..,1* .

a,. #.

= ..

a,. ,

a

Moreover, from the are in

2

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

and

A

Y con-

tains at most tinitely many irreducible compact analytic subsets of codimension

1

Z•

(here we use that the singular set of

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

>

0

B -.

co

Ic _

o

Then!

co-e.. given by r,

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

be so small that

biholomorphically to

.

e•

is compact in

•

Z

Y

-

N*

-r

and that

extends

7

Set

(Bc _€,). 0

and we can, by the isomorphism on

Y-

extend this function by putting it equal to

c o-

transpose the function

~

YCo-e.• €,

on

Yc

Let us o- e.

and

-72-

A . Andreotti

call

t

-r: Y

~

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

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

11)

iii)

on

Y-

yc. o

E-

°is

c

c

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

for any

z,

0

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

=

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

The c ase of di mE X L 3.

If dimE X L. 3

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

-7 3-

A. Andreotti

The proof of this t heorem i s r "ther c omplic :,t ed.cnu will be omitted. However t he interest of t he t heorem r es i de s i n th e f act th ctt it do es not hol l f or dima: X = 2. We will indic ate ho w to con struct * g equals

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

holomorphic an d the graph Denote by

rN

and

J'

0

f

g

the graphs

We h ave a commutative di agram

...

while

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

is analytic. 0

f

~

*g

and

g

respectively.

X

r --» I'

1;r 1

fT

X~y

where

X.

rr and rr are the n atural projections and where

= (1/>)( id) 1 '::' •

1

is

- 86-

A. Andreotti Since

ff.

and

tt

are homeomorphisms

4 is semiproper. the composed map

Horeover

r

7\. is semiproper bec ause

is closed in

y"

Therefore

It.

Y " It

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

By assumption

d>

* g is meromorphf.c

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

X.

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

V be a neighborhood of

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

Let gi: Y -;> Ie sider the map

be such that

p(K)

p * gi

= V.

= fi

for

b

with a

Select

1 s; i

s,

k,

Con-

x.:

defined by X(X) = (gl(X)" •• ' gk(X».

Let

L =

(V).

Then,

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

=

- 87-

A . Andreotti

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

z.

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

=

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

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

-8 8-

A. Andreotti

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

(iii)

f:n-'lY

is a proper map

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

for every

ye. f(U .. )

1

Vv ~ with

we h ave

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

V

v

n

f

-1

(y)

•

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

dim; X - dim; Y.

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

4.4,

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

remark 1, gives an example of a pseudo-

IS

- 89-

A. Andreotti

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

Xt

can be

2.

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

•

count abl y many copies Consider the sets

M

a: •

of

II: k

= ku.2Z

!.I k "- a: k•

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

where

t

k

~

.d

w

k'

k

* . eICk

gener ated by the

Define the ac t i on o f

r

on

S

to be

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

Indeed, let

f " /I(X) ,

holomorphic function expansion of

f

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

0 ,n

(t ) t

-1

- nk .

f

= n=s;

fk

k ~,n(t)

f +

=

then

£

w),

d

J

Ak

a. (t) ~K, n

H(4). fo r

;

= ~ Ui~ i G-I I

X

by open ,s e t s

then one can define a

with the prop erty

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

one defines a homomorphism of comp lexes

J

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

,T*

by sett1~g ( i f)

jO ••• j

= r q

Uj"{jo) •••1(j )

Vj

q

j

0·· ' q

This defines a homomorphism of cohomolo gy groups J

~ ..: v

One verifi es th at :r

H* (?(,

n«

'2.-'--

'1).

:J) -,) H* ( V,

d e oe nd s only on t he coverings

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

2{ , 2, Q-

One c an t hen

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

Hq(

lim -.;>

-z( ,c-y:: )

q

I

0,

~

7.{

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

(?!, Jr,

into ~

ii)

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

Homomo r phi sms o f

gr o up s

L)

v

X) ,

sh e ~ves.

( 5', {~"

is a c ontinuous map

u; o ¢

X

of

X.

Thi s is

with val ues in

"5-

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

X)

over

d

-:; -> ,~

X

a ho momorphism

0

s uch th at

tr

f or ev e ry x T::" f

~> 1/ -"? °

is called exact

°~

"T' J x

e:

if for each X

1fx

->

is an exact sequence. the sequence

°

(3 x

-;>

-:>

x

~

X

the sequence

Jlx-7> °

In particul :cr if

~~

!i:

-;>

1

fl7 -;>

is a subsheaf Of'; 0

is exact.

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

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

this induces a homo-

morphism of cohomology groups

t> It:

Hq(X,

T)

-> Hq(X,,lJ •

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

X

trr'

1

d.

""I' d:.

*

HO(X'J)

*

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

,.,>H 2 (X, err. j ) H°(X, T>

where

(1)

Hl(X,J' )

6

(3*

HO(X,J()

-:>

(3* ---;>

Hl(X,J!>

.~

~

*

is a "connecting homomorphism" which is defined in

the usual woy.

- 94-

A. A nd r eotti y

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

X.

sp ace

coverings

0

X.

f

U

i O " .i q

cofinal to all

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

i'

o

q

T ) =0

» > O.

for al

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

T

Let space Hr(U .

J. O•

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

X.

i , ~) •• q

be an open covering of

=0

X

such th at

iO, ••• ,i q, ~

for all

q

and all

r > O.

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

X.

covering o f

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

on an open set one has

i Ui S

by open s e t of ho l omo rphy .

is al s o an open set of holomorphy.

space)

"2{ =

be a c omp Lex manifold and X

U

q ~ O.

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

be an open

Th en any U i

A theorem of

O

••• i

q

H. Cart an and

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

":F'

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

Hr(U,

T)

=0

r >

if

o.

Thus coverings by

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

Ac yc l i c resolutions.

on the paracomp act

sp ace

Let X.

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

T

vr

be a she af of abelian gr01H!s

By a r esol u t ion

~ .r-J- 2

d

of

2

- '7

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

1-i )

=0

for all

CJ. )

0

and all

i 2. O.

T

we mean

- 95-

A. Andreotti De Rham theorem.

II

Gech cohomology of

X

(2)

is an acyclic resolution of

with values in "}'

".:r

~

is n aturally isomorphic

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

'10 )

i.e.

Hq (X, ar' j-)

'lJ

'1'

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

- Imtf(X,

7'q-1 )

Examples. 1.

lJ-

A sheaf of a b e l ian gro ups

open set

Uc

surjective.

Hq(X,X)

=0

X

:i.:.;;::-

is called flabby if for every

the restriction map .f(X,JeJ

For a flabby sheaf for all

q> O.

j;:...

-7

r

(U,

is

one has that

Every sh eaf

~ of a be li an groups

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

t- 0

= sheaf

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

'J-.

C?j-

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

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

2. A

A she af;:' c,

X

is call ed ~ if for every closed subset

the restriction map

For :~ soft sheaf

,X

r

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

one h as again

Therefore soft resolutions of a she af Let ·X

be a

different~able

is s ur j e c t i v e .

Hq(X,j.) = 0

st

f o r all q > O.

are acyclic.

manifold ef dimension

n.

Let

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

0

f the h omomo rphism

of c onstant fun c tions

IR

d:

"f 0 ~

j( 1

Since for e ach

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

is the she af r

t he s h t af

The 0

f germs

uf

r

-9.6-

A. Andreotti

One has

d. d=O.

Moreover the sequence is an exadt sequence

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

4. Let

Let

X be

a

a complex manifold of complex dimension

n.

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

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

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

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

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

°

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

°

°

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

Reference s:

[25J,

[30].

5

for any

k 2. 0.

-9 7-

A. An d r e otti

6.2

E.E. Levi problem.

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

I

X = tn

As for the case

we de f i ne

D to be an open set of

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

Let

n

2

2;

X=m

and let

1":

lIfn

-7

be the unit cube in

consider t he real torus 2n

/ 7Jn

X be the n atural projection.

m 2n t

Let

so th at 1'(Q) = X.

Consider the following set where

(I)

0



zq( Z'"-"

y)

defined by G) (

X2!~) ~ •••

(I

X:

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

rex,

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

where ,,( a) = '\

1

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

(3 (a & b) =

°

a/x

X

1° 2

q (1

~

0:

x2 ' e q )

-.;>

0

- bl x1ox2

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

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

(1)

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

-.103-

A. An dr eotti

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

o s:

111)

j

t,

j S.

with smooth strongly Levi-complex boundaries such

B - B c.c. U i_l i 1

i~t

and

1

any coherent

j,

o •

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

neighborhood such that

U=

~

=

~.

E

1

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

1v)

X

.!2!:

tha~

u,

of

U

L(+)x) 0

~U.} lSi~t

balls

B

=It ,

B () U

there is a

u I~

(x)


0

The covering

will be chosen with SUfficiently small coordinate

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

We now select

C

1 =~ -tlP l,

4>2

X,

on

>0 E

1

= Q- cl f'1 - t.2 P2 ' · · · '

have all their Levi forms

L( ~ i)X > 0

1

1!.. t,

5.

for all

€.

i

1s

with

a B.

and SUfficiently

;> 0

4>t

x

U

= 4>-E l Pl - · · · - Ct t' t

for all

x

€

U.

Define B

i

=B l x U

G

U

I

01,



t

wi th

It

U

U.

in

C

1

p

in

U

=~ -!.P.

>

0

A .. D

u

L(h)

with p

2.

TaIte ~ ;> 0

on

U.

X €

U

Set

~

d4

~ 0

in

U

and

We can write

~

C .,. function and

is a

U -? lR

> 0

L(~)

II zll

3.)

= 0 s(\

0, a upp

p

D

d~ l ~ 0

0,

on

OS.

¢l (x) -c

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

l(A, d1ma: H 9)

Write

=r

A = D


HO(D,

6)

for the sheE!-f () :

o -7

HO(A, L9

~ Hl(A, & ) -r

~

HO(A (\

u, e )

-?

HO(D () U, f) )

§..

-101;-

Consider the sequence of functions D

n U.

;lJ

A. Andreotti for

jJ

= 1, 2, •••

in

These are ho1omorphic and linearly independent over 1

~.

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

we can find constants c l"'"

1

~.

c r +1 m is a meromorphic function in A, holomorphic in D r+1 1 with principal part at Zo c i t1 For the function

Hence g

=hn

t

we thus h ave

I g(x)/ = 0".

Q..E.D

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

-101-

A . Andreotti

The proof is obtained from theorem (6.3.4) by the Mittag-Leffler procedure making use of the fact that for s;p ~ c : a: n ~ JB wi th the properties . (x) < Os be a strongly Levi-convex relatively compact subset of the complex manifold X. We have proved that for any coherent sheaf Hl(D,jf) is finite-dimen-

3.

°

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

=

xl

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

-.>

Hl(D

e

Moreover one

,"J')

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

y

,~

0.

- 10 8 -

A. Andreotti

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

~ II:

x

'"

HO(D ,(}) ~

I

Since

4.

Let

=

II:

t

t3

y

°

D

C .::>

3.

be as in }

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

1m 13 =

is an isomorphism,

C = ~ (x, y) .; D

Cl early

a

°

~) thus

c£ .

i s s urjective.

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

0

l'

D " D.

fo r every

1' '' # ( D ) •

Consider the closure

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

6.4.

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

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

A = U i

~

B.

=

Zo

zi -F

° wh ere

Yl = - t···, Yi - l = zi

we as s um e

.>..8

c oo r d i.nrt es

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

t ••• t

zn Yn = zi

-109-

A. Andr e otti Consider the hernd.tian form

dSl' =)"

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

~dS]

dsr

g

log

ziZi

where

o.J log (1 + j=lYij)

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

on

U i

rdYjdYj - (ZYjdYj)("iYjdYj)S

~dYjdYj.

2. (1 +iYi;-2

Thus

J.~

defines a hernd.tian metric on

Pn(t).

Its exterior

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

w),

(J'

zi zi

o

i..l ="2 a~ d

= aw).

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

Fn(E).

=~z2

B

g

fg

=;.

Jg

g

= 0

thus

d w",-

=0

and

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

get «J .~ l g

J

d=J.d ~

(l+yy)

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

0

!l(E)

the projective line

= zn = Os we

=

Therefore

A

Taking for

where

dsi

n -) log (1 + ~ YjY j •

W

>

0

2

where

Y

thus one can choose

=> C. (~)

The assumption

th at t h ere exi sts on

X

B

can b e re pl ac ed by the ass umption

a Kahler metric whos e ext erior form has

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

o

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

(1)

"dR".

-119-

A. Andreotti

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

2 sH (X, ~)

.H

2(X,

JI2(X, ~)

k



JI 2 ( X,

lC)

JtI2 ( X,

t)

1"'5

d

If

l'

2""

values on

sH 2(X, lC)

-y 1'1 2 , ,, , y

sH 2 ( X, ~) modulo torsion, a

is a basis of

cohomology plass of values on

lC)

d 1'S

d.1'S

is caracterized, viaP',

and i t is in the image of

l' '( 2"'"

are rational numbers.

tion shows that; the isomorphism

~'oct'o

which associates to a differential

T

j

by its

if its

A direct calcula-

is induced by the map

f>

2-form

the singular

cochain It follows that if a closed

2-form

co has rational periods i.e.

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

c

t: ~

i jk In this argUll8nt

(~)

(modulo coboundari es with values in

a:

can be replaced by

Define the sheaf ~

monic functions on

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

=c

II:

lR everywhere.

of germs of complex-valued plurihar-

X by the exact sequence of sheavea

~

It c

e

It

$' ~

11 -?>

and f.,(f Et

g)

0

=f

-

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

t).

Set for

we n ave the following

U

open in

X

g. C

fP

r

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

-J1.l-

A. Andreotti ~

!!ll

(6.4.1)

~

~n.

U

be a contractible domain of holomorphY

One has the exact seguence

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

d = ;) +;i

is the exterior differentiation.

Obviously the composition of any two consecutive maps in

the sequence is zero. resolution

°-> a:

1"\

-T .I~

Since

°d

nl

~ .1

..:>

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

•••

we get

Therefore (3)

s

k=O

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

.!2!:

q.>O

ther'l'!' eXist

k >.. k O'

ko

0

Hq(X,

» 0 such th at

fll

if

J-k) = O.

Now we apply this result to the exact sequence

o

7

tJ

k

fll F1/11

~

F-k -? Q 1./1F

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

~

0

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

J« X,

=

reX,

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

[ 27J •

-116-

A. Andreotti

Chapter 7.1.

VII.

Generalizations of the Levi-problem.

d-open sets of holomornhy.

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

cr

in

=

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

X

°

such that

D

=

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

;)WnD .. o,

c; is extendable over

In fact, if hood W of W e V,

Zo

Then ~

in

X

I W=0

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

I

thus ~ WilD

= 0.

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

~

€

U

W, '5)

-7

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

fl

Hd(D ('\ W,

W,"}')

'J')

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

r)

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

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

I

such that ~ D

= ~.

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

'J"'

- 11 7 -

A . Andr eotti

=

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

U.

We set

fl- = ~.x

€

U )

f (x)

5 I ~ (x ) = 0 S < 0

I

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

ur

=0

s > n-p-l or

{ 0< s < q.

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

Eo

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

'1

en

The number of negat ive eigenvalues of (*)

\

the number of positive eigenvalues of

L( ~)

ZIT

~

int 1. " OlD has a fundamental sequence of neighborhoods o U'" such that D (l V'Ii 1s an open set of holomorphy. We have thus to reinforce condition (*) assuming for instance that L( ~) T for every zo e:: ~ D is non-degenerate. An open set zo zo

I

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

Then one

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

~

~

l'

2.I! X.

7.2.

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

=

S H ( 0U7'v,

.-) =0, 'S-

if

s >d

=

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

and

En.

Let

on K

Moreover B.

Therefore

v (S

We can find a sequence En K,

A Ii B

be ~ oompact subset of

containing the origin in the interior aQd let

omorphic function on

any



1\

dt)

=

-128-

A . An d r e otti

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

If

D

the class e, as we did in

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

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

(8)

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

=

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

f

+

+

+

+

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

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A. Andr eotti

Chapter

VIII.

Duality theorems on complex manifolds.

8.1 Preliminaries.

v

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

u

d)

(V)

~ ~

(U)

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

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

C

and every open covering ?( sequence is exact. where

00

is defined by

and

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

X

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A . Andreotti

v

Cach homology. an open covering

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

of

= t .,)(U),

S

Given a precosheaf .

uJ

iV

and

one defines the groups

X,

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

by

aq-l1Jt. ""J.

=L(_l)h

i 1 O• • • q)

for We h ave

aq-l

0

q = 0

g

={gi 0"·. i q}I

for all

~ C (U, $).' q

.

and we put d -1 =

q 2. 1

We thus get a complex with differential operator of degree

o. -1

and an augmentation eX : C o(

We define

Hq(U, $)

21, S)

the q-th homology group of this complex,

=

H (0(,,$) q

If

V

=~ Vj1, jEJ

$ (X) .

""7

Ker

o

Im Q

9-1 q

is a refinement of

the refinement function

1":

J

'7'

I

U,

to each choice of

th ere is aascc Lat ed a

homomorphism

r * : Cq('Zf; £ )

-'I

Cq ('2{, ~ )

for every

which is compatible with the differential operator,

q

2:.

O.

Thus '1"*

induces a homomorphism

V u. :

'3'

Hq( V, {)' )

'7'

Hq 0

then the natural homomorphism

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

~

S>, S

Z/ =Z Ui~

i6 I

, ~II

be cosheaves on

be a Leray covering for

ku

and let de

an d ,S ~

II.

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

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

J'

X

-133 -

A. Andreotti

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

l

SI)

-7'

.s

(8.2.1).

~

h.

H (X, l

-7

k.

S) _

h.

H (X, l

,

S II)

Then one has

d

~

k.

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

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

t

__u

B'

tv 4-

~

C'

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

If

v

u

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

....:;. Hom cont

(~ Im u '

is a topological homomorphism then

0

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

£). is an isomorphism. with

b ' o u = o.

For every k £ ~:r ~ k = b + u(A) with v(b) = 0, we define = bl(b). This does not de pend on the choice of the represent ative b ' and b and thus it defines a linear map Ker v t. For every c > 0 the set Sl- b /0 BI I b l (b) I [(fI.,rr) -+ Co(1.(,C;) ~ Cl ('2(,"J') 7

~.

...

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

°~ "{(0-)

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

T.

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

cq-1Ctl, 1')

S

q::;

Cq(t{, T)

S

sequ~nces

'.F)

--$

cq+l('Z(,

'2(

77{ is a countable

(II)

where:F

is a coherent sheaf and

c.

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

-T3"6 -

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

we obtain

Theorem (8.3.1)

II

( a)

is a topologic al homomorphism then

cS q

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

H (X, q

Moreover

(b)

d q_1

If

cont (H Hq (X ,

'I) •

lC).

are separated. -

is a topological homomorphism then

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

Moreover

q(X,1"),

and

(Hq(X,

'1"'.),

Hq(X,"J")

lC) •

are separated.

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

0

f

H (X , q_ 1

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

q l

8.4. a) sheaf

'1.)

cr.) .

Hq+l(X,'T)

or, respectively,

are finite-dimensional.

v

eech homology and the functor Let

These conditions, are

T , Jj-

Jlom f"

be sheaves of

(:f ,f)

EXT.

e-- -modules

on

X.

Then the

is defined as the she af associated to

the preshe af U -r Hom

and

y

e)u

are coherent so is the sheaf

PI om6'

(1, f).

is a family of sup norta, we set

CJ

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

r

110m
(X:".t, ;J')-:> HOM (X; 'rl,-J)-;> 0

Le.

-137 ~

A. Andreotti

We have the following facts:

i)

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

T

ft

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

q-th

(T,fl2)?

cohomology group of this complex is denoted by

t

xt

q

tY

('7';').

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

Y-

•••

7~ 2 ~d' 1

be a resolution of

':f

7'/0 -

'Y-;.

0

(**)

be locally free she aves(l).

ApplYing

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

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A Andr-eotti

the functor

;fom

t9

.,~)

(

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

anoth ~r complex

we g e t

"

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

q:-th cohomology group is again

(. xt ~

('J' 'X ),

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

-;> • • •

as it

By this

G xt~(T ,~)

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

In particular if if

(~2'})

::J-

Note th at in any case

because

the sequence

o '-71fom t9 is exact as the functor

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

Applying to the resolution we obtain a complex whose

(*)

q-th

EXT~ (X; 'J' q = 0

For

the functor

'J', .)

,J).

~(X;

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

The spectral sequence of the double complex

P, K q d

~ (X;

we have analogously to the previous case EXT

and

HOM

cohomology group is denoted by

= ['4>

(X;

K ={KP,q, d}, where

110m c/ o(;p'j q»

is induced by the ma 's of the resolution

(*)

and

(**),

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

t. )

-- 'J EXTn (X; j-

( n = p+q)

where

b)

The connection of homology ani the functor

EXT

is

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

~

n.

-1 39-

A. An d reotti

For any open set of holomorphy ~

'T sa

the suffix ~.' (mcd" (..&p ,fq)}

EXT*k(U;

'1 'J ).

the se quence of she aves

7.Jf omt9 «6 p' /I )~ ;:{om& ()S p' ;/o ) ~;;f om£9 c.); p';

is exact as ~ ~

KP , q =

resolution of

p

~

1) -T •••

is locally fr ee and provides an inject ive .,

:/(omc: ( ,,(;P'i-).

Taking cohomolo gy with respect

to t he differe ntial co ming from th e r esolution

(- )

we get

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A. Andreotti

Efiq=H~(u,;fom§~p';»=o as

ffomt9(rPp'f)

is locally free.

EX~ (U;T

It follows that

EXTn~1 (U;~'Ji) is the

H~(U;

(0)

ql-n

if

i)

= 0

if

P < n

and

l-th cohomology group of the complex:

il om~ (rP 0' 1» ~ H~(U; .;rom &(06' 1 ,f)

-7

~ ~(u;..fom61(;: l'1)

-;> •••

In particular we get the exact sequence (1)

0

~ EXT~(U; j"f)

(8)

H~(u;;fomt' (~o'f) ~ H~(U;#om& rLl

7

'1»

Consider the exact sequence of spaces of Frechet-

Schwartz obtained by applying the functor

r

to

( •• )

••• .."r(U,06l)--- f(U,et'o)~ f(U,:(>-7 o . Exactness follows from the assumption th at of holomorphy.

By the duality lemma and

U is an open set (a. )

we get an exact

sequence (2)

0

~

(0),

Comparing

1"'.(u) = Given

T

U '7' EXT~(U;

f

H~(Ut.tom6'UO,jLn»~H~(u,ifom6'(.,tl'£\.n)

T.(U)-'!-

(1)

and

y.

T.;...)

for

!!ill

E-~·q = Hp(U

?1{

EXT~(U;j,nn) =

and

0 if

U

€;J1(.

q

I- n ,

We denote this precosheaf ~

We do not need to verify i t is a cosheaf,

(8. 4.2).

c

we obtain

coherent she avee , define a precosheaf by

Lemma

2{

(2)

EXT~(U;i',S1.n)

xt~( ·T.; ).

where

and

There exists a spectral sequence

,E xt~( 1, ~».

is a locally finite covering of

X.

- 142 -

A. An dr eotti ~.

Consider the doubl e complex

K-P,q = Cp(?(,J(omk( ·~';!q».

I f we t ake coho mology with

respect to the differential coming from

(*)

we get

7, )

I E-i' Cl = C ( 2{, { xt~( ~) and then taking homology with p v . respect to the Cech differenti dl we get

2Cl = H~ a:

~ Hl(X,D )

-t'

Hl(X ,6) (D»

-¥'

u V(ni 'Pi»

"=

°

~

~

°

the sheaf having

O-dimensional

Therefore we get, in particular:

dim~ HO(X,lJ(D» - dima: Hl(X, 6'(D»

= deg(D) + dima: HO(x,D) -

Hl(X,~ ).

- dima: Now: HO(X,e )

= II:

_1 I1(X,19)

as

X

is comp act thus

dima: HO(X,6J)

the space of holomorphic differentials on

r(X,C)(D»

g(X)

of

X.

= HO(X, e (K-D»

associated to the sheaf

called the canonical bundle,

where

Jt 1.

and

l K~

is the line

We thus have the following formula dim

IDj

=

K1..

is

= i(D)

dima: HO(X, [) (K-D» D.

(Riemann - Roch t heorem)

deg (D) - g(x) + i(D)

This formula can be extended to any divisor positiv~

~

The bundle

is called the speciality index of the divisor

if not

Its dimension

X.

if finite-dimensional and, by duality

v HO(X, Jt l(_D»

(1)

°

is finite-dimensional and, by duality, ~ H (X, _ed~&l-"O HO(X, a:) ~ a:,

HO(X,

°-r HO(X,JI.. 1) --,) H1(X,

e,

&) ~

we get an injective map

a:) •

This map associates to every holomorphic

I-form its cohomology

class as a closed form. Now of'

HO(X, J1,1) and HO(X, ji 1) can .b e considered as subspaces 1(X, H Their int~rsection is reduced to [01. In fact

er. .

if'

a e HO(, Jl 1 ) ,

g

is a

« = as-

HO(X,..ii

Eo

C ~ fun ction on Thus

'3 a g

maxi.urn principle Theref'ore

t3

1

= ,

° i ..e.

b

~

1(X)

0.., a: ~

we get an injection

and if

ft

g

is (pluri-)harmonic.

2 dima: HO(X, ~

G It

=t3 +dg

where

by reason of bidegree we get

must ve constant hence

dima: H ( X, a:)

We show now that

X,

1)

2g(X).

Jtl)

a.

i.e.

bl(X)

-7

~

2g(X).

From the exact sequence

e ~ ?1'~ °

0.., Hl ( X, a:)

By the

= 0.

l(X,t9 H ) It Hl(X,

&).

-148-

A. And reot ti

In fact HO(X, an ~ e, HO(X, (}) ~ maximum principle). Therefore dima: Hl(X, '~ ) ~ 2 dima: Hl(X,b!)

e,

HO(xdf) ~

di m

)KI

=

deg

(K)

=

D =

° we get

g(X) - 1

If' we apply the same theorem to

0)

(by the

bl(X) ~ 2g(X).

i.e.

If we apply the Riemann-Roch theorem to (2)

a:

D = K we get

2g(X) - 2.

t

To do this we have to know th at K 1 comes from a divisor. Now if g( X) ~ 1 this follows from ( 2). If g(X) then for any positive divisor D we have dim I D\ = deg (D) + i(D). But then necessarily i (D) = 0. There eXi sts on X a r ational function with a sing],e pole of first order. This function extablishes an isomorphism of X onto the Riemann sph ere and on this manifold on e verifies immediately that K = -2p, p being a point of X. In p articular for a divisor D !1!h. deg(D) > 2g-2 we must h ave i(D)

=°

= O.

As an exercise one can show now th at every compact manifold X of complex dimension one admits a porjective imbedding i.e. is projective algebraic. Indeed if .0' ..• ' St is a basis of HO(X, (D» th e map X -? Ft(l:) d e f i ned by is holomorphic everywhere. If x ~ (sO(x), ••• , St(x» ( D) > 2g t hen one verifi es by means of the Rieman n-Roch t heorem th at t he map is one-to-one and biholomorphic.

e

d) Let X be a co nnec t ed c omp act man i fo ld of dima: X = 2. Let D be a holomorphic divisor on X which will be s upp os ed o f "multi pl ic ity" on e a t e ac h point and non- s i ngul ar . We h ave now an exact s equenc e 0"";' O ~ 8 ( D ) ....,. 6) ( D) / D where

s

is a section of

-7

°

f DJ co rres pon di ng to t he di vi s or

D.

- 149 -

A. Andreotti

We get an exact cohomology sequence: 0--7 HO(X,l) ) ~ HO(X, & (D» ~ HO(D,e (D)

~ ~(x, e) j.... !?-(x, ~ ~(x, CJ) ~

2(X, H

e (D» t9

I D)

~ Hl(D, () (D) I D)

-j'

-;>

(D» -7 0.

We have dim(CI)HOCD, 6l(D)/ D) - dim£!?-(D, D (D)/ D) ::: deg

~D~ID

- genus of

(D) + 1

dim~H2(x,~ ) ::: dim£ HO(X,Jl 2) ::: Pg(X) ::: geometric genus of X dim~Hl(X,e) ::: hO,l

dim~H2(x,r:9(D» ::: dima: HO(X, S(K-D»

where tKS

denotes the

bundle corresDonding to the sheaf of holomorphic 2-forms S!.2 ::: () (K). This dimension is denoted by i(D) and called the speciality index of D. By the same argument used for dimension one we get now dim

I DI

4

deg 1 Dr)D - genus (D) + (p (X) - ho,l) - i(D) + 1. g

.

This is Castelnuovo's theorem. The l i f f er ence of the left and right side is dimll: Hl(X,8(D» which is called the "superabundance". If X is Kahler then q::: hO,l ::: hl,O ::: dima: HO(X,s&l) ::: number of linearly independent holomorp hic l-forms, and P g (X) - q ::: p a (X) is called the arithmetic genus. The inequality can be extablished without the r estrictive as s umpt i ons we h ave made on D. In particular for a multiple lD of D we get the inequ ality dim 11DI .2 1 2 deg iDS}n - (l(~-l) deg ~ D}ln + 1 genus (D) - 1+1) + ( p (X) _hO,l) - i( l D) + 1. g

If

D is positi ve and

1

large enough and pos i t i ve , then i(lDL ::: 0.

- J:? O-

A. An d r eo tti Therefore if

deg~DsjD > 0,

dim IID)

grows like

Therefore

II

t.

1

2•

°

X contains a d i vis or 0 with des OSlo> ~ trans. degree (X) 2. 'One could s how th at X i s in this

1o,o : U~

a:

I ~ °'°15 = °5 thus

QO,o(S)

°°

represents the space of C

.o

-:r

~. (Q)

, °(U) Let

a

~

u

€:

-"?

C* (U)/'1- * (U) -7 Q* (S)

For any choi ce of

-i>

-;>

°

0.

U and

S

the seguence

~O,l(U)

is exact.

1 o , 1 ( U)

-d O,O(U)

Assume th .\t the coefficients of .}pqllS =0.

with the groups

we may use the following exact sequence

'Y- * (U).....:pC* (S). '--i' Q* (S)

(9.2.1).

~

HO,s(C·(S»

ju

are fl at on

S.

Then

-156-

A..Andreotti

=pet 2

This means that q'l u

=f

2 Q'2.

thus

COiO(U)

thus

'

;1' Q'21 s = 0 . for some

Continuing in this

3

C(

E;

Co,O(U),

we see that

w~

u

d 0,°1'1°,°.

we have exactness at

«(3 )

E:

2

But then

= pc( 3

o: 2

a

for some

thus

= p3dy

u

must be flat on

To treat the general case we will make use

0

5

i.e.

f the

following fact Given on

S

a seguence

there exists a ok F

d pk

I5 =

C

f

f

O' func tion

d'

k

for

fk

f

l, 2 F 2!!

= 0,

2!

, U

C CI' func tions,

such that

1, 2, ..•

This can be derived as a particular case from the Whitney extension theorem. Let

Also direct proofs are available.(l)

e {jo,s(U),

f

s 21.

Then

= I'd. + ap

for some

/I (:,

and (3. Using the above remark, we can find a form CO,s-l(U) such that

r> 1

E.

131 1

5

akp

_l[

(*)

= 131 5 =0

ap k

S Thus we can write

k = 1, 2, 3, ...

for

a p" 13 1)

= (f -

f

as the coefficient of

Jp )\ 13 1

f -

satisfies the conditions (*).

= pQ'l

(1)

+

3p

/I

6

with

1

13 1

+

~u

"5p

/I ( \

vanish on

Let now

assume that the coefficients of u

f

= pGJ. S

are fl at on

a p " 13 1

while (31

u ~ ~ O t s( U)

satisfying

+

S.

and Write

(*).

In particular it follows from this lemm a th at the space

CO,O(S)

can be identified with the space

power series in

p

with

C

~

[

coefficients on

(S) ~ p } S,

of formal

-157-

A .An dne ntti

Then u - a(p(\)

Set y 1

=« i -

= peal 3~1·

- 81\).

By the assumption we get

Il=P~2+~f'/I(32 with u -

Set

-

2

o(I'f31 + tf' 132 )

r 2 =q2

--(2 = PQ'3 +

ap

- t~ e 2· /I

13 2

=p

2

satisfying (*).

~e

.1\

'(11s

= °thus

Then

-

«((2 - h~2)·

By the assumption

11 3 with (33 satisfying

a t'/\ (*).

y 2( = S

° thus

Then

Proceeding in this way we construct a formal power series in om

c;

=

wi th fl m satisfying

u -

_ m+l pk

a (~ i t 13 k) 1

1. 1

ilt'3 m

(*)

=P

and with the property that m+l

Y m+l

Using the remark made of the beginning, we can find f eo OO,8-1(U) such that () k'

T ~f'

t

Is =

e

k~

~

=0

ap

J.f

for

° s-l (U)

Set v = f then (l f f: -~ , has flat coefficients on S. ~ 0,8(U)/):'0,8(U).

k = 0, 1, 2,

. and we have that u This proves exactness at

(jV

This lemma tells us that the cohomology of vJ-l * (U)/'T * (U) is zero in any dimension and therefore the cohomology sequence of (2) gives a set of isomorphism HO,s(O*(S»

~ HO's(~*(S)

= HO,s(S).

-158-

A. Andreotti

Introducing this result in the sequence sequence

(1)

we obtain an exact

which is called the Mayer-Vietoris sequence for

U and

S.

b) The previous considerations can be repe ated replacing the space of C

functions on

°

s..Jl,

exists a satisfying the compatibility condition asu ~ ~t~h~e~r~e-=~~~=­ C 1» function u !.!! Jt+ which is holomorphic in Jl+ and such

!.!l!l

11 Is

(use is made of the assumption

=

u.

iii)).

-162-

A . A n d reotti

In connection with the second isomorphism (which is valid even if the assumption iii) is not satisfied) we remark th at iv)

01

1\ and in fact dim H01 ' (A ) = '-, provided.J~ lC is sufficiently small. In fact by a loc al change of holomorphic coordinates at a we may assume th at s..n. is strongly elementary convex (1.4, exerci se 2). Then the statement is a st~aightforward consequence of lemma (7. 2.2).

I.

H' (If)

0

An immediate consequence of this fact and the isomorphism established above is the following theorem first proved by H. Lewy [37]. Given on Lu for

any

!!

lR

3 the equation

~ oZl

point

iz

=

..L1!

1 aX3

a

~

lR

f

3 we can find a fundamental sequence of

neighborhcjods wiJ such th at for infinitely many f equation does not admi t any solution u E. C"'(w y ) .

Eo C"'( w >1)

9.5.

Cauchy-problem as a vanishing theorem for coho mology. Let us now consider the Levi form restricted to th e analytic tangent plane of the hypersurface S. Using the methods of proof of the vanishing theorem (7.1.1) and the regularization theorem of Kohn and Nirenberg, (s ee (3~, [321 and 5J) we obtain the following result

r

Theorem (9.5.1). For any point Zo ~ S at which the Levi form ~ p positive and q negative eigenvalues on the analytic tangent plane to S at zo' we can find a fundamental sequence of neighborhoods such that

l

tl zo'

U,)'v~N

all domains of holomorphY,

> n-q-l

8

or [

o

< s < p

•

-16 3-

A. Andreotti

Analogously one can find a similar fundamental seguence of neighborhoods ~ such that

tU " veW

s

)

n-p-l

2!

°

[

< s < q Moreover, if P > 0, we can select the seguence ~ U } v Way that the restriction

in such a

HO,O(U-J) ~ HO'O(U~)

is surjective, Le. U'l! is in the "envelope of holomorphY" of + U)I' AnalogouslY, if q). 0, we can select the seguence i. U'" 5 !B. such a w!Y that the restriction HO,O(U

is surjective, Le.

lI)

~ HO'O(U~)

is in the "envelope of holomorphY" of U~,

Uy

According to the remarks made in the previous sections this theorem tells us when locally the Cauchy problem for cohomology classes is solvable. Two special cases will serve as an illustration Case 1.

Assume th at the Levi-form is non de ge ner a t e with

°

Zp(w) such that r~ (4)v) ¢. i /BPC,l 1y»' i1)

or else

This rules out possibility

i)

for ev 7ry v.

Hence

is of first category. Therefore there exists an element every v i.e.

g

€

ZP(w)

such that for

such that the equation

3S }JV

=

g

cannot be solved in w)) although the integrability condition ~S g = 0 is satisfied in the whole of fA) • This shows that p nincar~ lemma cannot hold in dimension dimension q the argument is the same.

p.

For

Elcample. In the particular case 0 f the Lewy operator Lu = f in lR 3 it follo ws that given a €:iR 3 there exists a neighborhood lV of a and f -.

2,

is a domain with a smooth

if the complement of Jl

is bounded.

Then a smooth function

can be extended to a smooth functlon h

ls holomorphlc ln

f9:rall

is connected and

J1,

h

h

n.

on

if and only if

h

on

b

Sl.

such that

satisfies (14)

that satisfy (13).

al""'~

PoHowing Hormander

en]

(theorem 2.3.2') we can

prove the above by first constructing a smooth extension

H

such that ( 1.1.5')

j

Slnce the supported where

p

u

=1

a -problem for

0(

j

E

=

It •.. t n •

can be solved with a compactly

Col (([ n).

we set d.

in a small neighborhood of

outside a sllghtly larger neighborhood. with a compactly supported sion

..v

h = pH - u

u

= a (j'H)

bSl and vanishes Uslng o(=-:au

we obtaln the desired exten-

as in the previous theorem.

The function

H can be constructed by starting wlth any smooth extension

r

of

h

and noting that (14) implles that:

(1.1E! )

rewritlng this we have

(1.17)

near

bit

-1 87-

J . J . K ahn

which implies that

and hence (1. 18}

setting 2

H ::

(1.19'

f - fo r - f 1 r /2

we obtain (1. 20)

as required. It should be mentioned that the tangential CauchyRiemann equations (14) have been studied extensively (see Boc hne r Kohn

( 2)

[15]

,Lewy [2 21

[ n1

• Kohn and Rossi

, Andr eot t i and ~ il ~ [-1] • . e.tc. ).

In fact, the famous example of Lewy of an equation without solutions is one of these.

We will return to this

equation later. Our main concern will be to study (3) on a domain without any restriction on the support of

cc ,

Suppose, for

example, that the c( j E L 2 cD.. ) then ..we wish to find L u~L2(rt) satisfying (3), by this we mean that we want to find a sequence of smooth in

uy

defined on..fl

such that

L2 eQ ) we have u:: 11m u y and o(j :: 11m u yzj • Suppose that there exists a point P E. b{l and a holo-

morphic function such that

f

f (P) :: 0

defined in a nei~hborhood and

f ~ 0

in

U(\JL

11 of

- {p J.

P

Then

-1 88-

.T . J.

Kohn

we claim that if a solution of ()) eXists in there exists a holomorphic function that

h

is an open set containing

V

ists no holomorphic function

=h

p = 1

h

P.

cannot be continued analytically over the point

That is, if

g

L then 2(fL) defined on Jl, suet:

on

V (\

n. =

F

and we chose

au=cX

0(

f

N

Jf

Now we define

::

F by:

~- U F~L2(5'L). e{

J

We set

=0

u E L2 (.~\) then we define the desired h by: h

such that

Vf"IiL

(in fact,

L 2 (rt)

e C;( u )

So if there exists

P) •

(1. 22.)

Now

P.

N so large that

and note that borhood of

J pf-· in lOin

V such that

defined on

To see this let

in a neighborhood of

(1. 21)

g

then there ex-

P

0

Thls condltlon ls clearly lnvarlant and we say that ls pseudo-convex 1 f

(9) ls satlsfied for all

1l

}>l E. b.Q and

that lt is strongly pseudo-convex lf the Levl form ls posltlve deflnlte at all

P E bfl.

Conversely. l f i l p~

b{l

then there exlsts a holomorphlc coordlnate system

on a nelghborhood convex.

ls strongly pseudo-convex and lf

U

of

P

such that

JL {\ u

ls strlctly

To prove thls we flrst note that strong pseudo-

convexlty ls lndependent of the choice of course. that

r

satlsfy (1).

coordlnate system

r; provlded, of

Then we note that for any

z

r

can be chosen so that (r z (p) ) ls 1 j a posltlve deflnlte hermltlan form on all n - t .uples . (a 1 ••••• a n). 1.e. wlthout the restrlctlon (5). this we set (2.10)

r

=

e

't.H

- 1.

To achleve

-19 3-

J . J . Kohn

where

R

is any fixed function satisfying

a sufficiently large number.

( 1)

and

is

T

Then we have

(2.11~ _

we , e compos e an arbitrary

n-tuples

as follows:

(2.12)

where

so that (al •••• ,a n) satisfies (5). = t:

(2.13) .

Thus we have

L ~ z i zj (P)

+ 0(" 'fl

9

1a j +

2

'['2

(2. la i , 2)~.

The error term is bounded by: (large const. ~fi\ 2 + small const.

(2.14) -

Choosing

L:

lai l 2).

T sufficiently large this is smaller than

the first two terms on the ri ght of (13) thesis, the first term is larger than

- since,

''C const.

by

hy po-

L. I ail

2.

Therefore. we have (2.15)-

In z i zj (P)b;'b j >

const."[ I b

i\

2

as desired. We are now ready to prove the following classical result. Theorem.

If i1.

is a strongly pseudo-convex domain and

-194-

J . J . Kohn

P!C: bil.

then there exists a neighborhood

holomorphic coordinate system whose such that

U

f\ft

U

do~ain

of

P

And a

contains

is strictly convex with respect to the

linear structure given by these coordinates. Proof:

It suffices to find a coordinate system such that

the form (2) is positive-definite even without restriction

(3).

Let

u1, ••• ,u

P

with origin at with

(~u

i

U (0»

be any holomorphic coordinate system n and let T be a function satisfying (1)

positive definite.

j

Expanding

~

in a

Taylor series, we have

(2.16)

2 qe ([ r u (O)u i +

r =

r

i

lr

+

ui

u (O)UiU j ) J

((Ilu i \ ) 3).

U 11 (O)uiU j + i j

Setting

(2.17 )

Z

i

= ui

for i = 1, ••• ,n-1,

zn = 2 2 'ru ( P) u i

i

,.. 2l:r

and

( P)uiu j

uiu j

we have

(2.18) Since r

r = Be ( z ) + n u U i j

(0 )

L r zi zj (0 )

Zi

Z

,+

.J

is positive definite we also have ,r

positive definite and thu s b Y (1 8) the

~essian

_ (0) ZiZj

is also

positive definite. The

followin ~

classical

theore~

shows that in case the

Levi form is identically zero the d oma i n is al so locally convex. rhe proof of t hi s t.he or-en is less elementary than t he one a bove.

- 1 95 -

J. J. Kohn

Theorem.

If

r

If in

is real, analytic and B nei ~hbor hoo d

U

coordinates

zl''''' zn

on

P ~ bil

the Levi-

u n blL) then there ex-

form is identically zero (i.e. on ists a coordinate neighborhood

of

V of

P

with holo morphic

V , such that the set

IT/ (\ b fl

consists of the points for which The two theorems above make it seem plausible that whenever a domain is pseudo-convex then in a neighborhood of each boundary point these exist coordinates with respect to which it is convex.

However, this is not true as is

shown in the following example (see Kohn and Nirenberg [14]) Let

fL C( 2

so that near the origin the function

~

is

given by: (2.19)

r

2 8 1S 6 Re (w) + [z ] lWl 2 + /z{ +"7 IZ!2 qe (z ).

=:

Since complex dimension

2

dimensional the Levi form

the spacer-1,O(bjl) 1 K 1 matrix.

~

is one-

In the above ex-

ample it can be shown that the Levi form is larger than const. (/W1 2 + / z I 6 ) near the origin. Thus, by suitably extending

r

we have

Jl

pseudo-convex and strongly

pseudo-convex everywhere except at

(0,0).

Now if there

were holomorphic coordinates on a neighborhood

u /\ .n..

origin relative to which find a linear function the zeros of

ae

h

h

of the

is convex, then we could

such that

which are in

OJ

U

He h(O,O) remain

= ° and

outs~de

of

J1

That this is not possible in this example 1s shown by the following result.

-196 -

J .J. kahn

Theorem.

If

h

is a holomor?hic function which is defined

on a neighborhood

there exist points (z1,wl)'

= h(z2'w 2) = 0,

h (Zl,w1) where

~~

h(O,O)

of the origin and 1f

U

(z2'w 2) E

>

'T(z l , wl )

;J

=

°then

such that

°and

r (z2'w 2)




D

= 2 IT, we see that i f

I,,:. ( fd)! d

does not chan!1;e

= 2 11 which contradicts the above

inequa 11 tv , Proof of the theorem:

Suppose

defined in a neighborhood of h(O,w)

!!!

qu i r-ed ,

°

then

If

r, (O,w)

h(Z ,0)

!!

can be

\e

°then

sign by the above lemma. h

:

Deranretriz ed

h

is a holomorphic function

(0,0) (w)

with

h(O,O)

= 0.

If

which changes sign as re-

h(Z,O) = fo
~L 1--- ~

T*

0,1(f'\\h 0,2«('\\ .1""~~ .l&.J.

S*

These operators are defined 001

their domains (not on

the whole space at the beginning of the arrow).

The domain

of T, denoted by Dom (T), is defined as follows:

0.4)

Dom (T)

= {UE-L20'°

(b) and similarly (since T

wish to prove (b)==}(d) .

\(If'

Tu)B\

=

I(T*'f'

iff Dam

u ) A\ ~

thecl o ~ed

= T**),

g r a ph theorem

(b)~(d).

We

From (b) we obtain: I\T*lfIlA llu tlAf. cIlT*tIlA11ru\lB '

(T*) and u f Dam (T)

n [61(T*)]

hence

which implies ().12') since

\\ 'fIls = i nf 1(tp , IP ) BI lI ~llB The proof is then complete since (b) '

:::;> (a) and (d) ~ (c)

are clear. Now let C be a third Hilbert space and we assume that we have closed operators S : B ~ C such that

densely defined T : A ~ Band

-2 0 7-

J . J . K oh n

(3.1)

ST

=

0

We define 2{C.B by

(3.14)

~

= 1( (T*) n'l1.. (S )

then we have: Theorem.

A necessary and sufficient condition thatR(T) and

(R(S) be both closed is that

(3.15 ) for all 'f'~ Dom (T*) () Dom (S) 1fi th If> .1

for each 1r given by

(4.35)

T*NO(.

and L2 solution of the a-problem Now we have, if SO( = 0: 2

I\r*No
(T/,)

= (rr*N«, No£)

('tA)

(0(, Noc) (to£ )

(cont)nued

O~

next

Da~e)

~ 230-

J . J . Kahn

c:

C Ito(~,

-'t'-C

If (Az !.") 1 .1

2 (1:'>..).

ls the mlnlmum of the least elgenvalue of

IJ

we can ohoose

C = q

(J

-1.

In q:n

coord1nates so that the orl~ln "tlesln

~

=0

we can select the

Sl then settlng

d1ameter of..Q. (l.e. ~ = sup' \p-Q\, P, Q ~ SU

ohoos1ng .~_

(4.36)

= 1 z' 2, .~

_'1:'&2


C s

this follows from the inequality

u~C~«(l).

for all u~C""(.rL)

A corollary of this property is that

1f and only if

u~nH

CO...>.

1\ \\

and

s

By dual1 ty we define tegers as follows. define

1\ ull

s

If

s

UECOO(D.. )

H s

and if

for negative ins

- Itk

Under the samle hypotheses on

1, and if

au

= t::J..

1n a neighborhood

1n

L2

n

then for each

ap.q

if

ocE

k

there ex-

- 259-

J . J . Kogn

k _ ists

u

(fU

EC

k

such that

~u

k

=~.

Here we give a brief outline of the proof of the above theorem.

First we set

(7.12)

p,q and each of €:- L (iU

Now for each

2

q.,p,q there exists a unique lfrE-.u

such that (fA p, q

for each tj; f- !'Ove

that CP"

E H (St)

s

To

we use the method of

"ellipt1c regularization" (see Kohn and Nlrenberg lt9] ). Cons1der the form

where ~en,

r

>0,

!"f'L

the l.) ) are a parti tlon of unity and '1', v

clearly

Q

'"

is coericlve 1.e. (7.7) is satisfied.

Hence, 1f we denote By under

for all

Q~

""p,q

Z

the completion of ~ ..... p,q then the unlque /\ u«( .r) = (1 + I ~ I ) "u(~.r" b

where

S=

(~ ••••• S2n_1)

and

~(~,r)

denotes the partlal

Fourler transform deflned by

(v(f.r) \.- = 2f

~n-

where

and

-lx·f

1e

v(x.r)dx

1 2n-1 dx ,- dx • •••• dx

-265-

J. H . Kahn

on the follow1ng: suppose ~~b

is the solution of

(7.31) Then if "S ,:S'fC;(U(\n.) of :5"

and 1f

! k e Hs rn )

with

then

:S'

u- 2

J\

b

=1

on the support

)'fE- Hs~2 (Jl)

and we

have H.-2

11/\b

(7.)2)

There

S S of complex spaces is called stronglY

if

exists a

(p,q)-pseudoconvex-pseudoconcave if there

map f: X - > (a*, b*) C 1R. and there exists

a* < a# < b# < b* i)

nl (a

~ l' ~

such that bJ

is proper for

{f ~ b 1 = {SO < b J-

ii)

fj' is strongly

iii)

a* < a < b < b* •

b# < b < b* •

for

p-pseudoconvex on

ep is strongly q-pseudoconvex on

iv)

We introduce the following notations.

Fbr

a* ~ a < b ~ b* ' b

[a < 1 < b J

Xa

For a coherent analytic sheaf "] R)I(n:

v

b

"I

X , R (n a )*T denotes

on

)*(1I x: ) •

The so-called mixed case of the direct image theorem is the following parametrized version of the theorem of Andreotti-Grauert. (0.4)

Con jecture.

Suppose

n ; X - > S is a strongly

(p,q)-pseudoconvex-pseudoconcave holomorphic map

l'

and

a* < a# < b# < b*)

sheaf on

X such that

If < a#J. S

and

,)

Then, for

and

"l

dim S ~ n p

vb

Rn*"J-> R (n a

~

is a coherent analytic and

v < r -q-n, u-r

)* 7

(giv~n 'with

codh

7

R n* "l ,)

~ r

.

1S

£E

coherent on

is an isomorphism for

- 2 87-

Y- T . Si u

This conjecture so far has not been completely proved. The special case vex case.

{'f < ~ 1 = Rf is called the pure pseudo con-

The special case

pseudo concave case.

{'f

> b# 1 = f5 is called the pure

Partial results for these two pure cases

were obtained by Knorr [14] and Siu (24,25].

Recently the

pure pseudo convex case was completely proved by Siegfried

[21] by using the methods of the new proofs of Grauert's theorem and the pure pseudo concave case was completely proved by Ramis-Ruget [19] by using the methods of the new proofs of Grauert's theorem together with duality.

Unfortunately these

methods cannot be applied to the mixed case, because any induction on the dimension of the direct image is impossible. A partial result on the mixed case was obtained by Siu [26].

m these lectures we will prove the following improved partial result of the mixed case which is good enough for the known applications. (0.5)

Main Theorem.

n: X --> S is a strongly

Suppose

(p,q)-pseudoconvex-pseudoconcave holomorphic map (given with

If

and

sheaf on {'f

RP(n , ) * '1 a

p < r-q-2n

iv) for

P:i-))

is an isomorphism for

RP (n b)

a

'~

*'1

R'"

is an isomorphism.

-

(n~ )*1 is coherent on

s

< r-q -n-l •

Fbr the applications, only conclusions i) and iii) of the Main Theorem are needed. by the power series method.

The Main Theorem will be proved If we couple the power series

method with the methods of new proofs of Grauert's theorem and duality, we can improve conclusions iii) and

iV) to the

following, but we will not discuss it in these lectures. (0.6)

Theorem-

Under the assumptions of

p < r-q-2n , then, for S and a*

~

R,) n* 7

a < a#

-->

and

p ~]1 < r-q-n , R

11)

iii) iv) v)

'1

if

is coherent on

R'" (n~ )/1 is an isomorphism .f or

b# < b

~

b* •

The Main Theorem will be applied i)

(0.5)~

lIn*

to

the following:

extending coherent analytic sheaves blowing down strongly

I-pseudo convex maps

blowing down relative exceptional sets obtaining a criterion for the projectivity of a map extending families of complex spaces.

(For applications ii), iii) and iv), the pure pseudo convex case of the direct image theorem suffices.)

-289-

Y-T .~u

Fbr coherent sheaf extension, we will not obtain the best known result of extension from Hartogs' figures ~e

[23]~

will only obtain the result of extension from ring domains

(22] (which implies the extension across subvarieties

[29,5]).

The proof of coherent sheaf extension by means of

the direct image theorem is not the simplest approach.

A

very simple proof of the extension from Hartogs' figures was given in [27] which does not use the power series method of Grauert and does not use the method of privileged sets of Douady. The smooth case of the local result on blowing down strongly

I-pseudoconvex maps was obtained by Markoe-Rossi

[17] and the general case of the complete result was obtained in (25J.

The results on relative exceptional sets and pro-

jectivity criterion were due to Knorr-Schneider [15].

The

special case of the result on extending families of complex spaces where the parameter space is a single point was obtained by Rossi (20) and the general case was due to Ling [16l Now we give here a brief sketch of the main ideas of the proof of the Main Theorem.

In the actual proof, for

technical reasons, we use sheaf systems to construct complexes of Banach bundles to calculate the direct image sheaves, but, here in this sketch, for simplicity, we compromise the accuracy by calculating the direct image sheaves by the usual ~ech coch.in complex.

In this sketch there are

also other compromises of accuracy in some minor points for

-2M-

Y - T . Siu

the sake of simplicity.

The proof of the Main Theorem has

three key steps. The first key step is the existence of privileged sets for a coherent sheaf, i .e. if

s.

nt9P --> l)q

homomorphism on an open neighborhood

G of

0

is a sheafin ([n

which

is part of a finite resolution of the, given sheaf (wh~re nO is the structure sheaf of C n), then there exists an open polydisc neighborhood lowing.

If

P

of

in

0

G satisfying the fol-

s ~ r(p, noq) . i s bounded (in a suitable

sense) and the germ of s at 0 belongs to the image of s

and

is the image of a bounded section v

can be so chosen that

C-linear.

s

~>

e , then

over

v

of

v

is continuous

P

The existence of privileged sets can be proved in

three ways (some of which are valid only for certain senses of boundedness).

The first proof by Cartan uses the Weier-

strass preparation and division theorems and it is usually used in the proof of the Closure-of-M:>dules Theorem [$, II. n]. The second proof by Grauert uses power series expansion and it is used in Grauert's original proof of the proper case of the direct image theorem.

The third proof by Douady uses

holomorphic Banach bundles and it is used in his solution of the module problem [2].

In these lectures, we present Dou-

ady's proof, because it works for all senses of boundedness needed for our purpose and because the idea of Grauert's proof occurs in the second key step of the proof of the Main 'lheorem and we will see it there 8.I\YWay. leged sets gives rise

to

'!he existence of priTi-

'lheorem B with bounds and Leray's

-21]1-

Y-T . Siu

theorem with bounds, which are used, together with the bumping techniques of Andreotti-Grauert [1], to construct complexes of Banach bundles to calculate the direct image sheaves. The second key step is the analog of the Hauptlemma . of Grauert's original proof of the proper Case. is an open neighborhood of open polydisc in

«n

0

in 4: n.

centered at

0

SUppose

'l.t '" {U i} , l{ '" {Vj

tions of Stein open subsets of relatively compact in some

A (f)

f •

can be described as }

be suitable collec-

X such that each

Ui •

S

be the

with polyradius

Roughly the analog of the Hauptlemma follows.

Let

Suppose

Vj

is

Let

rUin n- l ( .6 (p») } {V

j

n

n- l (A(p»)} •

Then there exist

for some

fO

satisfying the following.

small (in a suitable sense), every l,

~

a

i'"l i

l,(i) +

when restricted to ~(f) , where

p sufficiently

~ l(?Jl(f} ' "1)

written as

t, '"

Fbr

67

can be

-292-

Y-T . Siu

ai

G:

r(A(p},

?

G: c"-l(\(f}, 7) •

nf!))

1tt>reover the bounds (in a sui table sense) of

ai

dominated by a constant times the bound of

•

~

and

we

7

are

will look

upon this as a generalization of the existence of privileged sets.

Instead of lifting bounded sections in the map

induced by 8 , in this case we consider the lifting of bounded sections in the map

defined by

->

~ a. l,(i} + i=l ~

6?

(restricted to l(P}) • It turns out that the ideas of Grauert's proof of the existence of privileged sets can be generalized to this case when we apply Leray's theorem with bounds.

For this we have to

use induction on dim S, but we do not need - any descending induction on -V.

The reason why we Can avoid this descending

v

which is so essential in Grauert' s original

induction 'on

proof of the proper case is that we look upon the analog of the Hauptlemma

as the generalization of tpe existence of

privileged sets and we use both the surjectivity and the in-

- 29 3-

Y- T . Siu

jectivity statements of Leray's theorem with bounds, whereas Grauert did not use the technique for proving the existence of privileged sets in his proof of the Hauptlemma

and he

used only the surjectivity statement of Leray1s theorem with bounds.

Our approach is simpler and gives the best possible

result in the finite generation of the stalks of the direct image sheaves. After the above two key steps there is still one obstacle to proving the coherence of the direct image sheaves. Suppose S is an open neighborhood of 0 in C n• To get the coherence of the direct image sheaves by induction on we need the following statement on global isomorphism. exists an open polydisc neighborhood

U of

0

in

n,

There

S such

that

is an isomorphism for all sheaves where

-§

of the form "l/(t n-c)m7, are the coordinates of (( n. The third

t

••• , t n l, key step is to obtain this isomorphism statement.

U:a lUi} set

Dof

is a Stein open covering of S let (,t(D)

define sheaves

)/

13

lUi

:a

v

n

X.

n-I(D)}.

D

1-->

D

1--> eV ( U(D) ,-9)

(§) ,

oJ

B

~ (9)

Suppose

Fbr any open subThe presheaves

(1t ( D) , ~ )

on

s .

We derive the isomor-

phism statement by constructing a sheaf-homomorphism

- 294-

Y . T . Siu

-e (7 ) V

- > ~v-l(7)

on

U which is

a

s

right inverse of

»

This right inverse gives rise to a right inverse 13"(9) _>fb l1- l C§ ) of () for sheaves .g of the form 7j'ctn-c)mj. f>v(§)

The existence of a right inverse

_>~"-l(§) of

~(U, ~)IC~)) v+l

R

.P.

n*~,

()

implies the vanishing of

which, together with t.he coherence of

R"Vn*

-g ,

_U-l .P. ••• , K n*~, yields the isomorphism

The construction of a right inverse

S :11("]) - > ~v-l (7)

of

()

is based on the generalization of the following observation. For an open polydisc G in a: n , a continuous lC-lin-

ear map

on

is induced by a sheaf-homomorphism and only if ~[tl'

~

••• , tnJ·

G if

is linear over the polynomial ring We show that, under the additional assump-

tion of the vanishing of

)1+1 )I+n_l "1 (R n* '1 )0' ••• , (R n*)O'

the lifting

in the analog of Grauert's Hauptlemma

can be done in such a

way that it is linear over the polynomial ring ([t l, ••• , tnJ.

When we have the finite generation of v l1+n CO (R n* "])0' ••• , (R n* trr~)O over no' we can apply the

above argument to a complex of the form

'& h ('])

Ph @ n([)

in-

-29 5 -

Y. T . Siu

stead of to

for

));;; p. :i-

~A('1)

v+

and obtain the isomorphism

n.

It is the third key step that makes the

additional assumption of

p < r - q - 2n

necessary, because

we need some room to get a right inverse of

6.

It is also

the arguments of this step that necessitate the introduction of complexes of Banach bundles for the calculation of the direct image sheaves, although such an introduction streamlines the presentation elsewhere as well.

In these lectures some tedious details, which are obvious and can easily be filled in, are left out, especially in

§ 3, § 4, and § 5 • Deta Us of this nature can be found in

[13, 18, 24].

There is an appendix at the end which deals

with homological codimension, flatness, and gap-sheaves. Consult the appendix when these concepts are mentioned or their properties are used. Now we list the notations we will use in these lectures. ~

the set of all positive integers

~o

N U IO}

~*

~ V ICll}

IR+

the set of all positive numbers

nfD

the structure sheaf of ([n •

-296-

Y- T .Siu

The components of a c;: q:m are denoted by a ••• , am. l, m a, b ~ IR ,by a < b we mean a i < b i for 1 ~ i ~ m

For

and by

a n

~

b

we mean

to

fRn ,p r: '=. +

.D.(tO ,

(tpt

p) O

~ ~

i

~

m •

~

a: [tl,

pa

v

r ..

b

0

n

v1

].

a: n

with

e

Pn

PI

+ ••• + )) n

II

1

1

()t

)1

n

.t d to, f)

~ IR~ , ~N(b)

with center

The

IlJn • in 1'rI 0

(1, ---,1) ,we denote

general, for

t

••• J

center to and po1yradius o lJ o lJ (tl-tl ) 1 ••• (tn-tn)n

When to .. 0 , we denote

IR.N

1

the open polydisc in

dt

(N

for

are always denoted by

,and

11> I

When

bi

denotes the polynomial ring

r: II"'n

'=. \L.

~

occupies a special position in these lectures.

coordinates of ([ n ([ t]

ai

n

simply by .6(P)

Ll (p) •

.6..

simply by

denotes the open polydisc in

and po1yradius

b.

Fbr

< b in

0 ~ a

,

GN(a,b)

..

{z

~

,6N(b) Ilzil

In

>

ai

for some

1

~

i

~

N} •

- 29 7-

Y - T . Siu

The closure of a set denotes the sup norm on

«N

r

,

L

2.(G,

JD)

functions on

If

G •

/loU

G

is an open subset of

G

2 L

denotes the set of all

holomorphic

G.

The stalk of a sheaf ~

7s

G-

is denoted by

G

at a point

'

rr

U is an open neighborhood of

then

fs

denotes the germ of

f

at

s

is denoted by

sand

f ~ nU, "]) ,

s.

A complex space may have nonzero nilpotent elements in its structure sheaf.

The structure sheaf of a complex space

Ox •

X is denoted by

ideal of the local ring

For

oX,x

, ...... X,x means the maximum and sometimes (when no confu-

x~X

sion can arise) it also means the ideal sheaf for the subvariety

[x]

of

X.

n: X --> S is a holomorphic

Suppose

map of complex spaces and

Then MN S,s means also X generated by the inverse image of

the ideal sheaf on

s

~

S.

S,s when no confusion can arise • For a coherent analytic V sheaf "] on X, R t denotes the yth direct image of

n.

..,.,."

under then

R"

Y is an open subset of

Ii.'

n.

cr. (71 Y)

is simply denoted by

Suppose U· lUi J and )( = (Vj

X and

CT'

= nlY

,

Il

R (1""* "].

J are collections of

open subsets of a complex space X and? is a coherent analytic sheaf on

Vl«)( means that each

X.

tively compact in some we can define

•* l, that

T

*l,

~

is also denoted

t;

=

7

on ?il

VT(i)'

ev nt, "J) by C, I U.

if

Ui

is rela-

For every t, ~ eliot,

'1)

by means of the index map For

T*l," T*?

V

l" 7 ~ e (1(', "l) , we IViI denotes Vi Ui.

"t •

say For

-298-

a complex space

Y,

Y x 7Jl. deno tes

{Y x lIt} •

When we have a sheaf-homomorphism analytic sheaves on a complex space same symbol

e

(X, "] )

->

. 7/J7-> jr

induced by

is an ideal sheaf on

e

--> -g

of

X, we sometimes use the

to deno te also the maps

r

(where

e : '1

r

(X,

.q )

-g/J§ X)

and other similar maps

when no confusion can arise.

- 2.99-

Y .-T . Siu

Table of Contents

§o

Introduction

Part I

Construction of Complexes of Banach Bundles

18

h

Privileged Polydiscs

18

§2

Semi-norms on Unreduced Spaces

31

§3

Theorem B with Bounds

40

§4

Leray t s Theorem wi th Bounds

48

§5

Extension of Cohomology Classes

55

§6

Sheaf Systems

65

Part II

The Power Series Method

75

§7

Finite Generation with fuunds

75

§a

Right Inverses of Coboundary Maps

107

§9

Global Isomorphism

111

§ 10

Proof of Coherence

119

Part III

Applications

135

§ll

Coherent Sheaf Extension

135

912

Blow-downs

141

§13

Relative Exceptional Sets

155

§14

Projectivity Criterion

157

§15

Extension of Complex Spaces

160

1

Appendix

164

References

178

- 30 0 -

Y-T . Siu

<XlNSTRUCTION OF <XlMPLEXES OF BANACH BUNDLES

PART I §l

Privileged Polydiscs

(1.1)

O D· A(t ,P)

Suppose

Define i)

B(D,EQ)

EO-vaJ,ued uniformly bounded hoIonor--

phic functions on the set of all

D,

EO-valued uniformly continuous holo-

morphic functions on iii)

the set of all

on

Eo is a Banach space.

as one of the following:

the set of all

ii)

and

D,

EO-valued holomorphic functions

D with

1I

where

0ll

Eb

is the norm of

B(D,E ) o is simply denoted by B(D).

EO

In any of these three cases

is a Banach space.

B(D,~)

If

neighborhood of with fiber pose

'1

EO'

n-

and

U is an open

E is the trivial bundle on

we denote

B(D,E o)

also by

B(D,E).

U Sup-

is a coherent analytic sheaf on an open neighborhood There exists an exact sequence

- 30 1-

Y - T . Siu

m~ m~ 0-> n1:1 -> ... -> n1:1 -> nVYO ->7-> 0

on an open neighborhood of Definition.

(a)

DI.

7 -privileged

D is an

neighborhood i f

the induced sequence

o _> B(D)

p

m

P

_>

B(D) 0

is split exact, (b) When

Coker CL (a)

-->

"lois injective. t

is satisfied, one defines

B(D,'])

as CokerCL.

This privilegedness is said to be in the sense of Cartan, Douady, or Grauert according as

B(D)

has the mean-

ing of i), ii), or iii). The definition of privilegedness and

B(D, 7)

independent of the choice of the resolution of

7,

is because,

by using Theorem B of Cartan-Oka, we can easily prove that any two finite free resolution of borhood of

n-

7

on a Stein open neigh-

become isomorphic finite free resolutions

after we apply to each of them a finite number of modifications

[8, Def. VI.F.I], i.e. after we apply to each of

them a finite number of times the process of replacing it by its direct sum with some finite free resolution of the zero sheaf which has only two nonzero terms

( of. [8 , p.202,

- 30 2-

Yo-To Si u

(1.2)

For Banach spaces

EO' FO we denote by L(EO' FO} the Banach space of all continuous linear maps from EO to F • O

Suppose

S is an open subset of ([ nand

holomorphic Banach bundle on we denote by set

U of

S with fiber

EO.

E is a For

s

~

S

Es the fiber of E at s . For any open subS we denote by EI U the restrictio n of E to

U• Suppose

F

is a holomorphic Banach bundle on

S with

FO• A map '(: E --> F is called a bundle-homomorphism if for every open subset U of S for which there are

fiber

trivializations a: El U

~ > U

there exists a holomorphic map

x

A(' )

EO

from

U to

L( EO,FO}

such t hat

for

s~U

denote by

and

tlu

x ~ EO •

U of

the bundle-homomorphism El u

induced by '(

For any open subset

.

For

s

~

- > Flu S we denote by

'Is

the map

S we

- 30 3-

Y-T . Siu

induced by 't • Suppose

e

o _> E(m) _> E(m-l) _' _> •.• _>

E(O)

is a complex of bundle-h9momorphisms of holomorphic Banach bundles on

S.

If for some

So

~

S the sequence

-> . .. _> is split exact, then there exists an open neighborhood

So

in

S

U of

such that the sequence

is split exact. To prove this, it suffices to prove the case where m· land

E(l), E(O)

the closed subspace of

are both trivial bundles.

Let

H

which complements

Let tr : E(l) ~ (S X H)

->

be the bundle-homomorphism induced by

E(O)

e

and the inclusion

map S

(j

So

x H '-->

is an isomorphism.

S x (H EB Im

es) o

=

E(

0)

•

Since the invertible elements of

be

-304-

Y -T . Si u

L(E~l)(f)H, E~O))

o

f'orm an open subset, there exists an open

0

neighborhood

such that a: I U

So

U of'

phism

(i.e. (~IU)-l

(1.3)

Suppose

is a bundle-homomorphism).

Q ) is an open subset of'

S (respectively

7

{.n (respectively ([, N) and S x .Q.

is a bundle-isomor-

is a coherent analytic sheaf" on

For

we denote by

7(s)

the sheaf'

••• , t are the coordinates of' {n. l, n be regarded in a natural way as a sheaf' on

where

t

7(s)

can

n.

For

p ~ 1 , we denote by

bundle on ([ n Let pose

s~ S

whose f'ib'er is

n: S dl and

'1

-> is

S

B(S2. 'n+P)

the trivial

B(.Q)P.

be the natural projection.

n-f'lat at

{s}x.Q

and '1

Sup-

admits a

f'inite f'ree resolution

on

S x.Q.

centered at z.

Suppose z

z ~.Q and

G C C.Q is an open polydisc

which is an '1(s)-privileged neighborhood of'

Then there exists an open neighborhood

such that, f'or any open polydisc

DC U

U of'

centered at

s

in s,

S

- 30 5 -

Y - T . Siu

D x G is an 1-privileged neighborhood of

(s , z) •

To prove this, consider the following sequence of bundle-homomorphisms induced by (*):

Since

G is 1(s)-priyileged and since by the

"1

at

{s} x Q

o ->

n+N'"

I

n-flatness of

the sequence

(rlm(s) -> ... ->

((ll

n+NI:J

(s)

-> ":1 .1(s) ->

0

induced by (*) is exact, we conclude that the sequence (#), when restricted to the singleton

{s}, is split exact.

(1.2), on some open neighborhood of exact and Coker a

B(D)(

G'n+Jl i )

(0

in

S, (#) is split

is a holomorphic Banach bundle.

that, for any bounded open polydisc B(D, B(G'n+NvPi))

s

By

D centered at

Observe s,

is naturally topologically isomorphic to

~

i :;:. m).

Hence, when

in a sufficiently small neighborhood

U of

D is contained s

in

S,

... is split exact and

to

B(D, Coker a).

B(D XG, "J)

is topologically isomorphic

To show the injectivity of

-- 30 6-

Y -T.Siu

->

B(DlCG,'1)

'1(S,Z) ,

it suffices to show the injectivity of

(where

is the sheaf of germs of holomorphic sec-

V(Coker~)

tions of the bundle

Coker~)

suffices to do the case

n f

Suppose

f

is nonzero.

integer

k

such that

~

~

and, by induction on 1. Ker

n, it

Take ~

•

There exists a maximum nonnegative

with

By

the

n-flatness of

"J, g

Since

G is an

lows that (1.4) subset {N+l ,..,

'1 on z

g(s)

Suppose

~

~

• z, it fol-

0 , contradicting the maximality of

7

and extend '

and if

Ker

F(s)-privileged neighborhood of

k •

is a coherent analytic sheaf on an open

S2. of a:;N. ~ x .Q.

~

7

If

G is an

IdentHy(N

with the subset

0 x (N

of

trivially to a coherent analytic sheaf GCC.Q is an open polydisc centered at

'1 -privileged

neighborhood of

z ,

~,

- 307 -

Y:-T . Siu

for any bounded open disc

DC ([ centered at

""

.2.!! "I-privileged neighborhood of

0 , D x G is

(0, z) •

Tb prove this, we can assume without loss of general-

ity that there exists an exact sequence

on

Q.

O _>

Define (0Pm (9

~l

"" Q.

(QPm-l m>

~l

-

by N

Q.

l

Q.. J

where

==

(Q.l,Wj

C

(_ljJ-'w ) (1

< j

~

m)

Q.j-l

w is the coordinate of

c

represented as a column vector, and

an element of Q.. J

N+lJP

is

is regarded as a

matrix of holomorphic functions which are considered as functions on (( x Q.

([ x.Q independent of w •

The sequence is exact on

Let

-> B(G,

lJ

p.

J)

(1 ~ j ~ m)

be a continuous linear map, which, when composed with the map

- 308-

Y-T .Siu

P 'l

B (a, N(D J- )

a. , gives rise

induced by

a projection

to

J

A corresponding

(where

where

p

a

,

can be given by

= 0)

-1

is independent of

t

and one denotes also by

~j

the map

it induces.

Hence B( D )(

"" a, "1)

B( a,

~

., )

and the result follows. (1.5)

Suppose

tered at

to

D is a bounded

0

pen polydisc in ([n

cen-

and suppose

0 ·-> I '

17"7

":r"

->~->-

->

0

is an exact sequence of coherent analytic sheaves on ' an open neighborhood of

D-.

If

privileged neighborhood of

D is an

7

t o ,then

,

-privileged and D is an

7 "-

'1 -privileged

Y - T. Siu

neighborhood of

to.

Tb prove this, we take finite free resolutions on P

,

a

,

m

0-> nI[)m_> 0->

" a "m Pm

->

(!) n

,

->

PI' a I

-> n " a" /API I -> nI;J -> (9

Po'

f!)

n

" //,\PO

I;J

n

-)

n-:

'1 - ) 0 I

"

->1 ->

°.

We can construct the following commutative diagram

000

o

J, 0->

,,,Pm

nI;J

~ ->

~lf\P

0-> nv

~

n

,,,Pm _

O -> nI;J

... v

->

->

~

o

where

i)

,

t

a

tf\PI

nI;J

n(!) I

1

->

1" P

J,

t

~

1

-> n~lo -> 7' -> 0

-> nV

a

m m ->

,

PI

->

,,,PO

nI;J

l

nf!)

n

P

->

tI7

iT

-> 0

I

~

0 -> "1" _> 0

10 0 1 01

II

l!'I v",Pj = ,I\Pj v w mPj ~ n n n

and, except in the last column,the

vertical maps are the natural injections and projec11)

tions. a j is of the form

(a~ oJ

being a sheaf-homomorphism).

- 310-

Y-T . Siu

Let t

p.

B(D, nlD J) be a continuous linear map which, when composed with the map

,

,.."

a. .: J

induced by

t

a..

J

,

(

B D,

p.

nl[) J)

-> B(D, n(!lj-l)

gives rise to a projection

Let

~~:

t

B(D,

be a similar map.

•

n(~lj-l) ->

Then a corresponding

can be defined by

Since clearly

o ->

, ) ->

B(D, '7

B(D,7)

->

" ->

B(D,'1 )

0

is exact, the result follows.

Before we state the principal theorem on privilegad polydiscs, we have to introduce a terminology. is a statement depending on for f

f (; IR~.

Suppose

We say that

sufficiently strictly small, if there exist

Sf

Sf holds

- 3 1 1-

Y - T. Siu

(.,)1 (; R+

1R +i-l

and positive-valued functions (.0i (fl' ••• , Pi-l)

(1 < ]. .

~

n)

Sr

sue h t h at

holds for f




'"'- I

X such that A.

1( refines Vi

Leray's theorem states that

the restriction map H

is an isomorphism.

L

(tt, 7) ->

Define

1.

H (1(,1)

cf"v (tt,

l()

as the set of all

such that

is skew-symmetric in (la, ••• , i O' ••• , i)l.

Define

I1

p

and skew-symmetric in

- 33 1 -

Y ~T .

ef'v (Vt, 1.() 6 1:

-> ef+1,v (1.t, l( )

62 : ef'» (U, l()

-> ef,Y+l (11, l( )

e1 : c

~

or ,F}

8 : efClJl ,F) 2

->

CO ,v (l7l, l()

-> ef,O.(7Ji,

I()

as follows:

. Iuo

.. c,.~O· •• ~)l

a.

('IV .

~O

Consider the following commutative diagram:

n···()v.~).1

Siu

- 3.32 -

Y-T. Siu

a

1

rex, "1 } ->

0->

0->

J

CO (1(,

a

1)

1"1 }

CO (Lt, 7

9

1 '1) -> co,

e2 'If1

°(lJl,

1(')

-2-> 6 ..J:...>

cl(Vl,

CO , 1

61 cl,0 (1t, J.( ) ->

62/ 1 6 (a, )() -> cl (a,

6/

1

'If

,1

J

l( )

61 ->

62

62/

'V

_6_> ...

e2'iI1

62

6J

el 0-:>- cl (1(', 'l) ->

a

'If

A sequence

is called a zigzag seguence if f*d f

JJ

(;

,f -))-1

zJ(l(,

(;

, i

* (; z

f I. ,

"l)

cv,l-v-l(lt,l()

(lot,

"l)

The proof of Leray's theorem consists in showing that the

- 333 -

Y-T. Siu

correspondence (cohomo.l rvy class of f*,t) (where

f*,£

Hi

Wi (l(, '7 )

This is shown in the following

three steps. a)

Fbr every

f*,L

~ zl{l(,"})

one can construct by the

Theorem B of Cartan-Oka a zigzag sequence with first term.

Likewise, for every

a zigzag sequence with bl

If

f l

f*,£ ' f )1.1-)1-1

sequence, then

,* (0

.

f l,* ~ Zl (Vl, "] )

there is

~)J

O}, where P '" P I X P II with PIlC 0, we can assume that t, 'fi ,

HI ,H" are so chosen that, H1'

C H'2 C C. P ' • Q2:

contains

P n{ ~l

t h e Hartogs '"

> E.}

I

for some open polydisc f igure

(pi X (p" _

(cr ,

H"))

U (H2 x p ll )

[1, pp , 219-220]) •

show that the restriction map

It suffices to

·342-

Y - T. Siu

is injective.

II

Suppose

P

=

N-p" TT P.

j=l

and

J

j-l N-p I " U .. P x TT P x (P - HjP TT PJJ j j /pI It ~"j+l

"

"

.. ..

lt1 1%2

"

{U~l) , U 1' {U~2) , U ' 1

An element C,i

< a < b < b*

with

respect

to

the

pro-

n: 6 x X -> 1:::.

a* < a < b < b*

let

x~

=

{a < " < b}.

Suppose

- 345 -

Y - T . Siu

n. c c c:

is an

pen neighborhood of

0

0

Vi, 1.(, Vi.' , )('

and

X , each of

are finite collections of Stein open subsets of

which is relatively compact in some (but in general not the same) Stein open subset of

X.

Then, for

< a l < a#

~

and

b# < b l < b* ' there exist a* < a2 < al and b l < b 2 linear over

Zl(A(tO'fpc?Jl.,'1)(BC£-l(A(tO,f)lCl(" ,7)

a:. [t] E.,=

Max

where

such that '1'([,) +

6'Y'(~)

on

A(tO,P)X

(1Irr([,)II U, t °, r ,IIJV(E,)/L, CIIE,1I)( ,t°, f " ,t°'F )~ .

C is a constant independent of

,

1"

(b) (Injectivity). Suppose 'Vi < < ~ , 1« jttj CC I~ , (SixX b ) n Supp"lCC QX\\fI, and

..

a

A(tO,f)

C

°

t , and

,

f •

< 'lk ,

a

(.Qx X~) () Supp"} CC

a borhood Q.

i

!2!.

n

Qxlu'l. Then there exists an open neigh.!.!:!..Q with the following property. For

° -

and

p
... ->1(, m -> ... ->R.1 ->R.a ->;r

0

X

and

1

is

is a coherent analytic sheaf on

We can regard

case, we identify

-c*-§

o ) ' ••• , 1:(iv)) •

the unique index map a- and obtain

(6.2)

->

as follows.

J~ta': 9+a t ->W~t

ii)

1: ': {I, ••• , k I}

define a sheaf system

"'90, C/L

J ~ 'a tl

i) '¢fa,::o

.Q: •

by an index map

I

on

1U ) T ~a

is a collection of open subsets

Ui C Ul: (i) •

(1, ••• ,k} , i.e. .p... (.f1. '1 ~a'

k'

{Ui}i=l

is a coherent analytic sheaf on

Then there exists an exact sequence

of sheaf systems on

?Jl:

::0

lUi }~::Ol

such that each

U

U i::ol i

R.. m

.

is free.

- 34&-

Y -T. Siu

Proof.

,

Take Stein open subsets

"

, k V~ = !UiJi=l

~~

k = {UiJi=l.

,~"

(1 ~ i ~ k)

and let

suffices to

show that, for an} '. """Rf system

on

1Jl'

1t." ,

and

~

-§ =

as follows.

.J - ,->.g Itt'.

f) :

a O'

Fix a multi-index

J

struct a free sheaf system and a morphism

(a)

e

R/

0

(a)

0

=

(

(§a' '!f~a)

on

We construct them

It suffices to con(a)

.J a

0

(a),

-> §IVl

0

It

(.Ja , cr~a)

there exist a free sheaf system ;/ =

and.an epimorphism

N

U. C CU. C C U C CU. ~ ~ i ~

is surjective, because we can set ~ =

(a)

,cr~a 0 )

such that

$Jl a

(a )

on

~

(a )

BI¥ O 0

0

and obtain

O

There exists a sheaf-epimorphism on

;/(a o ) a

,

=

(a.O )

0

a

tr (a O )

~a

e(a O ) a (a ) O

ea

,

for

Define

for

a

a

et a

O

O

C a

PI U~' the identity map of (Ox

~a

o:(a o )

p

(OX1ua

C/; a

=

0

=

Yaa 0 (7 I Ua )

for

a

O

a

0

for

"o

for

et a

•

The construction is complete.

a

O

C a

for

,

a

O

cae ~

- 3 50 -

Y - T . Si u

(6.3)

Suppose ~. {Uilf=l

of a complex space on VL.

X

and.q = (§a.' 'Y'pa.) is a sheaf system

Introduce the cocha1n group

= (where

is a collection of open subsets

At

is as in (6.1))

IT

a.(;A - "1

r'(u

a.

J

-§ ) a.

and define the coboundary map

by

where

fl •

;:...

••• , Ji J

SUppose

...

7

_>R,m

is a coherent analytic sheaf on

X and

-> ... _>R. l _>R.0 ->'7 ->

is an exact sequence of sheaf systems on £n , where each ~m

is free.

Consider the following commutative diagram.

0

- 3'5 1-

Y - T . Si u

o

o

o

o

-> ...

-> ...

-> ...

-> ... ...

where the horizontal maps are coboundary maps and the vertical maps are defined by the morphisms of (*). OJ IT cl+JJ (Vi, e" ,

)I

An element ~

">R+ l.l ,~ ~

~ of CR(a)

R+v

c

(If, 1{,

~

j.

=0

is given by

by

.

(~l+~ ,~):=o with

Define

d : C1 (Vi) ->

Let

Cf.+ 1 (Vi)

- 3 52 -

Y - T . Si u

where

Define

by

(6.4)

With the notations of (6.)), if each

Proposition.

is Stein, then the map e*:

H~C' (11») -> Hl~. (Lt,1J)

Ui

in-

is an isomorphism for all £

duced by ()

(This proposition can easily be proved by a spectral sequence argument.

However we prefer to present a more elementary

proof, because it can be carried over to Proof. B

,.,

(a) (Surjectivityj.

Take l,

t~ case

with bounda , )

I

~ Z (Vl,?).

By Theorem

of Cart an-Dk a , one can construct, by induction on 11 , ~ "1.+)) ,v

r:

~

CJ.+)) (, .., ~i) V(

"'

(0

:i-

)J

< lD)

such that d.l,OE,i,O

,. ~

? Then

N

and 8!," l, •

1) •

- 353-

Y-T . Siu

(b)

(Injectivi ty) •

E,

Suppose :0

(SL+v,11 )::00

is mapped to 0 in Hi (c'(~, 7)) ~ c;: C2- I(Vl,'1) such that

c;: ZJ (t!) Then there exists

By Theorem B of Cartan-Oka, one can construct, by induction on v , (0 ~

v < 00)

such that d£_I,O 7,-1 , 0

t;

(-It ~l-I+)l,)I-l +

d£_I+J', 117R._I+1J,11 :0

5.t-2+ 1J,"J-17R.-2+ 11,))_1 ()I~l)

because

It follows that

n{ : satisfies

a? =

.

l, .

( 7£-1+)),11)1:00 )00 c;: Q. E. D.

Cl - l (Vl)

,

- 3S4-

Y -T . Siu

(6.5)

For a given strongly (p,q)-pseudoconvex-pseudoconcave

holomo~phic

map and for a given coherent analytic s heaf

on the domain space, we are going to use the results and and

§5

§6 to construct a sequence of complexes of Banach bun-

dles which Can be used to calculate certain direct images of the sheaf. Suppose

I,

are as in (5.2).

b ,.,

1 ",.

p, q , r, Ia,Q, ai' bi (i = 1,2) We arrive at such a situation by consider-

ing the graph of the strongly (p,q)-pseudoconvex-pseudoconcave holomorphic map and the trivial extension of the.coherent analytic sheaf to the product after its transplantation

to

the graph. Let

m ~ 1.

Choose

Choose finite collections

Vlj «Vlj + 1 « 7.Jl of Stein open subsets of i)

ii)

~

d.

I

(1

~ j




of sheaf systems on Q)C 1.t where each

1?

,}

0

is free.

- 35 5-

Y-T . Siu

By (6.3) and (6.4), for

subset

'" of n

Q

,

1 ~ j ~ m and for any Stein open

we have a complex

(constructed from the co chain groups of

o ~ v < al)

whose 1. th

H't(Qx~j,l)

oJ

It IQ x 1tj ,

cohomology group is isomorphic to

(O~.e.Q .

Now we turn to the situation with bounds.

Fix

1

t < al)

and

Ua

Let

1tj

=

{Ui}~=l.

be as in (6.1).

i

~

C (Q x Vi., Jl) f

=

(fa)

C;(Q x z.tj ,

a ~ At.

(0

~

Define the subgroup

satisfies

for all

At

as follows.

J

belongs to

Let

Let

a

R)

~

j

~

m •

~(QxUj,Jt)

We say that an element

c;

At

c;

J.

C (Q x

oJ

Vlj , R)

i f and only i f

fa

c; r (Q x Ua ,(R/Vlj)cx)

- 356 - .

Y - T . oru

As in the case without bounds, we have a complex

As Q varies, we obtain a complex of sheaves

~ on ~,.

Since each

on"

is free, this complex of sheaves is

~

naturally isomorphic to

o -> ocE?) -> (9ci1:) -> ... -> VCEJ~) -> ... J J where V(E~) is the sheaf of germs of holomorphic sections ofa t'"

1

.

trivial Banach bundle E j on Iz , Let 1:'j be the i t h cohomology ,., dj ... sheaf of (#)j. Let 7rj : .Q.xXCj - ) Q be the natural projection map.

We have natural maps defined by restrictions:

').It. >'f'J

For

_> Rt Cn.) *;r "? -> ')/1. ,.,..J- 1 -> J

l. Cn . 1) * ..., ;r • J-

R

p ~ l < r - q - n , by the bumping techniques of Andre-

otti-Grauert [1] and by C5.2), one concludes that the maps RlCn j

): l - > 1.

9:tj

R£Cn j_l

).'7

L

-> ~j-l

satisfy certain conditions of surjectivity and injectivity. lliese conditions can be translated into relations between ~jI and

Rf Cn j )*1

.

llie coherence of

gated by working with

Wi.

RlCnj);:t will be investi-

lliese statements will be made

precise and presented in detail in Part II.

- 35.7-

Y-T.Siu

THE POWER SERIES METHOD

PART II

§7

Finite Generation with Bounds

(7.1)

In

(6.5) we constructed a sequence of complexes of

trivial Banach bundles and stated that the direct images, whose coherence we are interested in, are related to the cohomology sheaves of the complexes.

NOw we are going to con-

sider abstractly a sequence of complexes of holomorphic Banach bundles satisfying certain conditions and derive conclusions concerning the finite generation of the cohomology sheaves of the complexes. First we introduce some notations. trivial bundle with a Banach space ~(B)

For

When

(F,

Suppose

B is a

II-U F) as fiber.

be the sheaf of germs of holomorphic sections of to ~ A and

to

=

d ~N~

° ,~(B)(tO,d)

By identifying

Let B.

let

is simply denoted by

~(B)(d)

•

~(B)(tO,d) with its zeroth direct image

under the natural projection from .6. to

we can regard

V(B)(tO,d)

duced space A* .

as an analytic sheaf over the re-

(O(B) (to ,d)

is the sheaf of germs of holo-

morphic sections of a trivial bundle over A * whose fiber is

- 358 -

Y -T .Siu

TT

the direct sum of

d.foo

copies of

d.

1

1

For

F'.

with power series expansion

v

(t P to) f)l define

[r liB Denote by

B(to tdtf} f

with

0 d

tt t tf

..

the set of all

(D(Fb,)/-1 )(t0 ,d ) -> When

to .. O· , ¥~[ to ,d]

when

d

>

O

~

(Q(~){t

.

,d)

->

1I+1

(Q(Ea

a


"0

})

is simply denoted by 9?a[d] , and,

(0), ••• , 0)) , it is simply denoted by

For

0

) (t ,d)

'#a" •

let

~

0/

y

r~ ,~-l r~_1,~_2

For

When

to ~ A

d" (0),

and

d

~ N~ let

is simply denoted by

Y - T . Si u 11

~

.,)

(E) , (M) , (F)

Consider the following conditions

,

~

(B)n •

(Er

(Quasi- epimorphism with fuunds).

stant

C with the following property.

~~~(tO,d'f)

;""

with

6;;=0

There exists a conFbr

a < (:I ,

there exist

:

with

? such that

i) ii)

l'

iii)

~

0

,d,p

and

1f

a,(:I,t

0

,d,f

(Quasi-monomorphism with fuunds).

(M)

C with the following property.

stant

e; (;

a,(:I,t

•

~

0

(t , d , f) oJ ~

and

r(:la""

= 6(:1v-I ? ~

such that

i)

and

:

? (; Ft3

~l

0 ( t , d , p)

are linear over

There exists a conFor

a < (:I ,

satisfying

there exists

epa,(:I,t 0 ,d,p ( .."~

?)

r: ~

4: [t]

Fo"-l( t 0 ,d'f ) il.+l

-- 3,61- .

Y- T. Siu

l'

iii)

0 a.,P ,t ,d'f

( F).J

(

and

d ~Nn

is bilinear over It [t] •

) Finite-dimensionality along the Fi b ers.

Fbr

t

°

I~

~

there exists a commutative diagram of continuous

linear maps

°

~ 0 ...-> ~-l(tO,d'f) -> Eh(t ,chp) -> EttHI (t ,d,p) ->. ...

1

...->

1

F1>-l -> a.,t ,d,p

°

F-J a,t ,d,p

°

->

1

t

°

°

1

)1+1 ->... F 0 a,t ,d,p

1

"+1 0 1>-l( t ,d 'f) -> Eb.+l(t )I ...-> Eb.+l ,d,p) -> Ea.+l(t ,d,p) ->... where i)

11)

the composite vertical maps are

f'l'

°

a.,t ,d,p

-> Ff

a+l

(to,d,p) factors through a Hilbert

space iii) iv) v)

the middle row is a complex of Frechet spaces

"

. ° )
"=1 >..

t

(-!!)

>..-1

Pn

.

Then

Define

Then

Hence

M:>reover,

where

C"

f

is a constant depending on

Therefore (*) is proved. (II) Next we prove the following.

f .

It follows that

- 36 6 -

Y - T . S iu

For f

sufficiently strictly small, there exists

such tha t , i f there exists

c;

Y

>I

E

and 0o.l,=

Eo.(p)

~ * em ~ E~+l(P)

and

m~ 1,

p

then

such that

o~lX.+l (e m + r~+l a. ~em ~ 'A

a

D

)" -m

n) (t fn

A

'\ )

= a

(t)

is the power series sion of

E, in tn.

We are going to construct ease

m=1

y

Then 0ll+l L

em

by induction on

has been done in (*) of (1).

1:'"

or

or

a •

IJoe 11:+l,p We assume that

expan~

~ rlt+l,cx. Let

e

(

00 2: ~

h"'m

A

Suppose we

r~ve

,-m J + e m (fn)

y

,,; 1lE,II Cl,p

1s so small that v

1I"t11~+l'F ~

The

tn

< e + Ilemll:+l'f '"

fn

m.

2e •

Then

< or

e

+

fnD rr

e •

2 •

So

- 36 7-

Y -T. Siu

is independent of applying (*) of (I) to

1:

we obtain

,

'" 2

~

+ 1)e

" CC '" Cf{fnDf

...,

+ 1 )e

'" is a constant. C

where

1 }e •

By

~

(E)

,we can find

o

such that

-373-

Y -T . Siu A

where

~

C is the constant from

(E)·

•

Choose

so sma 11

D 'n

that

"( C fn D

2l" +

r

r

)

-BCC ,It" C_(~ D

r

fn

n

1 ~-2

+ 1)

f

Define Fi

,.

Let '¥ denote

(i)

a

(~)

11

1

~(~)

r a';3 ,a+27 +

'f(~)

,..., ~

qlo ••• o'J!

....

?

.

(). times).

Then

" (f D_ + 1) e + e • 2 C_

r

~

n

r

This is almost but not exactly the result we want concerning finite generation, because the equation is in of

Eh+l'

We are going to remedy it by using

Ea +

instead

3

(E)"

first

before using the argument to get the equation. (E)"

By . y

there exist

~(i) ~ Ell

a-2

(pO)

(1

~

i

A(i)_

° such that they. generate (We retain f Im(f-b:-l -> Ha_ 1) over ° and n-l °

with

°a_2 E..

"

~ k)

-

)I

I[)

k

.- 3 7.4-

Y-T , Siu

simply to avoid the introduction of more symbols and such a retention clearly does not result in any loss of 'gener al i t y . ) (E)"

· By app1 y~ng

to ~~

/\

,we can find

))

t, ~ Fu-2 (f)

/\

with

11

7~

~+l (f)

such that

11~1I:-2 ,p ~ Ce 11911:+ 1 , p ~ '"Ce

•

/\

t{i)

By repeating the preceding argument with t, , ~ of E"

{instead

~(i)), we obtain (for p sufficiently strictly small)

such that r

{where again

II

1\

J:

a.+1 ,a.-2 ">

" ' D f

Cp

..

are retained simply to avoid more sym-

bols and such a retention does not result in any loss of gen-

- 375-

Y-T. Siu

erality).

It follows that

~

Hence

is finitely generated over

~+l

nOO.

We have actu-

ally proved more than this, namely, we have shown that the finite generation is with bounds when generators are chosen in a certain way. (IV) We are going to prove the full strength of

,)

(B)n b)

by

invoking the existence of privileged polydiscs (in the sense of Grauert).

6~(i)

.. 0.

Suppose Let

~(il ~ ~(fo)

A be the

~

(1

k)

with

~:+1,0

gener-

and t h e finite generation

(E)"

By

~

n

oJ,

on A(pO)

n

l~i~m, l~j~£·

Et: (f)

Now. take .; ~ image of .; By

be defined by the matrix

(III), for

in

'N~+ I ,0

wi th

6~ ~

belongs to

=

A.

° such that the Let

e ""

11.;/1:. f

p sufficiently strictly small, there exist b

i n

~

r(A(p), neD)

~

( a (1) , ••• , a (k ) , 7 ) is linear over

a: [t].

Looking at the proof of (7.2), one

easily sees that this is the case if

i)

~

(B) n- I

.

has the correspond1ng property of

[[tl, ••• , tn_l]-linearity "

Im(Ha._l

)j

-> ~-l)

-oJ

for the

of "a.-l,O •

n_IOO-submodule

- 3.80 -

V -T o Si u

Coker Cf

ii) For

1

t

~

let

n

~

o.

is locally free at

1, ••• , 1) '----y--J

n-t Denote

Im(~[d(t)] ->~+l[d(.e)]) by ~

.

~~[d(t)]

which by (7.3) is independent of the choice of

The above condition By induction on

n , condition ~

ural maps from 9f [d for

1 < l

~

ii)

n.

(.l

)]0

ii) ~

y

A = Ha + 1 • is satisfied if the nat-

is satisfied if (

to 9:1- [d £-1)]0

.

are surjective

(A by-product of this surjectivity condition

is that all the constants i n the proof of (7.2) are independent of

p , because of the last sentence of (7.2)).

From

the exact sequence

(where

0-

is defined 'by multiplication by

tt)' we obtain

the exact sequence

->~:(d(t)) ->~:(d(l)) ->1J~(d(1-1)) ->Pirl(d (t)) _> ••• Hence the surjectivity condition just mentioned is satisfied if p.j~ = 0

for

v


~

,

t

°.

v+l ~ , Y 6 ( E, + (-1) o:a f ) a

By applying

From the defi-

~

,

0""a f )

t

E

to the equation (ob-

6

J,/-l~'

13

'-,

we obtain

such that

II

(-1) )l + 1 ()a" f , ,)

a,tO,d,p

So we have

I 'II" °

,l,

-1

13,t ,d'f

)

.

-

38 4 -

Y -T. Siu

The requirement on the estimation of the norm of the norms of

.; lB f

and

Now we prove a) , c,EDf ~

{

.;

,

Suppose

0 Ep(t ,d,p)

N~

6'"J 1; + (_l)H 10-;+ If

p

all+lf

E9 f

,

with '"

~" w lI> ~

f'

is clearly satisfied. a


a~

It . follows that

By applying property

l,

+ (-1)

)1+ 1" , r pa t, ,we

(Et 0

btain

of the complexes

E(

to

-

385 -

Y- T . Siu

with

such that

'!hen

,

(f, +f,)

(9

f

c;:

0 Eh(t ,d'f)

"'~

with

'6:((E,'

+ l,) EB f)= 0

?$O c;: i;-l +1

l,

",11

r~+l,~( Sf)

(' r~+ 1 ,a (l, + ~)$ f) +

~11

6;~f(7$

The requirement on the estimation of the norms of and

? EB

(7.6)

0

by the norm of

Proposition.

(Et, (M))I+l, (F)v

C; $ f

Suppose, for

0) •

,

(l, + l, ) Q) f

is clearly satisfied.

p:;;:

v :i-

hold for the complexes

s , the properties

~.

Then there

exists a complex

of trivial vector bundles of finite rank on ~(fO)

pO c;: IR~

and A( fO)

(where

C A) in which the maps are holomorphic

bundle-homomorphisms and there exists a commutative diagram

-

386

-

Y -T .Siu

x: ()~

/

\

\}

~

0"'~+1

~ -> (+1

r~+1,a

of complex-homomorphisms on A{fO) of

-

for

p

satisfy

0-"

a

>J ~ s •

~

Proof.

such that the mapping

We are going to prove by descending induction on

that, for

p

~

p

~

s + 1 , there exist a complex

of trivial vector bundles of finite rank on some ~(fO ) a commutative diagram

such that the mapping cones

for

P

~

V~ s "

p

N.

pEb, of

0"""

pa

satisfy

and

- 387

-

Y-T . Si u

The case #L• a 0.

#

s

a

1

+

To go from the step

serve that, by (7.5), #E~ (F)#-l.

is trivial, because one Can set

By (7.2), #~ A:

has property

is finitely generated over shrinking c,(l) ,

to the step

has properties

(H\I(CO(#~)o)

Im

..

#

->

nmO.

# - 1 , we ob-

(E)#-~(M)#, and

(B)~-l.

H)I((9(#~+l)O))

One can find (after

pO)

... ,

whose images in

A generate

#_lL 0 #-1

)/

#

~

..

rr~

#-1 u

(#

~ ~ ~

rr~ (l

Let

be defined by ~(l), ••• , ~(k).

Define

#-11:' ( - 1 )·#-1 r au 1 1

all - 1

1l-1

Define

LV

# 'i/ #

nCOO.

A over

s)

-

388 -

Y-T.Si u

Then the complex

~_lL· and the map

requirement. Q.E.D.

a-. satisfy the a.

~-l

-

38 9 -

Y - T . Si u

§s.

Right Inverses of Coboundary Maps

As in § 7, suppose

v

... _~ ~-l _~ ~

(a

~

l

a

~

(2)

6a

~

Ker((O(~)

v :;.

•

LJ..

Let

(0«))

6Cl> (O(Eh+ l ))

and, for any open polydisc Q C LJ., let

H~(.Q) Suppose that there

exists, for every

Cl

J

a sheaf-homomor-

phism

such that P- l 6a+2

ep (l

=

We are going to prove the f ollowing two statements f or any open polydisc

n C A.

-

3 94 -

Y - T . Siu

Let

Consider the following two statements.

If(S'2, E:)

1))1

k

~

->

If (Sl, S:+V_P+2)

1 •

»

2)

))

If(Sl, 'tal - > ff(n,

)I

k

~

has zero image for

~a+)l-p+2) has zero image for

1 •

First, let us show that

==}

1)

» The commutative diagram with exact rows

S"a

0->

(* )

)l ~

~)la

->

1 .

,..~

for

2)

JJ

p

~

v < s

."

-> ,#->0

~

II

o - > Sa+v_p+2 ->~a+}1-p+2 - >

,)

'#->0

yields the commutative diagram with exact rows

If ($1,

e:)

->

ff(Q,

1

i

...

~:)

->

If (Q,

1 "

II

ff (Q., S a+1I-p+2) - > Hk(Q, ~C1+V-P+2) - > ff (Q,

Since

9-i~ is coherent, ff(Q,

!result follows.

W'll)

=

W")

0

for

k

~

1 •

vi') . The

-

399 -

Y-T. Siu

Next, we want to show that

1)

2)~ ~

~+1

for any

«,

The commutative diagram with exact rows

0->

->0

yields the commutative diagram with exact rows

If(.Q,(9(~))

->

1

1

k(Q,

~

Ihe result follows from the fact that 0-

"

e,

/0-

'"

-> ~+1 (n, ~la+1I-p+3)

1>+1 (S2,6a+1I -p+3 )

~(.Q, (!)(E~+)I_P+3)) .. 0

i)

and

together with the vanishing of

for

k;;;- 1 , implies that

for

p ~ y

for

p ~ ~ ~ sand




r(Q,

T is surjective and

S

'# ) -> Hl(Q, B~+s_p+2 ) i)

follo~s.

we have to distinguish between the case s

> p.

W-(Q,S~)

->

The case

tence of e~+l.

s· p Suppose

Tb prove

s = p

ii),

and the case

follows immediately from the exiss

>

p.

The diagram

(t)~;i

yields the commutatiYe diagram with exact rows

->

Suppose an element r(Q, *s).

Since

1-

of r(Q, ~~)

is mapped to

Then

factors through the map

it follows from

such that

t.

2)s_1

that there exists

0

in

-

39'7 -

Y -T. Siu

Hence

ii)

is proved.

(9.2)

SUppose

is a sequence of complexes of trivial

~

Banach bundles as in §7. that the complexes p

~ y ~

Max(s,p+n).

~

Fix two integers satisfy

(Et,

s ~ p • Assume l, (M)lI+ and (Ft for

By (7.6) (after replacing

~(PO)) there exists a complex

L::. by some

L' of trivial vector bundles

of finite rank on A and there exists a commutative diagram

of complex-homomorphisms on L::. such that the -mappi ng cones

~

for

of o-~

satisfy

p ~ y :? s .

By the results of

§e,

(after replacing ~

by some A(PO)) there exists a sheaf-hpmomorphism

such that

are as in (7.5).

-

39 8 -

Y -T . Siu

For any object derived from

E~ "

we put a

'"

on top

of its symbol to denote the corresponding object derived from

~.

For any open polydisc Q C A and for

d ~N~

let

For

~.6

to

and

d

~ ~~

,

'£~

to

~ b. and

induces a sheaf-homomorphism

8~(tO,d) from

Im((O(E;~-I) (to ,d) ->

~~(tO ,d): to

(0(~;~)(tO,d) such that

... 0 on 6~(t ,d).

By applying (9.1) to the complexes of bundles

~(~}(tO,d) , we obtain the following.

associated to

.... 0 .... I 0 ~p[ t ,d], ••• , Ws - [t ,d] to

(0(F&)(t O ,d))

i) ii)

""v

0

->

~ (SL,t ,d) ...,~

Ker ( Ha, (Q.,t

0

r(Q.,

for any open polydisc Q with Since

dn

f

<Xl

d

n

"!

<Xl

..." 0 JH t ,d])

,

then, for

p

~ v ~ s ,

is surjective

.. " 0 -> rtc, 'N [t ,d]))

,d)

C

d ~N~

b. for all

are coherent on

~ A and all d ~ ~~ with

If

",, ~ 0 Ker ( ~(Q,t,d)

C

A

->

and for all

-~

0

~+)l_p+3(n,t ,d)

to ~ A and all

•

~(~}(tO,d) is the mapping cone of

)

- .399 -

Y -T . Siu

and f(Q, (9(~) (to ,d))

is the mapping cone of

we have the following two long exact sequences:

°

°

11 ...,~ ->~a.[t ,d]->9rJ.a.[t ,d]

- > P\lHl((a(Ll (to ,d)) - > ->

9J:+ l [ to ,d]

_>

°,dl - > ""~fb. (Sl,t°,dl

I\z. (.Q,t j)

- > l+l(r(Q, (!J(L) (to ,d)))

-> ...

- > th+l(Q,tO ,dl

From the first long exact sequence and

(M)~+l

(p ~ v

< s) ,

it follows that 9>/-P(OCL)(tO,d)) ->~.P[tO,d] ->WP[tO,d]-> ~p+l((D(r:)(tO,d)) _ > ••• _>,#s-l[tO,d] '_

>

w,s-l[tO,d] ->W-S(O(Ll(tO,d)) ->9f.s[tO,d] is exact.

From the sharp form of the Five-Lemma, we conclude the following. If ~p[ to ,d], ••• , Ws [ to ,d] are coherent on 4 for all

i)

for

to G;: A p ~

JJ

and all

d G;:f\ll ~

°

with

d n"

CD

,

then

v y 0 ,d]) ~ s , ~(.Q,t ,d) - > f(Q,W[t

is

surjective il)

for

p

< v ~ s , Ker(~(.Q,tO,dl - > r(n,W[tO,d]))

C Ker(F{ (Q,tO ,d) - > ~+)I-P+3 (Q,tO ,dl)

for any open

-

.4 0 0 -

Y -T . Siu

polydisc Q C A d

n

=I

CD

•

and for all

t

o G;:

A and all

-

401 -

Y -T . Siu

§10.

Proof of Coherence

(10.1)

Suppose

is a sequence of complexes of trivial

~

Banach bundles ~s in §7. s ~ p + n.

Fix two integers

~

Assume that the complexes

(M))l+l , (F)'J

for

p

< ." = < s. =

a)

and for 8ny

rm({(Q,tO,d)

,6.

Assume

n

91[ to ,d] with

d

n

w~thout

for ~

1.

Ker

-> (

)l+ 1 H~

(S1,t-0 ,d) )

°

fh)l+ 1 (Q,t

P:i-'"


The case

p

~ v
Het.;+ l (A(P),t0 ,d A)~)

is contained in

Proof.

Consider the following commutative diagram

~+l(~(P))

I

H: (.a.( p' »

which comes from the commutative diagram

-

403-

Y -T . Siu

°-~(Q(~+l) ...2....>(0(~+1) -~(!)(~+l)(tO,d>') -~ ° °-~

b AI

r~+l ,0.1r,..

(Q(()

2...> (D(~)

11\1

-~ (0(~) (to ,d>.+m) -~

°

where i)

I-

is the natural map

ii)

a

is defined by multiplication by

iii)

b

is defined by multiplication by

iv)

c

is defined by multiplication by

°>.+m

(t n - t n )

m (t n - to) n x (t n _ to) n

Let

~L\!+l

1 :

l'f"0

_

y

be defined by multiplication by

°

..... ::l/.)1+l >'f'

°

m• (t n - t) n

Let

.

Y v+l By (Bl +I (applied to A = 0), for r 0.+ I ,0. n f sufficiently strictly small, Ker J C Ker g • Since t n m ,#V+ I , i t follows that is not a zero-divisor of tn

be induced by

°

Ker

(* )

for

t~ =

°

When

t~ =

'f

C

Ker 1)'

° ,both

~ and 1jJ are isomor-

phisms and, hence, (*) trivially holds.

-

404 _

Y -T . Siu

One has

===> because

h = gf.

~

a

gf't

1.

a

hr

~

t

"te = a and

It follows that

The following lemma is in codimension

a

,

a Im

eC

Imo- •

strengthened form of (10.2)

Its proof is similar to that of (10.2).

Its consequence (10.5) will be needed only for the proof of the coherence of ~ p •

(10.3)

Lemma.

Suppose

s ~ p+2 , 1 ~ P,

d,t+l' ... , d n F\t(A( P ) .e

"

It

0

1 ~ ,d ))

is con-

01

f )) -> F\t(.6( P ) ,t ,d ) )l

It

we have the surjectivity of

It follows from the coherence of

~

W [to ,dm+l]

and the

Theor em A of Cartan-Oka that

,

;.s surjective.

Since W~o Lemma,

t

From (#) and (t) we conclude that

is finitely gener a t ed over

nVO' by Nakayama's

-4 11-

Y - T . Si u

It follows from (*) that

-, ~lIA( f

(II)

is generated by a fi nite

~i

be induced by

pI).

Let

(l:i- i

< k) •

Next we prove that the relation sheaf

16

), ••• ,

We distinguish

-,

f ) is coherent at, 0.

';kIA(

v>

between the Case in IR~

,

~l' ••• , ~ k ~ ~ (

number of elements

~i ~r(A(f''')~W")

v

W IA( p )

p

Y· p.

and the case

The case

~

to ~A(P )

for some


co , it follows from the arbi-

A that k

I

I!) Or(~(p i,f?)

n t

•

in

-

4 15 _

Y - T . Si u

Hence R is coherent on (10.7)

~(f

) •

Proof of Main Theorem.

So

Fbr every

one can find a proper holomorphic

~ S

map o: with finite fibers from an open neighborhood

So

into an open subset

G of

~ n.

We have

It is easy to see that an analytic sheaf ~

RO~*~ is coherent on

ent if and only if M is an

~S ,s -module for some

codh O

S,s

where

M

codh lD

S,s

U is coher-

on

G. MOreover, if

s ~ U , then

= codh lD

n a-(s)

M;;;- n , then

M

ncD0- ( s ) -modu.Le -

M is regarded naturally as an

particular, if

of

U

M is a flat

module (see (A.$)-(A .12) of the Appendix).

In

nWo-(s )-

Hence for the

proof of the Main Theorem we can assume without loss of generality that

S

=

A

and

'7

is

n-flat.

course of the proof, any replacement of

MO r eover , in the by

~

~ (f )

(with

P~IR~) does not result in any loss of generality. In (6.5) we have constructed a sequence of complexes ~

of trivial Banach bundles on A.

By (5.2 ) and the re-

sults of Andreotti-Grauert [1] these complexes

(El, By

(M)I+I, (Ft , and (l0.1) a), b)

(l0 . 6 ),

W"

is coherent on b. . for

p~ »

for p

~

~

)I


'1

f -

Supp Coker

Since

e

f

Xz

on

be de-

is nowhere zero on

are subvarieties of the

Stein space

and hence are Stein-

From the cohomology sequence of the

short exact sequence

e

0-> Kere -> 7-> we conclude that 8 l". ., .

Ime

-> 0 ,

induces an isomorphism

2"7 0 (R1 n._)

t

->

( R1 n. 2 (Im 8) ) 0t

Consider the following exact sequence

1 2"7

(R n• .T) 0

t

coming from the short exact sequence

o ->

Im e

c-> "1 ->

We are going to prove that ~ show that diagram

Ker

7 ..

O.

Coker e

->

is surjective-

0 • It suffices to

Consider the following commutative

-

4~O -

Y -T . Siu

(R

e

9 ,

where

1 2

n;l)

t

0

92

->

e

and g is the restriction 2 are induced by 1 map- Since 82 .. 1 and C, is an isomorphism it suffices to show that Ker 9 2 .. 0 - Since is an isomorphism on Xl ,81

c.

e

is an isomorphism-

By applying the Main Theorem to

the map n: X -> D together with the function

sheaf Hence

7

on

Ker

e2

X, we conclude that and C,

0

=

g

po

and the

is an isomorphism-

is surjective-

Since

Supp CokerE

is Stein, the image of the natural map

(ROn~(Coker e)) generates

(Coker 9)

0

t

0

0

(t ,z )

-> (Coker e) 0 0

(t ,z )

over

n+N({) 0

0

(t ,z )

Since

t,

is surjective, by Nakayama's Lemma, the image of the natural map

•

generates

"1

neighborhood phism

(J

0

0

By letting

(t ,z )

:n+P

f

U

of

->

to

t

on

in

zO

vary, for some open

D we can find a sheaf-epimor-

-

421 -

Y -T. Siu

some open neighborhood

U"

to

in

->

Ker

of

7 ,

Ker c: instead of

By applying the Same argument to

U'

for

we can find a

sheaf-epimorphism

-r : n+~

0-

on

< 1(t,z) < P" } •

{(t,z) ~ U" xa: Nj" a

or extends to a sheaf-homomorphism

By Hartogs' Theorem,

T: on

f (t,z) G

U• x

a: N

..

(

I

l'(t , z) < {3'}.

Then

Coker~ /

)[n+l]

/O[n+Z]Cokerl:

, '1 I U x GN (a' ,b)

extends

n+~q -> n+Nif

fl

"J I" U)C

and, hence, extends

(see (A.IS) of the Appendix).

GN (a, b ' )

By the arbitrariness of

to

and the uniqueness of extension, the special Case follows. Fbr the general case we use induction on be the set of points of

D x GN(a,b)

where

n.

Let

S

codh"J:i- n + Z •

Let TI Let

TZ n-flat.

..

{(t,z)

~

be the set of points of Take

a < a"

?

sjdim(t,Z)s n({t} x GN(a,b)) N

DXG (a,b)

where

I} •

'1

is not

< b < b in IR N • Let

By applying the special Case to

'1 I U X GN (a ' ,b , )

for bounded

- 422' -

Y-T , Siu

Stein open subsets

"11 (D-A) X aN(a,b) sheaf

'1

on

U of

D-A, we conclude that

can be extended to a coherent analytic

(D-A) x AN(b)

satisfying

7[n+l]

= :;.

Since

dim S :f n , rank ~l Tl U T2 < n (cf (A. 1)) of the Appendix). Since A = f! when n = 0 , the case n = 0 is proved. Take arbitrarily

t o ~ A.

After a coordinates transformation, we

and '( > 0

to = 0 and there exist 0 < a < ~ in ~ n in IR - l such that Lln- l (1') x al(a,~) is dis-

joint from

A.

can assume that

By induction hypothesis, the sheaf on

which agrees with

and agrees with

?

1\

7

on

on

can be extended to a coherent analytic sheaf 7* on Lln- l (,t) x Al (~ ) x d (b) satisfying (,:1*) [n+ 1] = 1*. The general case now follows from the arbitrariness of uniqueness of extension.

to

and the

-

.4 2 .3 -

Y -T. Siu

§12 • (12.1)

Blow-downs A holomarphic map n: X ---> S is said to be strongly

I-pseudo convex ~:

cf

if there exist a

X ---> (~,c*) C (~,ooi

function

and a real number

< c*

c

such

that i) ii)

n/[f:i- c}

'f is strongly l-pseudoconvex on

(When the additional condition c < c*

c < c*

is proper for

[SO:i- c}

[Cf> c#} •

= [1' < c}-

for

is added, this definition agrees with a special case

of strongly Fbr

(p,q)-pseudoconvex-pseudoconcave maps.)

:r S is a strongly

holomorphic map and

S is a single point, then

l-pseudoconvex X is holo-

morphically convex. Proof. in

Take

{T > c}.

{x~}.

c# < c < c* Let.J

and take a discrete sequence

be the ideal-sheaf of the subvariety

The exact sequence

yields the commutative diagram with exact rows r(X, Hl(X,5) -> HI (X,

0-1

~

HI (Xc ,9)

1

---> Hl(X c ,COX} 'l:

- 426 -

Y -T . Siu

S

Since

XC ,

= CD

X on

1:

is an isomorphism.

By the results

of Andreotti-Grauert [1], o: is an isomorphism. that

?

f'{x)/ )

->

is surjective. (J)

f~

r(X,OX)

such that

•

Suppose

Lemma·

(12·3 )

There exists

It follows

convex

map-

k~N

such that

Then for

n: X

-> S is a strongly I-pseudo-

s c;: S and

c# < c < c*

there exists

(ROn;~)s -> (ROn; (lOX/.w.s,s(OX)) s has the same image as

Proof. U of

Use induction on s

in

f(s) ,. 0 (R

l

and

n;1Js

m ~NO

ii)

dimsS.

S, there exists fs

For some open neighborhood

f c;: ,(U,lOS)

such that

is not a zero-divisor of ~S,s

is finitely generated over

Since

~S,s ' there exists

such that ·

fx

(when

f

r(n-l(u),lOx»

is naturally regarded as an element of is not a zero-divisor of

xc;:xcnn-l(U) • The commutative diagram

~~ for

- 427 -

Y - T . Siu

o -> ?m+l(9 _> 19. _> 19 /?m+lCQ -> 0 X X X X

II

1

-> (Ox ->

CDx/f~

1 o ->

f~

-> a

yields the following commutative diagram

We are going to show that

ba

~

O.

Consider the following

commutative diagram

where i)

~

and

ii)

e

is induced by the inclusion map ~'9x C->

P are defined by multiplication by

~l

tOx

- 428 _

Y -T. Siu

iii} iv} By

0-

is defined by multiplication by

~

~

is defined by multiplication by

f.

the choice of

m ,

~

is an isomorphism and

Ker a C KerO-.

It follows that b(Ker c}

C b (Ker 'ta-e~ -l) Hence

ba = 0 • By

It

follows that

o.

b (Ker b) 1m E, = 1m 7 .

induction hypothesis, there exists

kEN such that

and

have the same image.

(12.4 )

Lemma.

Then

Suppose

k

n: x - > S is a strongly

convex holomorphic map. s S is a strongly I-pseudo-

convex holomorphic map, S is Stein, and

c# < c < c*.

X ;s Stein if and only if for every compact subset

Then

K of XC

there exists a strongly plurisubharmonic function on an open neighborhood of

K.

-

43lJ

-

Y-T . Siu

E!:2.2.f. Only the "if" part requires a proof.

S is the coun-

table union of relatively compact Stein open subsets that

Sk CC Sk+l

has dense image. for each

k.

sucl

and the restriction map

It suffices to show that

Yk

Let

tion function on

Sk

Sk.

is Stein

n-l(Sk)

be a strongly plurisubharmonic Take

c# < e < b < a < c*.

exhau~

By

assumption there exists a strongly plurisubharmonic function

e

n -l( Sk+l)

n

Xa •

..2 Choose a nonnegative C" function a ~ on X whose support is contained in X and which is identically 1 on Xb • There exists a if function ~ on

on

(_ro,c*)

such that Supp o: C (c#, c* )

i)

the first and second derivatives of

ii)

are iii)

0-(;\)

(J

are

~ 0

and

> 0 on (e,c*) __> ro as ;\ -> c* froa the left.

For some positive number

A, the function

is a strongly plurisubharmonic exhaustion function on n -I( Sk).

(12.6)

Lemma.

Q. E. D•

Suppose

D is an open subset of ([n ,

[V J.. J.J. { I i s a locally finite open covering of D and ~

L.

J.

-

4 31

-

Y -T. Siu

{i

~ II

in

Ui

rf

is a

nonnegative funct ion on

such that all first-order derivatives of

on the zero-set of

l:: "to i ~ I ~

>

0

an open subset of

a: N

DxG

Then there exists

Al ~

x G.

U

i

G-->

f: D x

and let

i ~ I

Let

Suppose

be

G is

D

be a compact subset

K

be the natural projection.

1\: lR -->

lR and a function

~ l::

Let 1)1

(Ii (1 ~ I) is a strongly pluri-

and

subharmonic function on

van ish

~i

D•

on

D.

a strongly plurisubharmonic function on

of

D with support

("t • •

~

fie

IR

~

Mr. ~ + B{Y· f)

K

is strongly plurisubharmonic on some open neighborhood of

Proof.

For a

rf

x ~

Q and

and for

function

on an open subset .Q

h

a ~~m ,let m l::

(

L(h;x,a)

denote

m

l::

-

ah

i=ldz ~. Let in

"" -r.~ '"

<e n+ N

a ~ S

'1:'.

~

•

D

f

denote

~J

J

~

a(h;x,a)

a: m

a2h) (x)a.a_ .

i,j=ldZ .dZ'. and let

of

and

Y''" '" 'Y'

D

(x/a

i

.

s

and let

f

be the unit sphere

It suffices to show that for fixed

there exist

,

A (x s a ] ~

IR

and

,

x ~ K

B (. ,x,a) ~

lR

and

such tha~

- 4 32 -

if

,

A? A (x,a)

and

Y-T . Siu

,

B? B. (A,x,a) , then

Direct computation shows that

].

+

The first bracketed term is first

n

components of

ed term is

o.

a

> 0 when A > 0

are all zero, the second bracket-

When the first

all zero, there exists

n

components of

a

are not

B* (.) ~ lR such that the second

> 0 when B? B*(Al.

bracketed term is

When the

The third bracket-

ed term is at least as great as

where the only nonzero terms are those with O( S is a stronglY

Suppose

I-pseudoconvex holomorphic map and

i)

from

Since

X is holomorphically convex.

Theorem.

sheaf on

is Stein.

Since

(for the pro-

X'

and

W of

X

,

{ f > c# J biholomorphically onto its image,

1

By applying Grauert's direct image theorem (for the proper

case} to

I

lI

n' Supp R 0-* 1, it follows that, for

RO(n'}*(Rv~*1)

is coherent and

»~

1 ,

-

436 -

Y-T . Siu

for every Stein open subset

U of

S.

It follows that

-

437 -

Y- T. Siu

§13.

Relative Exceptional Sets

(13·1)

Suppose

variety

A of

above

n: X ---> S is a holomorphic map.

A sub-

is said to be proper nowhere discrete

X

n/A is proper and every fiber of n/A is pos-

S if

itive-dimensional at any of its points.

A subvariety

X which is proper nowhere discrete above exceptional relative to

S

A. of

S is said to be

if there exists a commutative

diagram of holomorphic maps

i

-> Y

X

s such that i)

~

ii)

is proper

every fiber of ~1!(A)

iii)

~

maps

o .

iv)

R ~*(DX

has dimension

X-A biholomorphically onto

~ 0

y- (A)

= COy •

The following result on relative exceptional sets is a consequence of (12.8). (13.2) and

Theorem.

Suppose

A is a subvariety of

crete above

S.

Then

and only if for every

n: X --> S is a holomorphic map X which is proper nowhere dis-

A is exceptional relative to s

~

S if

S there exist an open neighborhood

- 438 -

Y -T . Siu

U of

s

n-l(U) and

and an open neighborhood such that

Atln-l(U)

nlW:

w--->

W of

Atln-l(U)

U is strongly

I-pseudoconvex

is maximum among all subvarieties of

which are proper nowhere discrete above

U.

in

W

-

1 $9 -

Y -T. Siu

§14.

Projectivity Criterion

(14.1)

Suppose

n: X --> S is a proper holomorphic map and

p: V --> X is a holomorphic vector bundle. be weakly negative relative to exist an open neighborhood hood

U of

D --> U is strongly Let

Jk/Jk+" 1

as an

Ox-sheaf.

Suppose

herent analytic sheaf on

such that

N

J

V and

with

be the

and consider

X

It is easy to see that, where equals (Q((L*)k)

where

V

L* is

p: V ---> X is a weakly negative X and

kO ~NO '}? 1 and

S.

such that

is zero on

k? kO

K for

The case where

If

"1

is a co-

K is a compact subset of

S , then there exists

R~n*(7@~jk/jk+lJ

S is a single point was proved by

The proof of the general case is completely

analogous to that of the special case. coherence of

S .t here

L.

Theorem.

Grauert [7].

~

I-pseudoconvex.

holomorphic vector bundle relative to

Proof.

s

and an open neighbor-

Vln-l(U)

We identify

N •

is a line bundle, Jk/Jk+l the dual of

s

N be the zero-section of

ideal-sheaf of

(14.2)

S if for every

D of the zero-section of

no plD:

V is said to

R')1 [rt

0

pi DJ* [p*'])

It fol lows fro m the

and the fact that

- HO _

Y-T. Siu

is a subsheaf of and

U,D

(14.3)

*

R~ (n. p I D)*(p "]) ,where

are as in (14.1).

Theorem.

phic line bundle

)I

~

r~ 1, k \:.1'0'

Q.E.D.

If there exists a weakly negative holomor-

p: L

--->

X relative to

S

and

S

is

Stein, then for every relatively compact open subset T of S there exists a holomorphic embedding o: of n- l (T) into PNx T

tion

--

Proof.

such that the composite of

PN x T ---> Let

T

equals

Let 52

Stein open neighborhood of points

x,y

of

°->/Wo2X ,x'OIl.

101k

X.

~

in

be a relatively compact S.

Take two distinct

Consider the two exact sequences

->W.X .xfl)£k ->

= I#..X ,x {\ I\MX,y •

where """'"X ,x,y

and the natural projec-

n.

-

l. = (0 (L* J .

~

By

(ANY

/2

X ,x AMX ,x )~.c

(14.2), for

k

k

Rln*('MI ,x,yfl}

Van-

£,k)

ROn* ((........x,x/......i,x)@.,(,k) Since

Q

is Stein, it follows

Y -T. Si u

that there exist enough holomorphic sections of n-1(Q)

to construc t an embedding ~ .

Q. E. D.

L* over

-

4 42

-

Y - T . Siu

§15.

Extension of Complex Spaces

(15.1 )

Theorem. (O[n+l] = ~X and X

S is a Stein space of dimension

~

n •

n: X ---> S is a holomorphic map and

Suppose

X ---> (a*,b*) C

~:

X is a complex space with

Suppose

function such that.

is a strongly olurisubharmonic

(~ro,m)

nl la

~ ~ ~ b}

is proper for

X Then there exist uniquely a Stein space ""

a* < a < b < b*.

and a holomorphic map

n: X--->

S such that

i) (9~n+l] .. (9X ii)

branch of iii}

iv)

in

the restriction of

~~l]

=

n

to

"" X- l!f>a}

First we prove the case

is proper for

n = O.

It follows

Ox that, outside a subvariety of dimension

X, codh (Ox ~ 3.

such that let

X

intersects every

n = ~Ix

Proof (sketch). from

X which

X is an open subset of

codh (Ox ~ 3

X~ .. Ic < f < d}.

ideal-sheaf on

We can choose on Let

I R is an epi-

cOdhSM = codhRM when

M is

R is regular of dimens ion

codh M agrees with the maximum of

n-l

n ,

such that there

exists an exact sequence

o ->

It

- > ... ->

Suppose , space

X.

Sk(7l

codh

J.

;i. k •

->

lO -> M -> 0

•

is a coherent analytic sheaf on a complex

We define

codh"J x

Let

II

->

as the function codh», 1... I:IX,x X

den ote the set of points of

X where

-

447 -

Y - T . Si u

(A.2)

Lemma (Frenkel ).

,

~

a < b

D i s a Ste in doma in i n

is a non empty Stein subdomai n of

D

o

Suppos e

N

R

in

.

then, f or

I

1

~ ~

D.

< N- l ,

This lemma is proved by Laurent series ex pansi on. tails, s ee [1, pp.217-219].

If

For

de~

As a corollary, we hav e the

f ollowing.

(A·3i

Proposition (Scheja).

dimension

in a complex space

~d

analytic sheaf on

o

< r-d

~ ).I

for

0

~

.

Suppose

X with

He nce

v < r-d- l

~

X and

H (X,"] )

--->

.

?

is a coherent

Then ¥~A"J = 0 for H (X-A,?) is bij ective

codh"l ;;;- r .

A is a subvariety of

~

and in j ective for

)I

= r-d-l •

We can assume without l oss of g ener a l i ty that an

0

pen subs et of

cI: n.

When

"J =

n([)

and

X is

A i s regular, i t

is a direct consequence of Frenke l's lemma, and, when the

,

singular set

A

of

A is nonempty, it f ollows from the

long exact sequence

When

"J

r n'!)

,

we use a local finite free resolution of

"J .

Now we define relative gap-sheaves with respect t o a subvariety.

Suppose

A is a subvariety of a complex space

-

448 _

Y -T. Siu

1 C -1

X and

the sheaf

are coherent analytic sheaves on

7 [A]1 U

X.

Define

by the presheaf

...-> {s~r(u,.~)I(sIU-A) ~r(U-A,7)}·

The following is a consequence of the Nullstellensatz. (A.4)

Proposition.

If.5

is the ideal sheaf of

A , then

Q)

1[A1 = U ("l:J k ) -1

k=l

1

and is therefore coherent, where

is the subsheaf of ~ s ~~x

whose stalk at

such that J~s C

tx

Proposition.

(A·5 )

Proof.

x

is the set

•

We can assume without loss of generaltiy that

?

7

is

By taking a local finite

defined on an open subset of ~n. free resolution of

< m=

is a subvariety of dimension

and considering the rank of the matrix

defining the extreme left sheaf-homomorphism of the resolution, we see eas ily that

Sm(?)

dimension estimate, the Case ing

,Ia[ {x}]?

for

from induction on

is a subvariety.

m=

x ~ So (':1).

a

For the

is obtained by consider-

The general case follows

m and considering the quotient of

7

by

the subsheaf generated by a holomorphic function whose germ at some

iJ C 1.

x ~ Sm(7)

is not a zero-divisor of 'x

Now we define the

dt h

relative gap-sheaf.

Suppose

are coherent analytic sh eaves on a complex space

Define the subsheaf "}[dJ.~Of

1-

by the presheaf

X.

-

I

A of

Proposition.

coherent and Proof.

for some subvariety

U of dimension

"1[dJ~="1lSd(-§/7)J~

~

dJ •

Hence '][dJ~ is

dim Supp(7[dJgI11 ~ d •

Lemma.

"1 C.fj

Suppose

on a complex space

X

and

mary submodule o f ~x di mx

Proof.

Supp..gl1 ~

x

P •

Jk-lj C "J

are coherent analytic sheaves

c;: X such that

whose radical d

and

There exists

k

c;:

~

P

7x

is of dimension

d.

J

U of on

x

in

X

there

U whose stalk at pkg C ~ x

such that

.

x

Hence

on some open neighborhood and

Let

Y = su PP('7[d_ l J9 1:fl

Y.

Since

and l e t

!

dim Y < d , there exists

be the ideal sheaf of f

c;:

Jx-P.

Nullstellensatz,

f or s ome

is a pri-

(1[d_l J§ lx = ?x •

For some open ne ighborhood

exists a coherent ideal sheaf is

c;: r(U-A,':1)

Follows from (A.3l and (A.5).

(A.7 l

Then

-

Y -T. Si u

U 1--> {S c;: r(U ,$) (s] U-A)

(A. 6)

449

1. c;:

~.

Sin ce

f

Cl

P , i t f ollows t hat

By the

-

4 50

-

Y - T . Si u

(A.B)

Proposition.

on a complex space

is a coherent a nal yt i c sheaf

X, x (;. X , a nd

is not a

zero~divisor

dim x V(f)

n

V(f )

?

Suppose

7x

of

Supp O[k] 7 < k

f (;. r( X,(!)X) ·

some

Suppose

Supp s

fx

s (;. re U,?)

borhood of

i f and only if f or a ll

k (;. N* ' wher e

i s a zero-d ivisor.

with

x.

Let

°

, where Sx f k = dim x Supp s

.

Then U is

fo r ° an open ne ighf x sx

Then

C V(f) () Supp e lk] 1 . Suppose

f

is not a zero-div isor.

x

the kernel of the sheaf-homomorphism multiplication by U of

y~U.

If

By considering

7 --->7

defined by

f , we conc lude that, f or s ome op en neigh-

x , fy dimxV(f )

borhood

is not a zero-divisor of

n Supp 0l k]?=

for some open subset

W of

U

k

7y

f or some

k

7y

a non-zero-divisor of

Proposition.

on a complex space Supp Oed]

':1

Suppose X.

equals the

Equivalently, f or

for

for

,

then,

,

which, because of the Nullstellensatz, contradicts

if

fx

= Supp d

y~U.

By (A.$) , f y

°

is

Hence dim U n Sd ("]) = d

.

i f and only i f

k < d •

f ~ r(X,~X) , f is not a zero-dix i f and only i f dimx (Supp lOx/fC9x) Sk ("]) < k

Suppose

spaces and "]

0ld]"]=

42.£

Corollary.

visor vf

for

k

contains

k-dimensional

Sd_l (7/f"1) n U , contradicting

Corollary.

(A.II)

V(fj: = Supp (OX/fOX

such that

V(f)

un Supp

branch of

ule.

=

U by a smaller open subset, we Can assume that there

exists

for

dim U" Sd ('I)

For

n

rt :

X

- >

Y is a holomorphic map of complex

is a coherent analytic sheaf on

n-flat at

is said t o be

x~X

if

tx

is a flat

n- flat on (or at) a

X

.

7-

is

Oy,n (X) -mod-

subset G of

X

- 452 -

Y- T. Si u

7

if

is

n-flat at every point of

When t

r

to

Y

= 4:n , '7 is n-flat at x if and only if

is not a zero-divisor for

j

(t~, ••• , t~) (A.l) j

n(xj •

=

sheaf on a complex space that

7x

Let

Proof.

X and

is a coherent analytic n ; X -> <en

be the set of all points of

Z

is not

and the rank of

Hence

"JjjEl(t._t?j'Z, where x i=l 1. 1. x

Proposition. Suppose "]

phic map.

G.

n-flat. nlZ

is

Then nd+1 x O.

'1 ~ is coherent.

Since (O[d+l]7) [AJ(7[dJ)

Since the sheaf-homomorphism ~ ---> ~ de-

fined by multiplication by

t d+ I

Nakayama's Lemma that 10 = 0 in nd+lx O-A approaching 0

.

is Let Since

0

,

it follows from co

{x» lV=l

be a sequence

nd+lxo_A

is Stein,

there exists

such that

Sx

f 0 f or al l

V.

It f ol l ows tha t

s

defi nes

-

4 5 5-

Y -T.Siu

a non-zero element of 10 ' contradi cting 1 0 Proposition.

(A.15)

Suppose

'7

0

is a coherent analytic

sheaf on a complex space X. Then th e natural s heaf- homomorph ism"] > 7 Ld ] i s an isomor phis m i f and only i f dim 1k+2(7) ~ k Proof. d

=

~

d

i)

-1

The "if" part.

is trivial.

U - (Ansd+l(']))

Us e induction on

Suppose

in an open subset

Since

k < d •

for a ll

d.

The case

A is a subvarie ty of d imension

U of

X

Since

codh' ~ d+2

on

, i t follows from (A.3) that

dim Sd+l(7)

~

d-l , by i nduc t i on hypothesis

~ r(U-(Ansd+l(':t)), "]).

r(U,"]) Hence "] ~ "J[d] • ii)

The "only if" part.

conclude that Hence

Old] 7' = 0 •

0[d+l]7= O. Sd+l(?l

k > d+1.

x

f ~r( U,OX )

such that

Supp 0X/f~

k-dimensional branch of Let

-§

= "]If"J.

and dim un Sk+2 (7)

for

U of

but contains no

U n Supp O[k] '] for any O[d ]1 = 0

By (A.14 ), dim Supp O[d+l]1 ~ d.

x ~ X there exists

For

for some open neighborhood contains

From t he defi niti on of "J [d] , we

k < d •

Q. E. D.

~

dim 1k+l (~)

~

k

Then

-

456 -

Y -T . Siu

(A.16)

Proposition.

domain in q:n If "}

,

and

a

Suppose

,

D

~

< b

a

D is a D •

is a nonempty open subset of

Dx~N(b)

is a coherent analytic sheaf on

1[n-l]

in (RN

such that

';I, then the restriction map

:z

is an isomorphism. Proof.

For any open subset

U of

is injective, because the support s

of

from

Ker a.

D, the restriction map

V of any nonzero element

would be a subvariety of

U x aN(a,b)

and, by considering

UX 6,N (b)

disjoint

({xJ x AN(b)) (\ V for

x ~ U , we conclude that dim V ~ n, contradicting In particular,

e

is injective.

e

We are going to prove the surjectivity of duction on

n.

s t;. Im

such that

s!QXaN(a,bJ

r(Q x ~N (b)

element

Let.Q

x

,1).

,7).

e•

Consider first the special case where D x ~ (b).

by in-

s t;. r(D x aN(a,bl) U (D' x AN(b})

Take

We have to show that

on

O[nJo;= 0.

codh

1?

n+l

be the largest open subset of

D

extends to an element of

To show that Q

D

Let

D such that

x t;. P

of the boundary of Q

nonempty open polydiscs in

is closed in

There exists an exact sequence

in

D

,

take an

pep

and

,

be

P C Q.

-

4 57

-

Y -T. Siu

Pl

... -> on

n+ Nl[)

->

·..PO

n+ Nt}

-> "1 ->

0

It follows from

o that

P

C

Q •

Hence.Q

is closed in

Now consider the general case.

IRN • Let

Let

n: D xllN(bj

->

D and Q Take

a