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

Zo

there is a finite number of

t1 ..... ~ 1 such that q. = sup (~l""'~l)i there eXists an (n-q)-dimensional plane E through

Cd' functions

(ii)

Zo

such that

i E'

1

~

i

~

1.

8ubharmonic in a neighborhood of

b) n. 4> :

is strongly p1uriZo

in

E.

Let X be a connected complex manifold of complex dimension We Will Sa:! that X is q-pseudoconcave if a Cd function is given such that X +lll

(1) In this chapter the notions of analytic set, complex space. normal complex space, Stein space will occasionally be used although our main concern are complex manifolds. For the basic

definitions we refer to the fo110w1ng Chapter

5.

- 62-

A. Andl!eotli

(i)

for every

c

>

inf 4> the sets

I

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

5

there exists a compact subset

of

X

such that at

~ is strongly q-pseudoconvex.

Zo < i - K

every point

K

Remarks.

1.

For all

c> inf

4>,

0,

except for a set of measure

boundary aB = ii - B is smooth (By Sard's theorem). c c c Levi-form restricted to the analytic tangent plane at zo' aBc

to aBc'

c

<

inf 4>,

has at least

n-q-l

the The

positive

K

eigenvalues. special

Therefore if

ca...~e

n-q-l

L1

these manifolds are a

of the pseudoconcave manifolds studied in

chapter I. For this reason when we speak of a q-pseudoconcave manifold if

X ~

n

is not camp act, we will assume that

2.

Q

s:

Xt

q :So n-2 , hence

Indeed only the eigenvalues of the Levi-form in the

direction of the level surfaces of

~

do have a geometric

meaning; talting a rapidly increasing convex function of 4>

the

remaining eigenvalue can be forced to be positive on-any given compact set.

2.

The notion of

q-pseudoconcave manifold could be generalized

in the sense of the remark made in

a), above.

Examples. 1.

Every compact connected manifold

any 2.

q.

Take

Let

,Z

X =

z(i) j

,

ing at

I

q-pseudoconcave for

K = X, f = 1.

n ~ 2.

i -

'S. Xi

1s

be a connected compt;i.ct complex manifold of complex

dimension then

X

~ Xl t

j '5:. n,

Let ~ Xl' •• " •••

l

xk~

is

x k ~ be a finite subset 0 f O-pseudoconcave. Indeed if

are local complex coordinates at

we can select

e. >

0

xi'

Z,

vanish·

such that the coordinate balls

A. Andreotti

are rel atively compact 1n their coordinate patch and are disk joint. Take K • Z - U Bi and for ~ a C ~ function such i=l ~ ~

th at

€

K,

on

+I 3.

I z~i) I

=~

Bi

Z be compact connected manifold of Let Y be a complex submanifold of

Now generally let

complex dimension

2.

=

Z with Then

~

n

2

~

X

dim; Y q ~ n-2. Z - Y is q-pseudoconcave.

Indeed, at every point neighborhood

U(y)

y € Y

we can choose a coordinate

with coordinates

Z~+l

u(y)n Y =/Z€ U

U

Let

t

U t 1

••• ,

= j=t+ll

f P if

Let

= 1.

to the covering Then

4

2

Cover

>

0

compact in

on X.

b~

coO

a

4.

X

Let

and let

r

Z~

=

X

->

and for E.

Noreover

:it

H n

-I ~

~

c. ~

0

Pi'

t J,

~

)

Ui

l

+= ~

E

each point of

Therefore if £.

L(~)

•

is relatively Y L(4)

>

0

has

is sufficiently

be the generalized upper. half plane with

there is an integer

H n

and

has n-q positive eigenvalues.

oe the modular group.

th~t

o~.

with a finite number

1

Set

Zo

E

= •.• =

partition of unity subordinate

t UiS O~i~l

n-q positive eigenvalues. small on

Y

U such that zn

of these neighborhoods and let . U = Z - Y o

Ul

U

4o

.

Z~ 1

U zl"'"

1 / 0

the isotropy group l' Z o

From

0.4,

ex:))

n L 2,

i t follows

such that for every point

has an order

:£. 1.

A. Andreotti

Let

r

p

=

p). 1. Then the group cannot have fixed points(l). Thereis a complex manifold; 3.4.1 it follows that X is

be a prime number with

rg.f r foreg ~ IX (mod =

p) HnlJ'p

p From the proof of theorem

q-pseudoeoDcave with

= tn(n-l)

= dime Hn _l • Here the concavity 1s to be understood in the general sense of q

remark 2

5.

above.

More generally let

domain in

In

automorphisms of

o~ 4.2

q ~ n-2

Let

£

be an arithmetic. group of

A result of

without fixed points.

0

Borel shows that

be an irreducible symmetric "bounded

0

n ~ 2.

wi th

X = O/f

«( 17 J

A.

is q-pseudoconcave for some qt

).

The problem of projective imbedding of pseudoconcave manifolds.

a)

We have already remarked that if

manifold anI! X

T : X -+ P N(I)

onto l' (X).

(2.5.3».

If

then the map

Y

y.E"

Y

>: X

~

is a pseudoconcave

a one-to-one holomorphic map of is contained in an irreducible

then l' (X)

algebraic variety

X

of the eame dimension than is the normalization of Y factors through

is again a projective algebraic variety,

~.

X Y

Thus

X

is isomorphic

(1) 1

A'.I' 1

has a fixed point, for some

P

A '" I + pB thus 1 1 ( 1) pB + (2 ) p2B2 + + P 1B 1 =

• • must have 1

A

Y•

Moreover

to an open subset of the projective algebraic variety Indeed i f

(theorem

(cf. chapter 5),

• Y.

1

1

oS 1

Now

= I.

11

1

o.

we Bee that each entry b ij of B 2 A = I + P B1 , Arguing in the 2 same way we Bee that each entry of B1 is devisible by p • thns A = I + P4B2 • And so on. It follows that we must have B = 0 and thus A = I. Dividing this relation by must be divisible by

p.

p

Thus

- 65_

A. Andreotti

The problem of projective imbedding of speudoconcave manifolds can be loosely formulated as follows: to find some useful criterion to ensure that a pssudoconcave manifold X i8 isomorphic to an open set of 80me projective algebraic variety. We Will then say that X admits a projective imbedding. b)

It

is compact such criterion is provided by the theorem

X

[33J

of Kodaira

(4.2.1). Let X be a compact complex manifold. fcllowlBs.aae 'quivalent conditions I A. X admits a projective imbedding.

Theorem

B.

X

carries a K8hler metric whose aaeoclated exterior tora has integral periods. There eXists over X a holomorphic Une bundle F!!., X

c.

.hose total space

F

i.

O-pseudocoDcave.

There eXist over X a holomorphic line bundle th9t the graded ring )lex, F) ; 12'0 [' (X.

D.

Fl)

F ~ separates

points and gives local coordinates everYwhere on

Let

the

n

= di.eX;

to say that

)lex.

F)

X.

separates points and

gives local coordinates everywhere OD X means the following: (et) given x tl y X there eXist an integer 1, 1 > 0, and

so' sl e r(x,~)

such that

det (r> )

tl 0 X

s

such that

n

G

x l L(X. F )

there eXists an integer

for every

0:

(_l)i si

d

8

0

f\

•••

1 > 0

and

"

In otqer words the meramorphic function different ftvalues" at 81

S- ••.••

o

x

and

y

while the

Sn

s-0

(if

sO(x) tl 0,

provide local coordinates at

otherwise renumber the sections x.

s)

A. Andreotti We will prove here only the implications imR11cat1ons will be proved in chapter

D~

A.

The other

VI.

~. ~

D # A. For any two points X 1 = lex, y) and two sections

Y

X

~

q • (3

«(x),

Q(Y)]

1\ (x)

(3( y)

~

det [

Replacing assume

and II


th~t

we can find an integer ~ (X, F l ) such th"t

0

by convenient linear

et(y) = (l(x) = 0

and thus

d.

combin~tions,

(x) \,>(y)

~

we may

o.

Therefore also

:(X)

u

det

f

(l

~ 0

•

(x)

Set Al =

2(x,y)'.

X x X

I

V (
€

l rex, F )"

l

r (X, F ); C«X)(3(Y) -d (y) (3 (x)

By the above remark for any integer each

1

A l diagonal '" k> 0 : Akl

k> 0

is an analytic subset of of

#

X" X, Al •

If

Al ~ '"

=0 Xx X

A GAl' Now for kl containing the

then for some

Since a decreasing sequence of analytic

subsets of a compact manifold must be stationary there eXists an 1

0

>

0

(1)

such that

(11)

(i11)

x 10

Fy

in

Therefore 1

.c (X, F

the sections of

points that

A = h. lo

X.

0)

do separate any couple of

Similarly by a simpler argument one proves

can be chosen such that also

the sections

0

f

.r (X I

1 F 0)

have no common zeros,

they give local coordinates everywhere on

X,

-67-

A. Andreotti I

Let map t

{sO'···. sk} be a basis of [(X. F 0) given by X -+ lk (E)

and consider the

x -+ (sO(x) ••••• ek(x». This map is holomorphic by (ii). is one-to-one (by(i» and biholomorphic (by(iii». Therefore ; is an isomorphism of X onto -r(X) which is a projective algebraic manifold by theorem (2.5.3).

A ~ D. It is enough to prove that implication for X = Pn(E). Consider on !h(E) the line bundle of the hyperplane sections. this is given by

Z

~). zi

It is then

immediate to verify that this is a holomorphic line bundle A section on

Pn

ie given by

(Zo ••••• Zn)

= ~zO

~

(Zo.···. Zn+l)

Then f(Pn(E). F) + ••• + anz n points and gives local coordinates everywhere.

""ere

zn+l

Remark. D r.p A

ensure

If

F.

separatee

X is not compact the proof of the implication

breaks down, although one haa reasonable cond1 tiona to th~t

condition

D

1s fulfilled.

For instance:

if X = H/r is a quotient of a bounded demain H c. En by a properly discontinuous group (Without fixed points) then one can show directly, by means of Poincare series, that condition D holds;

_ 68-

A. Andreotti

X' is a complex manifold and on

if

X

there exists a

holomorphic line bundle such tha.t F is "positive" and t K-1 Itposltlve and complete l1 , K denoting the canonical

F

bundle of

X.

We refer for these statements to [13].

we

thus formulate the problem of projective imbedding in. the

following manner: ~

n

X

= dim~

bundle

F

X.

Assume that there exiets on

such that ..(( (X, F)

coordinates everywhere on imbedding?

~

X.

X

admit a projective

O-pseudoconcave .manifolds

[101 , (121).

(cf. 'Iheorem ~.'PoBe

(4.3.1).

Let

O-pseudoconcave manifold.

X

th ~.l.t there eXists on

Jt(X,

that

X

X.

X

~

We choose

=? x • X,

F

admits a projective imbedding_

(following an idea of (<>:)

a holomorphic line b'mdle

separates ooints and gives local coordinates

F)

everywhere on

Be

separates points and gives local

Solution of the problem for

4.3

o :;. qs n-2, X a holomorphlc line

be g-pseudoconcave manifold with

~

lP

~

Co

(x) .> Co

~

t

H. Grauert).

inf ~ X

<

c 0

inf d K

such that

has a smooth boundary.

is compact, we can choose an

1

Because

sufficiently large so that

o the sections of I (X, F1 )

have no common zeros, separate points

and give loc-~ coordinates everywhere on

~

Co

•

This is done by

the same argument developed in the preVious section. (~)

Let

N+l =

bo.sis of .f (X, Fl ).

dim~

f

1

(X. F)

and let

We consider the map

..-

:

be a

-69-

A, Andreotti

defined by

x f-:l> (sO (x), •••• SN(X».

an irreducible algebraic v
(-r)

of

Because

~

Zo

A '

It:

1.'

- Bc

X

~t

must be

61 A

Then

t

0

Ap

EI

E

A

Let

=~X~

~

15

<.

: D

X where

~

such that

~(Dl)

= O.

t

n = dimE X. X

o

is of positive dimension, we can find a

It)

having a maximum at ()

A

has a maxilllum at some point

C

Ao

is a 8ubharmonic function on

I-'

A

O-dimensional.'

has an irreducible component

I-' (0) = zo'

Then

the same

0

A

one-to-one holomorphic map 1 D =

is contained in

o

Since

o

of

Z

)

O-pseudoconcave any analytic subset

is

X

Assume if possible that of dimension

Co

X.

contained in

X

~(B

By theorem (2.5.3)

and one-to-one.(l) dimension than

This map is holomorphic

I

for all

(~(_l)h

D

l

,

non-constant and

This is a contradiction ..

Consider the subset of 8

i h

~ i o , •••• in~

dS

<:.

This 1s an analytic subset of

i

"

••.

1\

dS

0

~ 0, ••• , N'§ X

i

X :

1..-•• /\ dS

1 )x

=

0

n

h

•

contained in

X -

Bc '

Thus

o

it is a discrete set by

i

(Y),

Obviously

extends to a holomorphic map 1"

which i,a of all

thus at most countable.

poin~.s

of rank

ran:~

n

over which

X-A --> Z

at; each point. ')

We agree to delete from

may be extenjed by a hololllorphic map

n.

(1) We can also

assur~e

?( (X)

t ••• ,

A

-1Q-

A. Andreotti

We Will call this set

A again.

(e) If we can show that A is a finite set then one co~­ pletes the proof as follows. We can replace 1 by a convenient multiple of 1 so that the corresponding new map is of rank Ie.> n at the points of A. Let Z -'!> Z be the normalization ot Z. It is knowU that Z* ie aleo a projective algebraic variety (c t. t52], (531). Since X is normal the new map ~: X Z factors through the normalization Z ot Z by a holomorphic map

.

~

N*

•

"'.

Since T is ot rank n everywhere, and since Z being normal is locally irreducible,) must be one-to-one and therefore an isomorphism.(l)

(s) We are thus reduced to show that A is a ~inlte set. Let T ~e the closure of the graph T ot ~ in X]I. lEN(C),' The set T must be an analytic set as it is the irreducible component containing T of the analytic set

l

(x, t) ~ X

"'.

)<.

:IN(a:) I

l!Ii(x) t j - Sj(x) t i

=0

for

0

~ i,

j ~ N •

Let ~ X-A normalization

Z•

be the factorization of 3'" through the ". be the closure of the Let T 1"'. graph 0 f T . This is also an analytic set as it is contained in the analytic set (I ~ ~)-l(T) which has the s~~e dimension .... n than ". T. Let ex: : iI* T -7' X, I?> : .T......:;> Z be the natural ""* projections. NoVl let a ~ A and set T (a) =l3 d -1 (a). Let U be a coordin'~te ball centered at a and not containing any other point of A. 7'

Z* ~ Z of Z.

(1) is of rank }(X)

n

everywhere

= lI:

sl

(-,

So

~d

... , sN So

generally one-to-one as ).

-71-

A, Andreotti

I

is normal; so ~ -1 must -1 d(U) be one-to-one as it is one-to-one on c! (U-a) and since

Now

(l/

'T'

must be one-to-one as

-1"

Z,

Theretore.n. = 1\" -l(U.) is

mu"t also be one-to-ons.

op~ ~!')thUS a neighborhood in Z· at ,(a). Since J1, is n01'1lal, the holomorphic map A " ,,~-l I Sl- T' (a) ~ U extends to a holo.orphic map A I JL ""U. We claim that each component cimension

z·.

set ot

1

B

T' (a)

at

which is not at co-

ot the singular

must be an irreducible component In tact it

B

contains a non-singular point

b.

z·

and it B is at codimension 2. Z, then A must have· a nonvanishing Jacobian at b. ThUll A. -1 ill detined near a and -1 ,..,. ).. (a)" b i.e. T must be a local isomorphism at a. Hence

a" A.

(~)

Now we remark that

compact.

Thus tor each

«

Z*

is a proper map because

N* a" A T (a)

many compact analytic subsets of

is

is a union of finitely

Z• -) *(Bc

=Y

)

of codimension

one and finitely mshy irreducible componenti ot the singular llet

Z· (which must also be contained in

ot

"J.

~

a

Z

~.

then

T

.

r,;*

("J.)

Y.)

Moreover, trom the

"J.'"'2

previoull ariument it tollows that it

are in

A

(l T (a ) = ~ • Z A ill a finite sst it we show that

We will prove that

and

Y con-

tains at most tinitely many irreducible compact analytic subsets

ot codimenllion

1

(hers we use that the singular set of

Z•

has a tinite number ot irreducible components). Let e

>

0

be so small that

biholomorphically to

.

B

Co- £

is compact in

. tV.

Y

4> and that

d.

1 4 > c oc a -e = '

Y e• Z c o-

Then

> itt

- T (B

extends

7

Set

co-~

).

and we can, by the isomorphism

f

on

Y-

Yco-e.•

extend this function by putting it equal to

c o-

€- on

given

transpose the tunction

Let Ull

Yco-e.. and

-72-

A,. Andreotti

call

t

the continuous function thus obtained.

.".: y ~ lR, i) 11)

The function

just de fined, h.....s the following properties

..,.

is continuous; Yc ={'« cJ cc Y

on

11i)

Y - y,

Co - E.

for any

'is

cO"

c

<. C

0

:: sup 'f;

Y and strongly plurisbharmonic.

Then every compact irreducible analutic subset of Y of dimension ... 1 must be contained in 1'c _ ~ by the same o "argument given in (y). Moreover Y is holomorph1cally convex as it follows from i), ii), iii) and the solution of Levi problem (cf. chapter VI). It then follows by the reduction theory of holomorphically convex spaces (cf. [20J) that there is a compact analytic subset C ~ Y of dimension 1 at each one of its points and such that any irreducible compact analytic subset of Y of dim~l

ension

?

and 4).

must be contained in

Since

of codimension

1

C

(see section 6.3

remarks

C has finitely many irreducible components the theorem is proved.

only if dim= X = 2 this result can be considered s~tisfactory as the only type of pseudoconcavity available is O-ps8udoconcavity. Remark:

The case of dim= X L 3.

4.4.

If dim= X L. 3 in the previous theorem the assumption that the ring ~(X, F) separates points can be dropped; one has in fact the following Theorem (4.4.1). ~ X be a O-pseudoconcave manifold with dim= X ~ 3. Suppose there exists on X a holomorphic line bundle F such that the ring df(X, F) gives local coordinates everywhere on of

X

so that

X. X

Then

J('(X, F)

dOES

also separate points

admits a projective imbedding.

-73-

A. Andreotti The proof of this theorem is r':..ther complie :...tcd ......rH1 Vlill be omitted.

However the inttrest of the theorem resides in the

fact th"t i t does not holl for dim X = 2. t We will indic ate how to construe t "counterexam,le based on the next

(c f. [10])

Lemma.

W <.c.... U

Let

V

be

0.

connected comnact manifold.

~e1nrrr open subsets of mani fold and let 11':

be an holomorphic map making

V.

~ ~

Let

be a connected complex

X-?V-W

X

into a ). -sheeted non-ramified

covering of V - Vi. II dim V ~ 2 and if V is simple cont nected either A = 1, or else X ca_",not be compactiti ed (i.e. X cannot be isomorphic to an open set of a compact complex snace). The reason for the validity of this lemma can be said briefly as follows.

If

X

would compactify into

that the holomorphlc map ,..., N rr:X...;a,V. Since

~

somewhere in

V.

then one can show

extends to a holomorphic map

V is simply connected the map

..l > 1,

"-'

X

fl'

must be ramified, if

But the rami fication set must be anal-

ytic and contained in U i.e. it is a compact analytic subset of a Stein manifold. Therefore the ramification set consists of a finite set of points E. Since V is simply connected, non-singular, of dimension ~ 2, also V-E is simply connected. Therefore _ff-l(E) is an

X

irreducible non-ramified covering of

V-E.

But this is possible

only if ). = 1. Remark. One could also assume that X is ramified over V along an analytic set F c V-U such that V-F remains simply connected.

(1)

A Stein manifold is a holomorphic convex manifoll on which

holomorphic functions separate points.

-74-

A. Andreotti Construction of the exarn.'Jle.

T = r,2/r

Let

be a.n algebraic

torus of complex dimension two .lefined as the quotient of

r

by the group

2

translations gener&ted by the vector columns

f

0

a::

of the matrix 0

r:

~,]

zll

1

zZl

where

t: T -. T

group

=

zzz

is a point of Siegel half plane Let

Z

t

HZ'

Zu

ZlZ

z21

zzz

z = Z,

1m Z

) o.

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

its inverse.

T

has 16 fixed points. space of dimension points.

Let 2

K

with

= TIT; 16

this is a complex analytic

conical, non-degenerdte double

A theorem of Kummer shoVis that

algebraic surface

"0

K

of fourth order in

is isomorphic to an

F (a:)

With

3

isolated. n?n-degenerate, conical singular points. disjoint small balls

~~

of fourth order in

It is known that

simply connected. i)

U(. =

ii)

'" -U ~

There fore torus of 1" give

lP (a:)

3

close to

.0

and han-singular.

q,E. -U E..

lP (a:) , 3

is

is a Stein open subset of

is di ffeomorphic to

(.

16

1 S d. ~ 16, centered at the 16 U = U lI. • Considsr a surface d q =1

and let

<1(., as a non-singular surface of Moreover

~<.(1U

T where



0

-

U

adrni ts a non-rami fied 2"*sheeted covering, the

16

small neighborhoods of the

have been removed. X

1I'3'

U" in ~0

singular points of

16

Select

Let

X

16

fixed points

be t;!is double cover'::'ng.

We

the complex structure that makes the natural map

1r : X - " -Us holomorphic. Since /be -UE: is O-pseudoconcave

Moreover if

F

X

is also

O-pseudoconcave.

denotes the holomorph1c line bundle of the hyper-

plane section of ¢

!

t

J~(4 -U t 7 t c

F)

gives local coordinates,

-75-

A. Andreotti

everywhere on f -U. The SaMe is therefore true for ~ 1"q eX, rr *F). But the preceedinc le'11ma tells us th,"~t X be comp;.),ctified, in particuLJI

X

cannot

cannot c.umit a projective

imbedding. Remarks.

1.

We have constructed an example of a

O-~seudoconcave

th,,"t cannot be comp ;,ct1 fied; actually if l'Ie let on a small disc

U :" ~ X

t

1t'€ A

0

around. f:

..1

f

t

e

manifold

vary" 11 ttle

we h:ive constructed a fD"inily

2-d1mens1onal mCJ11 folds. all

O-pseudoconc ave

and no one of which can be compactified. 2.

Making use of the remark made after the lemma one can show

that example

3

complex structure

of 00

(2.3)

gives also'" oon-comp,ctifiable

1'2(11:) - ~Of.

_76-

A. Andreotti

v•.

Chapter

Meromorphic functions on complex spaces.

Prel1m1naries a) Let Jl be an open set in r.", A closed subset A ~ Jl • called an analytic set if A is locally the zero set of a fiJl1 te number ot holomorph1c functions 1.e. tor every Zo E. A

there exi· ts a neighborhood U f1' ••• , f • i"(U) such that k An U =~z~ U

I

fl(Z)

0

f

in it

Zo

and

= f 2 (Z) = , •• = fk(Z)

=05.

On A we dsfine as the sheaf 0A of holomorphic functiona, the sheaf of germs of restrictions to A of holomorphic functions in some open set of 11. A mOrphism (or holomorphic map) from the analytic set (A, ~A) to the analytic set (B. (9B) 1": A

is a continuous map 7' •

~

(9 B. i(x) ~

B

such that

t\,

x

for every

x. A.

The notion of isomorphism is then defined and then the notion of complex space is obtained replacing in the definition of complex manifold open sets of a numerical space and isomorphisms between them by analytic sets and isomorphisms between them. The "structural sheaf'" will be,denoted by & . A point x ~ X of a complex spuce X is called a simple point if some neighborhood

U

of

x

in

X

is 1so:'liorphlc to some

open subset of some En. Let SeX) be the set of non-simple points of X, this is called the singular set of X. This is an analytic subset of X as one can prove. A complex space X is c'llled irreducible if X-SeX) is a connected

m~ifold;

The dimension of

f~

its dimension 1s called the dimension of arbitr~y com~lex

space

the maxim::l1. dimension of its components. sp:l.ces ti;en complex sp.:lce

dim Y 5. dim X. X

we Jefine

If

x

dim:x!

E

X

X If

Y c. X

dim seX) < dim X.

are complex

is a -point of the

as inf. dim U(x)

describes a fundamental system of neighborhoods of Always holds:

X.

is, by definition,

when

x

in

u(x) X.

-77-

A. Andreotti

b)

The Dotion of meramorphic function on a complex apace is

defined in the same way as for complex manifolds.

complex space

X

Given a

we can then define two rings of meramorphic

functions: ~(X), the ring of meromorphic functions which are quotient of

two global holomorphic functions on not a divisor of zero.

X,

the second of which is

Jt(X), the ring of all meromorphic functions (which are locally the quotient of two holomorphic functions, the second of which' 1s not a zero divisor). If ~

X 1s irreducible, as we will alwaYs assume in the seQuel, (X)

and

J\(X)

are fields.

0) A complex spaoe (X,I9) :l.s called normal if for every x. X the ring cr is integr_illy closed in its quotient ring. x A oomplex space (X.6') is called weakly normal if every conf: U ~

tinuous meramorphic function

set

U ~ X is holomorphic on

U.

a:

defined on an open

Given a complex space

(X,crX)

there exists a (normal) complex space (X, &' X) and a morphi8ID o f' : X -) X which is sur jeotive and such that for any normal complex space (Y, 19 Y) any morphism Y --> X such that the im ..... of any irreducible component of Y is not contained in SeX) factors through X A

(X,Cf~)

The space

is uniquely Jefined up to isomorphisms and

is c illed the norraalization of

(X,I9). Analogous 'statement holJs repl.~cing the

normal". of

(X. (3- ).

morphic.

VlOrd

"normal" by "weakly-

We thus obt:odn the notion of we:.lk-·normalization space A space

X

and its we."k norlnalization are homeo-

A nor:!tol space is

we'i:~ly

normal.

-78-

ILAndreotti

Examples. 1.

The space

x _ 1 (x, y)~ a: 2 I ~

weakly-n<>mal as the function gral over the local ring of X 2 the equation (1 ) _ x _ 0 on

x

x 3 _ y2

l

is neither normal nor

1s eontinuous on

X

~d 1nte-

at the origin as it satisfies

X.

Its normalization and weik normalization 1s the complex space with a:.? X defined by the map t ~ (t 2 , t 3 ).

a:

2 2 The space X -1. (x, y) ~ a: I x _y2+y3,. 0 5 is not normal as xy is integral over the local ring at the origin but not 2 holomorphic; it satisfies the equation (~) + y - 1 • 0 on X. 2.

y

Its normalization is t '-I (t(1_t 2 ), I_t 2 ). g

a:

with

However

p

1:->

X

X defined by

is weakly normal.

is holomorphic and continuous near the origin on

must be holomorphic

in the neighborhoods of

and assume the same value

ex.

at

t

~

1

and

In fact if X

t = 1,

t = -1.

goP t = -1

One

verifies th,tt any holo""rphic f;:nction of this type can .. written as a series of the form 0. + 1 1 a t r (1_t 2 )r+s, but this

r+s=

rs

is the pull back by P of the holomorphic function near the ~2 ct· + ~ r s ortgi n i n \II r+8=1 are x y •

3.

Simpler examples can be ootained by tacing into consi~erattn reducible spaces. So for lnst3nce -)(x, y) E a: 2 \ xy = OS is not normal bu.t weakly normal; f (x, y) " a: 2 1 (x_ y 2) x = 0 S is not weakly normal, its weak normalization is isomorphic to the preceding set. As re ference one can consult [29 J,

L1,2 J

we~~

normalization.

5.2

Pseudoconcavity for cOffillex spaces.

The notion of pseudoconcave :n'..lIlifoll s paces with the fol1o\'/"'.nc jefinition.

C9.Il

and

L 8]

for the

bG: extended to complex

-79-

A. Andreotti

The complex space sutset

Y

#

~

(X, (f)

X

of

is caLloeel

pseudoconcave if an open

is given Bllch that

yccX;

i)

=y

:11) For every Zo Gay sequence of neighborhoods

-------

(U,,!1 y)u

lU"

- Y there exists a fundamental

l

of

Zo

in

X

is a neighborhood of

Buch that

zoo

\J

Here (U:;;-y)U y

= ~z

€

Uy

I

If(2)1~us~~

If I

v

for all

f€ #(U v ) •

This is generalization of the notion given for complex manifolds. in fact if

X

is such a manifold and

boundary and if

V zo' 3 Y

analytic tangent

p~ane

Y« X

haa a smooth

the Levi-form restricted to the

has one negative eigenvalue. one can

construct, using the disc theorem. a fundamental sequence of coordinate polycilinders holomorphic function on Uy

= p"

P v centered at Zo such that every P v n Y extends to P. Taking

the above condition

ii)

h~

For a pseudoconcave space one

theorem

(cf.

(2.5.1)

!heorem (5.2.1). n,

If

then the field

is then fulfilled. the equivalent of

[1]).

X

is a pseudoconcave space of dimension

'J«X)

of meromorphic functions on

an algebraic field of transcendence degree We remark that from condition

ii)

d

Gt (X)

!!.

n.

it follows again

holomorph1c function on a pseudoconcave space Therefore

~

X

X

th~t

every

1s constant.

= 1:.

,

5.3

The Poincare problem. a)

It we drop the assumption of pseudoconcavity there is no

hope to obtain a st3tement of the

n~ture

have already remarked that for

= E,

e" , e z2

,

...

are algebraically

X

of theorem (5.2.1);

.e

the functions

~ndependent

and, therefore, the

- 80-

A. Andreotti

transcendence degree of as grounl field hopeful.

}(eG:) inst€~d

1jeX)

is infinite.

I f .. ~.

Theorem e5 •.3.1).

~

of

First of all one h ,s the

However if we take

the situation is more

followin~

useful fact

is a normal space, then

tieX)

~

X ex).

algebraically closed in

The proof is the same as the proof given for lil:mifolds etheorem e2.2.2».

Moreover the previous

counter-ex~aple

disap-

pears as one h.1S: Theorem e5 •.3.2). (a) II X is a Stein sni;lce(l) then (b)

!!

X

[( (X) = J{(X).

is an open connected subset of a Stein manifold

~

~ (X) = :reX).

(1) A Stein space complex space.

X

(or holomorphically complete space) is a (with countable topology)

satisfying the

following conditions (i)

H(X)

separates points

eXi"ats an (11)

tEo H(X)

X. in which

ally convex. A sheaf is called coherent U

there

~xi~ c.

X

there eXists an

such that

I t(xi) I = IY'.

sup

neighborhood

x I: y, x, y ~ X,

if

fex) I: f(y).

with

for any divergent sequence t e !I(X)

A space

1.e.

of

(ii)

~

is satisfied is called

of 19--modules on a complex space

(ct. [28J) x

ho1omorphic~

if for each

x ~ X

X

there is a

and an exact sequence of the form

eP~eq~~ ~ U

In particular the sheats

U

e P,

U

P 2 1,

0 are all coherent.

Kernels, cokernels, images, coimages of homomorphisms of coherent sheaves are coherent. Stein space

X

the space

f (X,

For a coherent she'·t

'J)

generates

-=; x

'"!

on a

over f) x

A. Andreotti ~.

J

(a)

Given

= [f

<:

13

h = £

Locally

I

~(X'>

E

fxh x

with

q

zero divisor.

h

ex

<0

P

and

£

19, q(9

/J X

= r9 X

O s ~ ['(X, j) sh

=t

•

r

If

with

(X, <9 ).

&/qe

X o

q

not a

S

e/q 6> 5

are coherent,.it follows th at

There fore by theorem

h A

is holomorphic there eXists

o s(X ) f. O. o Thus

= is

h

and

s _ 0

(since

X

is

h. 6J(X).

Let us "ssume first that

X

Given

.

h : X

-;>

~

is an open set of h'" .H(X)

and denoting

7~ the sheaf of germs of meramorphic functions on

defines a section

(;J ,

But, by the definition of

(a particular Stein manifold). by

(mod q)

0

!!

is a point where

X

Eo.

assumed irreducible), hence (b)l

!P

<9- -.14

and thus

1s coherent.

we have

X •

6

holomorphic and

q

eI

.. Ker ~

{J

x

for

Thus locally

!) = U Since

consider the she"f

77( of the stac.>.ed

B~3.ce

In t

h

-:J?'( over

the open set x. Let X be the connected component of heX) in;r,r. Because the natural projection rt: 7Jf-i' a;n 1s a local homeomorphism, X acquires a natur~ complex structure of an n-dimensional complex manifold. This mwoifold is Hausdorff as the stacked spJ.ce

7?( 1s Huusclorff.

for every

x c X

(theorem

By the very me::l111ng of

A of

subspace of the numberical space

H. Cart;m). Every complex is a Stein ap .ce. Every

a: N

Stein sp;.lce (0 f finite dimt:;1s1on) a:lmi ts hololliorphic

.

X 1 th .. re

r,w.~) into ::1 numeric:u sp.,ce

;l,

a: N •

proper one-to-one For a Stein

mani fold th.lt m '.p can '"ue chosen to be an iso:7lorphism onto

its 1.m.:'.ge.

- 82-

A. Andreotti A

A

eXists a meramorphic function A

h

I h(X)

• ~ It" h:

X

A

Therefore

A

-log

on which

6(~)

6 (x).

from tbe boundary of

X

is a plurisubharmonic

[31l).

X

is a Stein manifold. We can then apply to A A A argument developed in (a) and write h = l!. with p, A

rr•

A

X

l!. " q

Then

and

is an isomorphism between

X

the A

q

q

A

holomorphic on

h

polycilindrical

A

x

(cf.

,

Hence

is the

x.

A

distance of

hx

the germ of

X

is the largest domain spread over

can be continued. function

such that A

A

X

on

at each point of

germ represented by Now

h

•

h(X) .. ". h

and

h(X)

and since

we get the desired

result.

(b)2

Let u& now consider the general case, in which

an open subset of a Stein manifold for Stein manifolds

(cf.

Y.

we may aosume that

~N.

1s a Bubmanifold of some numerical space

([23])

Coquier-Grauert U

Y

of

~N

i.n

is

By the imbedding theorem

r311)

[281,

X

Y

By a theorem of

cone can find a connected neighborhood

and a holomorphic retrilction

i : U ~> Y, 7 (y) = y for all y E. Y. Consider the meromorphic • -1 function r- h on T (X). By the speci~ case (b)l we can find two holomor~hic functions such that index 111.1 <

a.

€

Id I •

q T ·h = p.

JI'

such that

d

q ~ 0

Since d

JI

D qlx

D" (q . T h)

Then

p,

=

Dd

0

p.

q

q ~ 0

on 1" -l(X)

there exists a multi P while D q\ X "0 if Restricting to

d

D al X h = D P Ix

Vii th

X

we get

d

with

D q X 'JI 0

as we wanted.

The previous theorem gives the solution,

cases of the so-called Poincare problem: ~

X,

is not

a1w~s

do we ho.ve

@ (X)

= :1(X)?

when, for a complex

Of course this problem

solvable, for instance for

irreducible algebr<:dc varieties we he.ve

in some particular

comp~ct

projective

~ (X) = II:

but

- 83-

A. Andreotti

~

J!(X) #

if the variety is not a point. One could hope that !?(X) = 1{(X) i f II(x) separatea points of X, but this in general 1s still an open question. b)

For a complex sp.3.ce

sep["~rated"

tR

reI '-l.tion

X

which 1s not "holomorphically

one 1s Ie "d n.ltur:1l1y to consider the equivalence on

X

iff for all f ~ ..v(X). rex) = fey) iR. Let Y = X//2 be the topological quotient space, and let p : X -7 Y be the natural projection. A set U c Y is open iff p-l(U) is open in X, thus Y is a Hausdorff space. We jafina for U open in Y p-l(U) _> ~ f y(U) = {r : U 7 ~ I f continuous, p " f x..-vy

holomorphic We give in this way to

y

the structure of a ringed space (a

"pace with a sheaf of rings of germs of continuous functions) and the problem arises to see when

If this is the case, then

space and

p: X

7

em then consider

G:

which contains

Y

is

3.

complex sp ace.

is a hololl1orphically separatec

is a holomorphic surjection.

Moreover one

as an exte~sion of the field

71 (X) (X)

(I,e)

(Y, e)

and may be equal to

~ (X)

p "7\.(Y)

if on

Y

the poincar~ problem 1s solvable, as one ob~ously has

~(X) = p"qJ(y). To decide when

(Y,<9)

has the structure of

~~

complex sp:'.l.ce we

h ve the follo,nng sufficient criterion (of topological nature): Proposition (5.3.3). com~lex space !! (i) (11)

Y = X/cR p: X K c... Y p(KI)

Then

(Y t

()

-j>

~

X

be an irreducible weakly-normal

is locally compc,ct, Y

is semiproper ,

i.e. for every

there eXists a comp,",ct set

Ie' c-- X

= K. is a wealdy nor;~alcomplex

3D .lce.

como~~ct

set

such that

-84-

A. Andreotti

Example.

Let

X

be a weakly-normal,

convex

holomo~hica11y

space. p-l(K)

Then p is a proper m.J.p. IndE:ed if K c=:.. Y 1s compact, must be comp-,ct t othErwise we c:m select u divergent sequence ~ xiS c. p-l(K) and thus Fnd an f e JI(X) with

~p

I f(X i ) I

=

But

If'.

sup I fl p-l(K)

= sup

where

p.f

Y = X/If! whose pull ba ,c by is continuous this is impossible.

function on p • f

Moreover such that

Y

is locally comp cct.

p(x) = y.

Since

X

p.f

1s the

K

is

can find a finite set of functions

Let

p

is

y'" Y

f.

Since

and

x'" X

holomorphic~ly

f , ... , f l k

convex, we

in H(X)

such

that the set

has a compact closure and is contained in

K'.

p(K')

Then

p(U)

is a neighborhood of

y

which is compact.

We satisfy thus the conditions stated by the orev!ous proposition and therefore

(Y,

e)

is an analytic space which is also weakly-

normal.

But on

Y holomorphic functions separate points and holomorphic

convexity is inherited by

normal Stein space.

Proof of proposition (<%)

Y

from

X.

Thus

is a weakly

In this case we thus have

(5.3.3).

First we remark that, given a compact Bet

find finitely many holomor·,hic functions H(X)

(Y,S-)

such that: ift

f ,. '" l

KG X

tk

in

we can

- 85-

A. Andreotti

ce

Indeed the equivalence relution

is represented in

X ,.. . X

b,y the analytic set

Locally ,Y('

can be given by finitely many of its defining

equations.

Since

K

~

K is compact finitely many of these

.tJ2f!

defining equations suffice to define

K

(~)

Secondly we show the following fact:

Let

X, Y

Let

g: Y (i)

~ ~

if

(11) if

t: X

be complex spaces and let be continuous.

>..

K.

~

y

*g

g

is meromor9hic so is


is semiproper and surjective then if

meramorphic

60

is

be holoillorphio

Then ¢

¢

*g

is

g.

This fact 15 proved making use of the following remark. A continuous function

X

h : X

1:,

-,>

.lefined on a complex space

1s meramorphic iff the graph of

h

is an analytic subset of

X ... 11:.

This is a p',crticuLx case of a general theorem prop.

(cf. fllJ.

3.12).

In this instance if

X

is weakly normal there is umost nothing

tQ prove; in general a space anJ its weuk norm;.u.1zat1on have

the 1!same lt meramorphic functions.

Making use of this remark,

st"tement (i) then follows from the fact that the graph of 4> * g

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

equals

holomorphic and the graph Denote by

r:

and

r

0

f

g

while

the graphs of

We have a commutative diagram ~

r

X --i>

4'id: X x ~ ->y x ~

* g

and

g

respectively.

r

lfr lIT X-~-H

where

rr

:X= (~x

and

,7

id)!j::'.

1s

is analytic.

are the natural projections and where

- 86-

A. Andreotti ~

rr

Since 4

and

are homeomorphisms

iT

is semiproper.

is semiproper because

A.

y,.. e.

C is closed in

Moreover

Therefore

thE: composed map

x

Y ,

is semlp~oper and holomor?hic. so th ,t [' is analytic.

A theorem

0

f Kuhlmann (c f.

By assumption " g

L40 J.

tion of a theorem of Remmert

,.

a:

[111,. a generaliza-

se also

(45J)

is meromorphio

states that the image of a

holomorphic semlproper map between complex spaces is an analytic set. By virtue of that theorem 1m therefore g must be meromorphic. (y)

b

Let

£

compact closure

Choose

K c. X

fIt ••• , f k

..

and let

Y

X. =

[

wust be analytic and

be a neighborhood of

V

b

with a

~.

p(K)

compact such that It'(X) such that

V.

Select

iff

Let gi: Y -;> sider the map

a:

be such that

•

p gi = f i

for

1

<;

i

<;

k.

Con-

'1-..' defined by x(x) = (gl(x) ••••• ~(x». being continuous, X is l!ijective

L

X

must be compact as

Let

V

L =

(V).

is compact.

must be a homeomorphism of

V

onto

Then. Since L.

Therefore Z = XCV) 1s open in L and there eXists an Q)en set k k il c. a: (open in a: ) such that Z = L In particular Z is closed in

n.

Set

W = p-l(V)

"n.

and consider the composite map

- 87-

A. Andreotti This is given by

x ..... (fl (x), ••• , fk(X»,

therefore

holomorphic map which is also sem1proper as ~

By the Kuhlmann theorem

(Z,8)

Let

(W)

p

is a

J,

is semiproper.

= Z is an analytic set.

be the weak-normalization of

Z.

We get a

A

bijective map '" V'" z. We claim that'!' is an isomorphism of

(V,8

v)

(Z,&).

onto

(~) and the fact that on a weakly normal space continuous meromorphic functions are holomorphic.

This follows directly by application of statement

5.4.

Relative theorems. The previous considerations lead us naturally to the following

situation. X, Y be irreducible complex spaces and let

Let

a holomorphic surjection. p'

p';t(y)

For instance

Y

X""y

This gives an injective map

0A'(Y)...,. Jf (X)

and we want to investigate when extension of

p

1(X)

considered assn

is an algebraic field.

could be the separation space of

favorable conditions we would have

It (y)

=

& (y)

X

and in

:;:~(X).

In this connection one can prove the following: Theorem 7(X)

(5.4.1).

is a prooer holomorphic map then is an algebraic field over .p * jf( Y) t 0 f transcendence ~ dim~

degree Example. Let

Let

If

dim~

X X

p: X...,.y Y.

be a weakly-normal holomorphically convex space.

Y be the separation space of

space,

7f (y)

= d;l(y)

following result as

and

X.

Since

p' ~(y) :;:~(X).

p : X -> Y

Y

is a Stein

We thus obtain the

is proper:

For a weakly-normal holomorphicallY convex suace

It (X) ~ dim~

is an algebraic fiel.d over

X-

~imt

Y, where

Y

q (X)

X

the field

of transcendence degree

is the seuaration snace of

X.

If in theorem (5.4.1) Y is a point we get back theorem (5.2.1) for a compact space. One 1s lead to conjecture an analogue of theorem (5.4.1) also for a pseudoconcave situation. This actually is the case. For this purpose we introduce the following defini tion.

-88-

A. Andreotti Let

X, Y

f.: X -» Y

be irreducible complex spaces and let

be a holomor9hic surjection.

We say

th~t

map if an open, non-empty, subset (2c...X

is a pseudoconcave

f

1s given with the

following properties

E ...

f :

(i) (ii) (iii)

is a proper map

y

for every

ye. Y

intersects

a

for every point

z0

.. ao.

=Q_G.

there exist two

fUlldamental sequences of neighborhoods Vv

Co

Ul.l'

ye. f(U,)

,

!!.i

~

(5.4.2).

X

Uyi,

1 Vv \

with

we have

~ (U"nn f (y))Unf-1(y) Theorem

i

such that

for every

(~ )

f- 1 (y)

every irreducible component of

:>

V v

n

f

-1

(y)

X, Y be irreducible complex spaces and

be normal.

p : X .. Y be a pseudoconcave map. !!!2 ;J( (X) algebraic field over p' j(Y) of transcendence degree ~

!!...J!a

! dimE X - dimE Y. Remark.

For the validity of this theorem actually one needs only

to suppose that A c Y (43)

with

~

is peeudoconcave over some open set

and such that

for every compact subset

K' c. X

of

p

(d)

A"

such thJ.t for every

p-1(y)

intersects

p-1(A)

K c::: Y

y

~

K.

is normal,

we can find a compact set

every irreducible component

K'.

The proof of thie theorem is ratber complicated. It can be found in [111 with all the material_considered in this chapter. Remarks.

1. Theorem (5.4.2) For example the family

is more general then theorem (5.4.1). V ~1Xtf • where lJ = rt< E I It I t·1l

constructed in

4.4.

remark I, gives an eX&mple of a pseudo-

<:

1~

-89-

A. Andreotti

concave map in which no one of the fibers compactitied.

Xt

can be

2. The assumption of pseudoconcavity in theorem (5.4.2) cannot be relaxed. Here 1s a counter ex:mple due to Kodaira and Kas (cf. [111) Take count ably many copies A of the unit disc .1 in 11:, and k cQuntably many copies II: • k of II: • Consider the sets M

=

U

k.~

11

k

~

II: •

k

s = k~~ fo} x ~ On M-S

consider the cyclic group

r

generated by the

automorphism

w • II: * • k k

where

Define the action of ['

on

s

to be

the identity. One verifies th ,t X = Hie is a two-dimensional m,mifold. The ""onifol1 X loolcs like A • 11:* in which [0\' 11:* is replaced by S (cf. (111). We have a natural holomorphic map

p : X

-;>.1

and one h

!I(X)

Indeed, let

f • h'(X),

'.6

= p' H(t.) then

f / Ak

wn •

holomorphic function expansion of fk+ l

f k ~,n(t).

(t, w)

= a o ,n (t) t -nk .

is given by a

#(A). for

in the Laurent

Now t I< 0,

therefore

This merms that if

n

# 0

"'o,n

has

-90-

A. Andreotti

a zero of infinite order at

and therefore in the expansion of of

w

reduction space of On

X

6i

th~t

It follows then

f

f
i.e.

appears

a= 0 if n # 0 u,n only the term independent

thus

t = 0,

(X) = p*t;; (Ii)

and th"t

11

!J

IC * O

consider the functions given on f

is the

X. 0

,

by

= w ;

One veri fies easily th"t these define meramorphic functions on X. 3i7 Set t e with 1m /- 0 since I t\ <1. Set also 2ffi z e , then g is transformed into the Jacobi theta series w 2 :>. e(ri(n +2nz) 19(71Z) = n'7l: For any value of T, 8( r,z) has an inJt:ini te number of zeros of

=

=

Zo(7) + 1 + k 1" ;

the form

We show that q;(X).

and

g

are algebraically independent over

If, on the contrary,

dependent over in

f

I, kG 24

X,

Y,

61 (X)

F(t;X.Y)

f

and

g

were algebraic ..11y

there would exist an irreducible polynomial ~

to be holomorphic in

0

with coefficients that can be assumed ~ ~,

t

such that

F( t; f, g) ;; O. Now for some

t = t 0

#

0

F(t o ' X, 0) ~ 0,

but this equation

f(t o ' X, 0) = 0 should be satisfied for all values 2k 2ffi z t 0 k E 7l: • This is impossible since a X = e 0 p olynom1al in one variable h

,5

only finitely many zeros.

conclusion is that the transendence degree of J«X) is

2

2,

while if theorem

(5.4.2)

should have a transcendence degree

is holding for ~

1.

over X,

The

e;:> (X) one

- 91:-

A Annreotti

Chapter 6.1

E. E.

VI.

Levi problem.

Preliminaries. a)

v

each cohomology with values in a shea!.

Let

sheaf of abelian groups over a topological space

11

U 1~I 1

q C (Z(.1- ) where I

be an open covering of

=

and where we have set

~t

O

1

••• i

q

Obviously'

)

j'(UiO ••• iq

i q)

runs over all

(1 , ••• , i ) q 0

Thus an element l i

n (i O•••••

U

i O" .i q

f • Cq ( l( • '"1)

where

fi

Cq(Z(. 'J

)

0···

i

X;

q

~

7f

X.

be a

Let

we set

.1)

for

q

.>.

°

(q+1)-tup1es extracted from

= Ui

(l

U·

i1

(I • • •

flU

l

q ° is given by a collection

f(U i

0···

i ' '1). q

is an abelian group.

We define

1 Oq : Cq(U .1-) -> Cq + ( 2(.7) by the formula q+1

'= j=O

where

is the n ltural restriction homomorphism. One verifies tho.t Oq+1 a a = 0. therefore the sequence q of abelian.groups and maps CO(

tI. "])

30

->

18 a cochain cemples.

denotee by

Hq(Z(. ~).

1 C (U.

.

'J)

\'1

->

C2(Z{ • 'j )

"2 ->

Its cohomology in dimension

q

will be

-92-

A. Andreotti

Note that, since

is a sheaf,

= r(x.?-)

HO(Z(,t) = Ker 00

eo that the above complex can be completed with the augmentation map

°

The groups

r;-I € -?.f(X,J) --;)

Hq(Zt,~)

of the covering

U

are called the eech cohomology sroups

with values in

V = ~. Vj~ j~ J

If

refinement function means

0

f

~,.* :' .~( 21,

by settir;.g (i f)

"?- •

is anothEr covering

U

which is a refinement of

a.v

°

-L C (V,'T)

= ~ Ui~ i"'1

J : J-»1

0

X

by open .sets

then one can define a

with the property

one de fines a homomorphism

J

f

Vjc-U)('j).

f complexes

0

1) ~C*( z'/'. '1) UT(jo) ••• 1(jq)

= r

jO ••• j q

Vj

0··· j

q

This defines a homomorphism of cohomology groups

--r ..,) H*v H• ('l(, J) ( ,

J z(V-:

One verifies tha.t

j

u

?~

1-) •

de~ends only on the coverings Z{, 2.,"--

but not on the choice of the refinement function.

One can then

define Hq(X. "1- ) =

11m --;>

Hq ( -Z{,7),

q

~

0,

"2-(

the direct limit t3kcn over all coverings U called the b)

Homomorphisms of she:-J.Ves.

(CJ, Jr

eroups into i)

ii)

v

q-th Cech cohomology group of

$-

t

X).

(5 ' c~',

is a continuous map

u-o¢

X) cJ

X

of

X.

This is

with values in

'1

Given two she ',ves of abelian over

-3-

X

a homomorphism of ~

-> ~ such that

".

for every x f!:- X the induced nap group homomorphism.

i' x :

Jx

->J:'

X

is a

.

- 93.-

A .-.Andreottj

~ 1s injective vj can be considered as a subsheaf of ~ and one can then define a quotient sheaf ~ 1'1. as the quotient

If

rt

space of

by the equivalence relation

J

iff

the topology being the quotient topology.

(J 11-)x

=

One hos thue

fxl'Y- x

A sequence of sheavesand homomorphisms (1)

0

f ::,

->:r~ ~

is called 0""

1/..,. 0 ~

X

~ }/x'"

0

if for each

'1x d ~>

:;

x

is an exact sequence.

x

In particul ".r if

the sequence

o .."

the sequence

"f.... :;;

->

if 1 r

1 -;>

1s a subsheaf of 0

ie exact.

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

~:

f""

y-,

this induces a homo-

morphism of cohomology groups ~':

aq(X,J')

->aq(x,~

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

X

(1)

of sheaves over a paracompact

then one has an exact cohomology sequence

d." o -> ao(X, "Y' T)

--;>

al(x, T) "'-?'" al(X'J) ~ al(x.J!)!.

",H2(X,'T)

where


8 is a lIconnectlng homomorphism rt which 1s defined in

the usual way.

J!

- 94-

A. Andreotti v

Cech cohomology is space

p,~ticu1ar1y

useful if the topo10gic31

admits a system of coverings ?(d.

X

X

coverings of

co fin31 to 311

for any

U

,T)

i

i O " .i q

= 0

r> O.

for 31

q

One has in fact the folloWing important theorem of Leray:

T

~ space X.

be a she'_lf 0 f abelidll STOUpS on the paracompact

Let '2{

be an open covering of

X

such that

for 311

and all

r > O.

Then the natural homomorphism q

q

H (2(,7)-> H ,X.'1)

is an isomorphism for all

q.L O.

Example. Let

X

covering

0

f

X

by open set of ho1omorphy.

is ,usc an open set of holomorphy. J. P. Serre (lITheorem BII)

on an open set space)

'-{ = 'iui\

be a c"mp1ex manifold and

one has

U

be an open

Then any U 1 .•• 1 q 0

A theorem of

H. Cartan and

st~~~_~ha~ for any coherent she~f

'1='

of holomorphY (or more general on a Stein

Hr,U.

T)

= 0

if

r> O.

Thus coverings by

open set of holomorphy are Leray-coverings. c)

Acyclic resolutions.

on the paracompdct

space

Let

X.

an exact sequence of sheaves dO ( 2) o~T~1e -> '1"1

"T

be a sheaf of abelian g ro1Ps

By a resolution

~

~

"1 2

d

of

2

--,

the resolution is called acyclic if and all

i}' O.

'j

we mean

- 95-

A .. Andreotti

II

De Rham theorem. cech cohomololjY

0

t

(2)

1e an acyclic resolution ot

with values in ~

X

or

!!!M

1s naturally isomorphic

to the coholomogy of the complex l(X. ']0)

i.e.

Examples. 1.

!J-

A sheat ot abelian groups

open set

Uc

surjective. Hq(Xi,X) = 0

X

is called tlabby it tor every

the restriction map ; (X'JL) ....

For a tlabby sheat tor all

j;..

Every sheat

q> O.

r

is

(U.

one hos that

"j:'

ot abelian groups

can be considered as a subsheaf of the tlabby she at

C- 0

= sheaf

of germs of arbitrary (not necessarily continuous) sections of

1-.

"Y'-

Malting use of this fact one realizes that any she __f

admits a flabby (thus acyclic) resolution

2.

A sheaf;:'

is called

the restriction map

A c. X

For '. soft sheat.x

!!.2..!!

r

if for every closed subset

(X.;.)~ [(A • .}C..)

one h~s again

is surjective.

Hq(X,j) = 0

Therefore soft resolutions of a sheaf ~

for all q> O.

are acyclic.

Let ·X be a differentiable aantfold 6f dillension n. Let denote the she~f of germs of real-valued differentiable C~ ~ r f'r+l forms of tY~Je r (r = 0,1 .... ) and let d :c./t -.rr be the sheaf homomorphism induced by exterior kernel

0

f the homomorphism

of constant fun~tions

lR

d :

vf 0

differentl~tion.

~ ~1

Since for each

15 a soft sheaf we obtain a sequence

is the sheaf

r

The 0

the sh~af

f germs

uf

r

- 9.6-

A. Andreotti

One has

d. d=O.

Moreover the sequence is an exact sequence

(Poincaf~ lemma): We have 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 differential t>rms.

4. Let

Let

Jl r,a

X

be

a

a complex manifold of complex dimension

n.

C ~ dif-

denote the sheaf of germs of complex-valued

ferentiable forms o~ type (r, s) i.e. of those forms
Let J be the operator induced by exterior differentiation with respect to antiholomorphic coordinates. We obt,un in this way a sequence

o .......:'t r j; .fl..r,O ~ J{rtl ~.> in which ftr :; Ker

~J1r,n

~J/.r,o __ ,,1r,1~

->0

is the sheaf of germs of holo-

morphic forms of degree r. The sequence is a complex as ad and exact (Dolbeault lemma). We obtain in this way for any r..L. 0

=°

a soft resolution of the sheaf J'l. r.

In particul;u- for

r =

of t he structure sheaf

°

)l0.= (9

{}

of

= Kerfrcx.jO,k)

ImtL'(l!:.,Il°,k-l)

References:

[25J,

[3c].

X..

and we get a soft resolution We have thus

~r(x.flO,lt+l)f

j

C(X,tA°,k ) ~

for any

k

2.

O.

-97-

A. Andreotti 6.2

E.E. Levi problem. a)

Let

X

~

be

n-dimensional complex manifold.

To

clarity the exposition we will restrict our attention to open relatively compact subsets We can thus assume that

g1.ven a that

C' D =lX

6

I

X

As for the case holomorphy whi~h

I

D

~

function

D

of

with

df;i 0

we define .D Zo

<

aD D

the analytic tangent plane to ;) D

(theorem (1.4.2». open set

D

and such

to be an open set of

1s

Zoe

f

G

1,1 (D)

In chapter

open set of holomorphy,

In

the Levi-form at each point of 0 D

(0)

We have

on d D

there eXists an

1s not holomorphically extendable over

we have also proved that if

then

bound~y.

05,'1).

X = En

if for every

with a smooth

1s defined as follows.

X -) lR

b(x)<

X

restricted to

hus no negative eigenvalues

The Levi problem consists in asking if an

with a smooth boundary and relatively compact in

X

which verifies the condition (0) is a domain of holomor~hy. Vfuen no additional conditions are put on the nature of the m~ifold

X

the answer to this question is negative as the

following example of Grauert shows.

Let

Example.

n

2

X

and let

,.

2;

consider the real torus

= lR 2n / .,jJJ.

nfn~ X be the ndtural projection.

be the unit cube in

lR 2n.

so that

Let

X.

l'(Q)

Coneider the folloWing set Q

=

Lx •

Q

I '''11

<: £

.5

where

0

<. t.... <.

i

(1) It is easy to verify that if

D

has a smooth boundary aD

this can be also defined by a global equation.

- 9R-

A. Andreotti

and let D =t(Q£). Then D 1s an open subset of X, relatively compact (as X is compact) With a smooth boundary given by ?(\X

n ).

=!

l

Consider now an

so th~t

lR 2n

1R -linear surjective map A. : lR 2n.-,. 1°

inherits through A

the complex etructure of

En.

~2n is a group of tr~81ationBt these are holomorph1c

Since

for every.A

we consider.

Therefore

provided with a complex structure. structures is Levi-flat ~.

X,

and thus

Df

1s

D in any of these

Moreover

for every

is identically zero. In fact this is a local property is a local isomorphism it is enough to verify this • condition on T -1 (J D) = 1s Xl = ! £S. But now we can take for 'P the function 4> = Xl :;: E- which has identically zero Levi-form.

On

The set Y = '(lx l = is a real compact submanifold of of real dimension 2n-l. Let A -1 be given by

X

= (A,

A)

In particular IX

I

=

01

D

= Z Re

Xl

is the union of a cone - parameter family of complex

linear sUbspaces

t . lR.

Let

c

= t (0,

be a rc.J. vector in f Xl 0 J such are linearly independent ov~r the 2n Since the raatrix A has to s~tisfy the only condition c Z'· .. , c Zn )

th·s-..t the numbers rationals.

det

(A,

A) :# o. c

t ;

1R

<;

c 2t ... , c

we can choose

f ":1.1 zl

1"m:n~l)

+••• + ":1.n zn

A

so that

= O}.

Then for every

must be everywhere dense in

Y.

Moreover

A. Andreotti y = U t.m

(E n l ).

t

We can now show that 1/(0) = E. Indeed, let f ~ #(0) and let !r(zo)/ =lDalC If I. There exists Zo be so chosen in Y so th"t y ) a t < m such that Zo·T(En-l t 0

.

0

must be constant on ~(En-l) as to l n • But ,. (En-I) being dense in for must be constant on E o to It t must be constant on Y because f is continuous. The set of zeros of a non-constant holomorphic 'function on a complex connected manifold is of real dimension ~2n-2. Therefore f must be constant on D as Y has real dimension = 2n-l. Since #(0) E every holomorphic function is estendable at By the maximum principle

f

t

=

each boundary point of

D

and therefore

D

cannot be an open

set of holomorphy. b) If we wish to obtain an affirmative answer to the Leviproblem, one need to reinforce the Levi condition (*). For instance one could assume that the Levi form restricted to the analytic t:~gent ~lane is nowhere degenerate on aD. Then we h~ve the following Theorem (6.2.1). (H. Grauert. [26]). I f 0 satisfies condition (*) and if the Leviform restricted to the analytic tangent 'Tlane to ~D is non-degenerate (thus is positive definite everywhere on JD) then D 1s an open set of holomorphy. lM1en the conditions of this theorem are 8 :.. t~_sfied we say that

has a Gtrongly Levi-comvex boundary. The proof of this theorem will occupy the next section. divide the proof in several steps. 6.3

D

We

Proof of Grauert's theorem. step 1. A criterion for finiteness for cohomology. Let X be a complex manifold and let '}""" be the she,;,f of germs of holomorphic sections of so~e holomornhic vector bundle F.

-100-

A. Andreotli For every coordinate patch

U where

Flu

is trivial, we have

r(U, 1() ~ #(U)r

where r is the fiber dimension of F. Now #(U), and. therefore #(U)r has the structure of a Frechet space. r; can transpose this structure on r(U,'T). Banach'e open mapping theorem ensures us that this structure 1s independent of t he trivialization chosen for F over U. If V is 6pen and V G U the natural reetriction map r(U,'T) ~ (V,')')

U

r V

1s continuous, and, if

Vitali's theorem.

V

c.<::..

Ut

then

r UV 1s a compact map by

Similar considerations could be repeated for

any coherent sheaf even if it is not locally free.

Since for

the sake of the proof we need only to consider the locally free case the corresponding argument 1s omitted.

70=

Let

1Wi' 1<1

be a countable covering of

X

by coordin-

ate patchss, then the ~ech cocha1n groups

a B a countable product of Freehet spaces have the structure of a Frechet space. Since the restriction maps are continuous,

the coboundary operator

S

1s continuous.

Therefore in the complex

the spaces are Frechet spaces and the maY'S are continuous 1.e. it is a "topological complex of Frechet spaces".

Lemma i)

i1) : •."

(6.3.1). B

GG

~

A ,B

be open sets of

X such that

A

~.: Rq(A,~) ~Rq(B,Jr) is surjective ~

dimE

Rq ( B,'1 ) <.

cP.·

-101-

A. Andreotti ~.

U = lU~ ~IN ot

Choose a countable covering

(<()

A

by open

eets ot holomorphy contained in coordinate patches on which

B!l U # ~ tor only Unitely i tor 1 ~ i ~ k,

ie trivial.

we ma;{ assume that

sany

Sa;{

Ui' s.

#~~

B n Ui

In each of the open seta

set

U i

U , 1

11k,

we can take an open

such that ,

Ui ,,- C Ui'

B

It i~

Co

Ui

We now chooee a countable covering Z/ = l Vj~ B and such that

zr

ot jolomorphy of the set

lUi

ot the covering For each (~)

F

let

j'J by opsn sets is a refinement

n B i<.i$k'

-r(j)

be such that

Vj c U,(j)'

Ws now considsr the cochain groups q C ( Z( I q C (

".n

=

V, j'") =

n r(u io ' .. i

nf(Vj

j

(:r"t)j

j

0'"

q

T

for

A

l"r)

tor

B •.

t

q

0'"

and the rsstriction map detined as tollows:

I"J)

q

•

which is

= t:r(jo) ••• Jejq)

I

Vj j 0'" q

tor

It we consider the Frechet structure on the cochain groups, then "

is a compact

map since

Vj c

Uje j) <:~

\e j)

and since

thers ts only a tinite number ot groups [eu,(jo)."7ejq)' ~),

(y)

Consider the map <.J: zq(z.(,"1) ~ cq-leV',

T)

-> zq(Z'--, y)

deUned by GJe«~{3)

where

Z

denotes the space of cocycles.

By the Leray theorem

-102-

A. Andreotti

HCl ('<:(, "'J) = HCl(A, 'f) and HCl(ZI"-; '1') = HCl(B,'j::). By the assumption the map r AB induced by '1'* among these groups we have

is surjective; hence

is.


Since J

*

is compact, &

has a closed image of finite codimension by a theorem L. Schwartz. (1) step

Hence

'dim~ Hq(B,~)

~

Mayer-Victoris sequence.

2.

<

0

=tJ.:- T* f

oD •

X

be a paracompact

topological suace and let

Xl' X be two open subsets of X 2 X2 ' ~ 'T be a sheaf of abelian groups One has the following exact sequence

such that

!!!.

X.

X '" Xl

° ~ HO(X, <7f) ~ 1

~ H (X,

~.

'f)

'1') ~

HO(Xl' 'f) • HO(X , 2

-7'

1 1 H (Xl' ";) • H (X ' 2

~

oS )

° -?>~-.>

eO ~

1

Cl-'

~ H (Xl (\ X ,r) ~ •••

2

~

Choose' a flabby resolution of

on

X:

e,l_::- •••

One has the following exact sequence for every

°.. . r (X, eCl) ':" f (Xl' ~Cl) • where

d.

(a)

=~

rcX2' t:; q)

Cl 2 0:

~ r (Xl (1

X ' 2

e q)

..,.. 0

1

Taking the t1rect sum over all of complexes.

x2 ,1=)-?

HO(X () l

q

~

we obtain an exact sequence

0

Writing down the corresponding exact cohomology

sequence we get the desired result. Step ~

3. The bumps lemma. (6.3.2). ~ B be an open relatively compact'subset of

the complex manifold

X

with a smooth, strongly Levi-convex

boundarY'

(1) Let

E, F

ous linear surjection and let

map.

Then

(cf.

[281.)

u+k

u: E..;> F

be Frechet spaces, let

k: E

~

F

be a continu-

be a compact linear

has a closed image of finite

codimensio~,

-.103-

A. Andreotti

Z.(

The" there exist arbitrarlly fine finite coverings

2! dB o s: j s:

i11)

and for each ~

a sequence of open sets

Hr(U

i

n Bj."f)

!l!.!.!! ~.

j

with smooth strongly Levi-complex boundaries such that

t.

.!2!:

Bi - Bi _ l "'''' Ui

iV)

=~ U1 Hi"t

B •

~

=0

1"- i s , t

for any

and for all

and

i

r> 0

any coherent

j.

.

Without loss of generality, we may assume that on a

neighborhood euch that

B

U of fl

U

B

=ZZ ,

there is a

C" function

U , ~ (x) < 05.

4:u-"lR, tor every

(d4)x,s. 0

u. L(t)x > 0 for every x" U. The covering u= 2UJ lS1<.t will be chosen with sufficiently small x •

balls

U c U such that U I'l B in the coordinates 1 1 elementary' convex and thus a domain of holomorphy.

We now select ,,,uPPP1~cU1

C


functions

and such that

Pi';"

0

Choose e 1' .•• ' E: t successively with small so that the functions 'PI .. ~ -tlPl'

4>2

on

LP1(x) >0

L( ~ 1)x

>0

4>t

f

1" 1!o. t.

for all

E1 "> 0

= Q- clPl-<2P2.···.

have all their Levi forms

X.

0

coordinate U 1

is

with

x,,-aB.

and sufficiently

= 4>-€lPl-···-E t Pt

for all

x

€

U.

Define for

1 < 1

s t ,

B .. B. O If the G1 are chosen sufficiently small then

B will be smooth for 1 S 1 So t , 1 B (l Uj will be convex in the coordinates 1 an 0Den Bet of holomorphy.

~

0

f

Uj

and thus

-104-

A. Andreotti lie then have

(i) (ti)

(iii)

(iv)

B = BO " ';. c. ••• BO "-"- Bt

Co

Bt

since

~O > ~t

since

0 2 t> i

~ ••• .>

q, t •

on ilB

Bi-Bi _l c"- Ui since q i-1-1 =(i Pi hee support "-"- Ui r H (B () Uj , "f) = 0 for r > 0, all i, j and all i coherent sheaves "1' since B n Uj is an ope.n set of i holomorphy, by the theorem of H. Cartan and J.P. Serre.

step ~

(6.3.3).

complex manifold

~

X

B

a relatively compact open subset of the

with smooth stron51Y Levi-convex boundary.'

There exists an open set

(i) (11)

A

X such that

~

B <:e,.. A

for any coherent sheaf ~ restriction map Hr(A, 'f}

2!t X and any r.>- 0, !h! Hr(B,"f} is surjective.

Applying the previous lemma, taking A = B , it t sufficee to ahow that for r > 0 the restriction maps

~.

r H (Bi

,'1}

Hr(Bi_l''J}

are surjective tor each

Write B = B _ i l i sequence:

U(U

i

k,

1

n B } i

~

and

1 50. t. a~ply

the Mayer-Vi.toris

" Bi'~} = 0 = Hr(Bi _l n Ui ' ~}, t~e conclusion i follows from the exactness of the sequence. As a corollary of the above considerations we get the following

Since

Hr(U

Theorem

(6.,:',.4)

boundary. dim/t

~

B

"-Co

X

be open vlith stron51y Levi-conve>

Then for any locally free sheaf Hr(B, "Y) <.:

&'

for every

r? O.

~

2!l

X

we have

-105-

A. Andreotti

!2!!.

This theorem is true for

(6.3.1)

~y

coherent sheaf as lemma

can be carried over to this more general co)Se with the

same proof.

step 5. Solution of the Levi problem. ~

a divergent sequence

we can find

x,\<. D

converges to a boundary point

Choose a coordinate patch

zl' ••• Zn'

zero at

D~ U where

°

in

• (z)

U

If

6

= lz

around

L(h)}

a

with coordin&tes

C'" function with

d4 #

°

in

and

U

f(z) + L(4)zo (z) +0(llzI1 3)

f'

1n

U

°

on

with P

Talte '-

>

2.

1f =

°

0, Bup9

Os (\

D

p«

= fl.

Choose a

U, p(O)

so small that

d~l

#

> 0,

°

on

U

U.

A-DC'lXEU

4>l(X)


has a Bmootq strongly Levi-convex boundary and

holomorph1c on A n U. d1"~ Hl (A,8) = r < "'. Wr1te

z

such that

is suffic1ently small, then

A

~D.

We can write

and set .b 1 = ~ - ~P.

T_.n

U

zo';;'

!I(U).

C co function and Set

given

such that

Ul¢(z)<. 0S

is a

U.

= Re

zo'

~

U-,>]R

4>

L(t) >

with f

th~t

f .. H(D)

sup f(x) =.,. (condition D of (1.3.1)). v ~ 0 c.c.. X it is not restrictive to assume that the sequence

Since x

It 1s enoufh to show

f

1s

In particulor by theorem (6.3.4)

A = D U (A n U)

and apply the Mayer-V1etor1s sequence

for the sheaf,g : 0"7 HO(A.&) .... HO(D,I9)

.... Hl(A.tl ) ."

~

HO(A" U,e)

-,>

HO(D,' Ute)

§,.

-10·0-

A. Andreotti

1 ;lJ

Consider the sequence of functions

D

n U.

Since

~

for

= 1, 2, ••.•

in

These are holomorphic and linearly independent over dimE Hl(A,f:t) = r

E.

we can find constants c l , ••• , c r +l '" I:

not all zero such that 1

1

&(c l

1 + ••• + °r+l fr+l

) = O.

Therefore there must exist holomorphic functions • !f(D) and h E hI(A U~ such that d AOU

h

r+l !::

'::

1

+

°1

Hence

m 1s a meramorphic function in A, holomorphic in D r+l 1 wi th principal part at Zo For the function i

i

g

=

~

c

r

we thus have

I g(x)/ = "'.

Q..E.D

Remarks. 1. C

Let us call a manifold fP

q

function

(i)

X

O-oseudoconvex

X ..:y. lR and a compact Bet

for every

c

~

Bc = ~ x ~ X the Levi form

~

X

such that

l!l the set

I

¢

(x) " c

1s relatively compact in (11)

if there exist a K

L(4))x

S X

of 4>

at every point

x", X - K

is positive definite.

A O-pseudoconvex manifold for which we can take also called

K empty is

O-complete.

It can be shown

th,~t

theorem

(6.3.4)

extends to

O-pseudoconvex

manifolds as follows: Theorem

(6.3.5).

For a

~ £!!. X dimE Hr(X, 1') < ~

coherent sheaf

O-pseudoconvex

m~~ifold

one has

for all

r > O.

If moreover X is a-complete then Hr(X,r) = 0 for all r> O.

(cf. [2J l.

X.

and any

-101-

A. Andreotti The proof is obtained from theorem (6.3.4) by the Mittag-Leffler

procedure making use of the fact that for

restriction map HO(B

c

has a dense image.

2.

Let

,'1) -i'

HO(B

2

s';:P ;

c

<

~n

°

E

°

~

I ~ (z)

O~,

<

.p

Indeed, replacing

D

1 ~<

L(
constantS c C

!

ec

;>0.

Then

by an increasing convex function of ..

mey assume that there exists on

°

with a

function

Cd'

dop "" on a 0, L(4').>. on some neighborhood 0 f is an open set of holomorphy.

L(f) >

the

with the properties

o : tZ

Then if

z

2

D be an open relatively compact subset of

.p : En ~ Il

<. C

,t)

c

smooth boundary and such that there eXists a

..,.: l> +

1

0

and with

a function

°

<j>

0 we

0 -." lR with

:

outsi:le a compact set.

is a large positive constant the function J

has the property {'l" < constants

Zil2

on

O.

It follows then that

0

0,

e.G

is an increasing

~on

of open sets of holomorphy and thus a set of holomorphy. One could also apply the previous theorem of vanishing for cohomology to derive the f4Ct that

3.

Let

0 = ~ x eX'

.p (x)

<

Os

0

is an open set of holomorphy.

be a strongly Levi-convex

relatively compact subset of the complex manifold

0e

= ~ X"

If l.

xl

>

°

We have

Hl(O,~) is finite-dimen-

proved that for any coherent sheaf sional.

X.

is suffioiently small then

.p (x) < - e.? has the same property.

Moreover one

can show that the restriction map Hl(O, is an isomorphism, ( 2). #(0) = HO(O,S)

t)

-> Hl(0e. ,'T)

It follows in particular that

separates points on

O-Oe'

In fact let us consider the sheaf ~

of germs of holomorphic

functions vanishing at the points x, YEO-De" coherent sheaf and .e have the exact sequence

cy .~ o.

:J

is a

-108-

A. Andreotti

From this we 4educe the commutative diagram with exact rows:

nO(D,~) ~ \I nO(D

I

Since

4.

Let C

,e) ~

is an isomorphism, D

3.

be as in

=1(x,

y)

C;>

Clearly

Ex

0

D

~

=°

1m {3

thus

is surjective.

<£.

Consider the analytic set

= tty)

t(x)

D

the diagonal

A

ot

D

~

t •

for every

#(D)

.

Consider the closure

D.

=

S C-A of C-A. This is an analytic set. Therefore Ii n f>. is an analytic set. Any compact irreducible analytic subset ot

d of dimension since N(D)

~

1

must be contained in

sep·u-ates points in

D-Dc:

S

11

A.

the set

Moreover

4

S n

must be

compaet. Therefore:

if

D

is a strongly Levi-convex relatively compact open Bubset

or a complex manifold s~

A

X

there ext ts in

of dimension.Ll

a compact analytic

at each point such th,;.t anY irreduc-

ible compact analytic subset of

in

0

D

of dimension

~

1

is containec

A..

This argument 6.4.

CDn

be carried through for complex spaces.

Characterization of projective algebraic manifolds,

(D: J).

Kodaira's theorem

As an application of the previous considerations we give here a ?roof of theorem

(4.2~1)

of j{od9.ira.

We have already given the proof of the implications We will give here the ~)roo f

A = } B.

I t is

zO····· Zn U = zi ~ i

= -zi .... ,

enou,'~h

f the implic ations A. =? B =

to prove this for

be homogeneous coordin::ltes in

°

X

= Pn(~)' j'n(~)

=l

Yi-l

=

zi_2 zi

,

Yi

=

zi_l zi

,

Yi+l

=

Let

and let

zi+l •••• zi

zn

I

Yn

D.

C ='1 D

where we assume .,s coordin ..tes

Zo

Yl

0

A <

= zi

-109-

A" Andreotti

Consider the hermitian form

A. lR and ). > O. Then dsi z,l and by direct calculation we get

~

dS;

= (1

+}:yij)-2 i(l +

if -2

a.a

~Yij)

g

log

log (1 +

whsre

3zl y i j )

on

U i

(~YjdYj)(~YjdYj)S

;fdyjdY j -

defines a hermitian metric OD

del

zizi

~ dy jdY j

L (1 + i Y

Thus

i!.~

de1" z,\.

Its exterior

form is g1 ven by the 2-form

w.l.

z

iJ. ~a ~

f

log

o

=;;"';... = 0

By the way it is constructed d"';>.. therefore the hermitian metric Now in

H (:F (I:). 2Z) 2 n Pn(I:).

Taking for

where g

g

dw,\." 0

is a projectivs line

and 11(1:)

the projective line ).d d-

g . ~z2 • Therefore

= 2Zg

thus

is a Kahler metric (cf. (511).

de;'

= zn "O~ we get OJ;'/g" ~ 2

where

(l+yy)

f g '",l =J. Jg "'1'

But

Jg wl

>0

thue one can choose

.l. to make that period an integer (1) B

=>

C.

(~)

The assumption

that there exists on

B

X

can be replaced by the assumption

a Kahler metric whose exterior form has

rational periods as we can always multiply the metric by a positive integer to make the periods of the corresponjing exterior form integral.

Singular homology or cohomology based on differentiable sing. v ular simplices will be denoted by a au ffix "Stl, eech cohoilology by the suffix "v" and the cohomology of the de Rham complex of 4i fferenti ab1 e fgrms by the suffix

(1) We can take

1.

1

"ii

IIdR".

-l1P-

A. Andreotti

We then have the following commutative diagram of isomorphisms and inclusions:

c(

~)

2 SH (X,

2 sH (X, Ill)

1'5

d 1'5

J!2(X, ~) 1'1'5 It!2(X. 1:)

J!2(X, Ill)

d

If

l'

2....

is a basis of

cohomology ~lass of values on valuss on

sH 2 (X. l!Z) modulo torsion, a

sH2(X,~) is caracterizsd, via f3',

-y I' Y2.... Y I' '( 2' •••'

and it is in the image of are rational numbers.

j

by its

if its

A direct calcula-

tion shows that the isomorphism 13'00:'0 T is induced by the map which associates to a differential

2-form

to

the singular

cochain It follows that if a closed

2-form

w has rational periods i.s.

fYi"'~ III then it is repressnted in esch cohomology by a 2-cocycle ~Cijk~ c ijk ~ III (modulo coboundaries with values in 11:). In this arguaent I: can be replaced by lR everywhere.

with (~)

Define the sheaf?-f

monic functioDs on 07 I:

X

~

of germs of complex-valued plurihar-

by the exact sequence of sheaves

~ 1/7 0 (3(f. g) = f

&. "$

where <1.(c) = c • c and - g. denote the sheaf' of genae of c ~ complex-valued forms r,s(U). of type (r. s). Set for U open in X r(u.JI r,s)

Let ~ r,8

we have the following

=JI

-J 1.1-

A. Andreotti

&!!!.! !!

~

(6.4.1)

CD.

U

be a contractible domain of holomorp!lY

One has the exact aequence

° ~ r.Ii o,O(U) ~ Al,l(U) ~) l,2(U) !h!!:! f!22!.

d =
a

•

Jj2,l(U),

is the exterior differentiation.

Obviously the composition of any two consecutive maps in

the sequence is zero.

Since

U

is a domain of holomorphy the

resolution

is acyclic, lli

denoting the sheaf of germe of holomorphic

i-forms.

U is contractible the complex of sections on U

Since

°

°

d II: --?it (U) -->

~

is exact. Let fO,O • • O,o(U)

q

satisfies
°

U.

=°

i.s.

fO,l

is antiholomorphic.

with

fO,O _gO,O

fO,O = gO,O + hO'O,

We have then

and a holomorphic function on

afO,O = fO,l

Then

U.

= hO,O

anti-

thus is holomorphic in

a sum of· Hence

Moreover

:l.Il..

gO,O •. J'!)(1t)

afO,O = JgO,O,

Hence i.e.

d' n.'2 (11),'d •••

4'.

with" :ifO,O = 0.

fO,l = dgO,O

thus

holomorphic in a(fO,O_gO,o)

°

· n 1(11)

81

fO,O

U.

utiholomorphic <;

r (U ,'!if).

This

proves exactness at J?O'oCU).

Let Now

i , l • ./ll,l(U)

3g1 ,l

= 0,

such that

Hence

h 2 ,O

°

and

h1,O _k 1 ,O=3kO,O

gl,l

=

3h l ,O

Al,l(U) •

=ahl,o, thus

= dkl,O

ahl,O. dkl,O

di,l

thus there exists a

gl,l

Now ~ ahl,O =

with

=

because

(1,0) U

ahl,O = h 2 ,O

with

kl,O

J(hl,o_kl,O) for Bome

7(h l ,o_k1 ,o)

=°

l.e . . . i , l form

on

U

is holomorphic and closed. U.

Thue

Consequently

C""' function on

= 3JkO,0.

hl,O

is a domain of holomorphy.

holomorphic in

= 0.

= ° =5 i , l

U.

Now

This shows exactness at

-112-

A. Andreotti

Let

=i

(,)

1. g

dz A dz p be the exterior torm of the '" 7: d Kahler metric. Let 'Z( = ~ Ui~ be an open covering of X by coordinate balle. Since dw = 0 by the previous lemma we can writ. (y)

"" Iu i (g.. i)

Since

= ';Wi

is herait1an.

and therefore

tJ

Iu = t i

'"

is real. so that",

Iu =;;

~ a(W - Wi). i

We can thus assume

= 21

= Pij

aWi' Wi

to

be purely iaaginary. U n Uj i

Then on

Wi - Wj where

n Uj

Pij , U i

~

1m Pij

- P ij •

C ie holomorphic and deterained up to

addition by a real constant. On

Ui (\ Uj n Uk

•• must have

= c ijk ..

lR.

c ijk - c ljk + c lik - c Uj

=0

Pij + P jk + Pki

Moreover

Hence ~ Cijk~

represents an element ot

l. Cijk~

110. the cocycle

is the

wlu

i

~Wi - ~Wj

Ui

vH 2 (Z{,

n

Uj n Uk () Ul •

1Il).

~ech represent ...tive of the

de Rhea cohomology class represented by Indeed we have

on

4l

(up to a sign).

= d(iwi ) •

= -dP ij

• -dPij •

-(P ij + P jk + Pki )

=-

c ijk '

which is the string of homomorphisms that make explicit the isomorphism of the d. Rhea theorem: By the assumption that

~

has integral periods, by

derive that we can assume that the real numbers integers, (1)

P ij

+ P jk

+ P

ki

• c ijk

E ~

c ijk

(a.)

are

we

-113-

A. Andreotti 2~iPij

e so that ~j Bocause of (1) we get

(0 l Let gij holomorphic.

~j gjk = gik

Therefore the collection

on

l

Ui

is

Uj

A

Uk •

A

~j ~ is a set

0

f transition functions

F 2; X.

tor a holomorph1c line bundle Now

where k

=n

Let

-1

~ II:

(Ui l -::: Ui

and let

the fiber coordinate.

and

is a well-defined function on

z

i

= e

-2"iW + i : U -. lR • i

be the base coordinate

One verifies that the

F

funct'~n

as on

ki(zl/SiI2 = k j (zl/SjI2 • Indeed

j...( z, E, l

Consider the tube of radius Br

For every

=~(z,l)l

r? a

to

rl

r

around the O-section of

Br is relatively compact as the base space Moreover the Levi-form of log ;X(z. C;) restricted

3Br

J.;; log f
z..

i

F:

:x(z,;l< rS'

is compact. to the analytic tangent space to

X

r.

defined a hermitian metric on the fibers of

dZ" +

coordinates ·along

1

1';i th,~t

dSt

JB

r

reduces to 2/TL g _ dz dZ",. Q3 d.

is given by as

=0 ,

plane.

the analytic tangent plane to

He'JOe the Levi-form restricted to

B r

is strictly positive definite.

rhis shows th:.i.t the complex space

F

consider inste,?.d of the bundle

the bundle

F

is

O-pseudoconvex.

F-

1

If we

which is

-114-

A. Andreotti given by traneition functions fg~~f, then b, a simple verifica-1 tion we Bee that the space F 18 O-pseudoconcave. C •

_> Let

D.

J

bs a sheaf of ideals on

X,

for instance the sheaf of

germs of ho10morphic functions vanishing on some finite set S c: X,

Consider on

F

the sheaf

'3' ="* J • <9 Xt9 F'

In

the previous instance, this is the sheaf of germs of ho10morphic functions on F vanishing on the finite set of fibers n -l(S). Every

f c.

r (Vi' ':J)

has a power series expansion

Set

so

,1)

which i6 independent of the This gives a filtration of .r (V i choice of the tiber coor.i1nate ~ i' Similarly one filters the groups [(Vi i ,"1), therefore

o

we get a filtr~t1on of the eech complex

q

C*

(r,

~1)

Which is compatible with the coboundaxy operator. q '" f(d.) (z) I f If E c (V;1) and f, =~q~ i O" .i q one verifies that, for given a ,

where

F-

denotes the

~

the bundle

(2)

0

-T

F-

d

she~f

of germs of holomorphic sections of

Therefore we get a (split) exact sequence

C~+l U Y , '1-) ~ c~u~,1l

. ,. Cq ( 1.(, J

II !-k).,> 0

-.1)5-

A. Andreotti

* .Since the filtration of C (11-', oj-) is compactible With the coboundary operator, we obtain a filtration of cohomology q( 13HO v , =-') j while from (2)

'?~ = Hq( (/,

•.-,-)

j-

::>

Hlq( <.I}-,.J'r)

~

J ..-') H2q( Z', J

::>

•••

we get

Therefore

1

k=O

=

From theorem (6.3.5), if q) 0, Hq(?.I', j) Hq(F, .y) is finite-dimensional as 2: is a Leray covering of F. Thus if q > 0 in (3) the left hand side is finite-dimensional. The same must be true for the right hand side of (3), therefore

.!2.!:

q ,> 0

the~

k. ~ k ' O

exist

Hq(X, .~

ko

> 0

e

such that i f

J-k)

= O.

Now we apply this result to the exact sequence F-k ~ F-k -,) Q ~ 0 -v,r where Q) is the sheaf of germs of holomorphic functions vanishing at two distinct points or vanishing of second order at a given point. Writing down the corresponding cohomology sequence fP U one realizes that the ring ~!? (X, F-1 ) = k-o r(X, F-k )

o ..,. tJ e

1AI

separates points and gives local coordinates everywhere on

X.

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, thua avoiding the use of the Mittag-Leffler arg~ent 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 holomorphy.

Let X be a complex manifold, let D be an open relatively compact subset of X with a smooth boundary <md let d be an integer, d ~ O. Let ~ be a locally free sheaf on X, for instance ":( = 8 or ":f = Ctd. \lie repeat the considerations of the previous chapter replacing the ~pace #(D) = HO(D,8') by the cohomology group Hd(D,~). Let Zo 6 oD, an element E, ~ Hd(D, '1) is said to be extendable over Zo if there exists a neighborhood V of Zo in

Ad"",

and an element E, co H (D U V,

X

Rem ark I

d > 0

if

~

an element

D =

such that

the necessary and sufficient condition for

Hd(D t

E

~)

to be extendable over

that there eXist" a neighborhood

Zo

2! Zo !!!.

\II

aD

E:

!!.

such that

X

~I\IInD • O. In tact, if

C,

hood

Zo

\II

of

is extendable over

IW= 0

Then ~

W c:. V,

zo'

we can find a neighbor-

X which is an open set of holomorphy with

in

I

thus I; W,D " O.

Conversely suppose there txists a neighborhood

W of

Zo

such

that ~ \ W n D " O. holomorphy. By the Mayer-V1etor1s sequence we get the exact We may assume that

W is an open Bet of

sequence :

'f)

Hd(D v W, Since ., in

s

,

Hd(W,~)

-;>

= 0,

Hd(D" \II, "J)

E Hd (D (1

w, "r)

We say that

the sheaf

'J'

Hd(D,

'en

~ Hd ( W,

r) ~ Hd(D

(I

W, 'J').

because d> 0, and since the image of is zero, i t follows that there eXists

I

such that £, D = \'. A

D

1s a d-open Bet of holomoruhY with respect to

if for every

which 1s not extendable over

zo.
E

Hd(D, "f)

- 117-

A. Andreotti

For

d = 0

the boundary of

we realized D

has to have a

that result we are leaj to

a- = ~.x and we assume th.t on

a-

S

<;

p_'~'.1'ticular

~nsider

a: n

Let U be an open set in function on U. We set

S

(Theorem (1.4.2) of E.E. Levi)

u

= 1x E

(x) < 0 U

In view of

the following situation:

f :

and let

If

shape.

that

U .... lR

be a

C

~

5•

I f (x)

= 0

S

d

~ ,I

0,

so that

is a smooth hypersurface, which constitutes the boundary of

in U. Let us consider the Levi-form of f restricted to the analytic tangent plane to at every point Zo € S, and let us C$sume z • that at a point S it has p positive and q negative o eigenvalues (p+q .< n-l). Then one can prove the following

Theorem (7.1.1). There exists a fundamental sequence of neighborhoods u. of Zo ~ U such that. for !nY locally free ~:r on U we have s

Rs(a-."

= 0

it

> n-p-l

or { 0<

B

< q.

These neighborhoods can be chosen to be open sets of holomorphY.

r

(cf. 2] ). The proof of this theorem is rather tedious and will be omitted. As a corollary we get the analogue of E.E. Levils theorem: Theorem (7.1.2). (E.E. Levi). i! D is a d-open set of holomorphY for a locally free sheaf 'i then at any point

r (0)

~D

0_'"

of negative eigenvalues of

the number of positive eigenvalues of

L(~)z IT o

is

< d

1s

< n-d-l.

Zo

LW z o IT Zo

-118-

A. Anrtreotti

One can then formulate the analogue of LeVi-problem for Hd(D. The example of Grauert shows that condition (*) is not sufficient to 'ensure that it is solvable as in the example any JIXlint "0 ~ olD has a fundamental sequence of neighborhoods U v such that D (l V" is an open set 0 f holomorphy. We have thus to reinforce condition (*) assuming for instance that

:r).

L(~)z

o

IT

for every

zo" aD

ie non-degenerate.

An open set

Zo

D of this sort will be called strictly LeVi d-coavex. has the following

Then one

Theorem (7.1.3). (Grauert)~. !! D is strictly LeVi d-convex 1!!!!!l D is a d-open set of holomorphY for any locally free ~

T

2!! X.

7.2.

Proof of theorem ( Grauert) d • (cf. ( 8]). The proof follows the same p~ttern given for d = o. It is based on the following remarks. (" ) 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)

HS
=

,'.f>

z

0,

if

s

>

d

=

since (n-p-l) n-(n-d-l)-l d. Here ":F is any loc ally free sheaf (although this part of the vanishing theorem holds for any coherent sheo.! Y. on Xl. Koreo\~er one has to real.ize that this stutement is stable by small defo~ations of the boundary, precisely

given any sufficiently small neighborhood which is an open set

0

p : U 7' lR with P L o. p(zo) > 0, Co > 0 such th<\t if we set 4'1 = 0< <. < £0'

U/S) =

<:~

supp p

zo.

of

UV

f holomorphy a..'1d givan any

C

U\J'

t - 'pwi th

~x

Eo

UJ4 1 (X)

<.

O}

then we still have for

l5

> d •

-:F-

cD

in X

function

we can find

-'119-

A. Andreotti

(~) This enables us to repeat the proof of the bumpe lemma with condition iv) replaced by iv) ar(U () Bj,j:') • 0 tor any i, j, any loc ally free i

sheaf

"J'

and any

r <. d ..

Then the analogue of lemma (6.3.3)

and the criterion of finite-

ness gives

Theorse (7.2.1). Let B open relatively compact in X with a smooth boundary on which the Levi-form restricted to the analytic tansent apace has at least n-d-l p08itive eigenvalues. Then for any locally free (or coherent) sheat "3" 2!! X we have

Cy)

We now have to construct for every d

Z

E:.

0

d>D

a sequence

of coho.ology clasees Sv Eo H (IV (l D,5') defined in a fixed region If n D where W is a neighborhood of Zo in X, such that i) they are linearly independent over ~ every non-tirvial linear combination of them is not ii) extendable over Zoe Ws postpone the proof of this fact to the end. We remark that to achieve this purpose we will make use of the assumptions that ~ is locally free and that the Levi form on the analytic tangent plane at the boundary of D is non-degenerate. (8) Let us suppose for the moment that point (~) has been settled. Then the proof proceeds as in the case d o. Let A be constructed as in the solution of (Levi-problem)o making a bump on D to swallow the point Zo e "D ,

=

A ,. D U (A t'I U).

We then apply Mayer-Vietoris :.>Hd(A,"f) -,load(D,g) • Hd(A n U,"f)7 ad(D

~Hd+l(A,

or) '" ... ~

n u,1")~

_IZ0-

Hd(D n V, ';')

C, ..

We consider the classes d+l nLet r = dim; H (At~)

C1 ' ••• t Cr +l '" ;,

We can find constants

constructed in (')').

(which is finite by theorem (7.2.1».

6(Cl~ + .,. + c r +l

not all zero, such that

Sr+l) = O.

Therefore there must exist cohomology classes and

h

•

UU

u, 'T)

n

Hd(A

ho. Hd(D,

~)

such that

r+l ho =

f=1

ci

Si

+ h AQU

g = ho • Hd(D, 'F),

We set

neighborhood hAnuJw = 0

W of

Zo

thus

D () U.

then in any sufficiently small

which is an open set of holomorphy r+l

holw=i~l is not extendable over

Zo

cisil w ' because of property

ii)

(1).

of

I t remains to prove statement ('Y) :

(e)

Since in

on

'J'

is locally free, in a small neighborhood

X,;: -::. (} p

for some

statement for the sheaf

':t

p.

W

of Zo

Thus i t is enough to prove the

= [}.

The proof can be based on

the following ~

(7.2.2).

forms of type

;n _

Consider in

1. 0]

j 1:(-1)

(n ~ 2)

the differential

,

(0, n-l)

"'j+l Zj

_dl+l dZ A l

_dj+l . . . ,dZ A j

_
. . . ,dZ

'I'd = - - - - - - - - - - - - - - - - - 6 j\1

(rZj for set

lin.

"U" in

TlI'ese forms are

;n-lo}

n l !1J. '= H - (U,19).

If

"j+l

Zj

)n a-closed therefore for any open

they represent cohomology classes U

contains the closed half sphere

-121-

A. Andreotti ~.

The fact that the forms

'fd

are a-closed can be

verified by direct computation. ~

'1

Let

[

cd

'1'&

be a !1ni te linear combination of those cd e 11:.

forms with coefficients for some

C&

form f'

of type

(0, n_2)
If

~.

En

we have the generalized Cauchy formula: 5

dZ1A ••• Adzn •

Jf

W A

'1

<1 > 0

Let

~

=a ,.,

'1

Let us assume that U.

t

then for any holomorph1c function

!2"ij;

on

where

n-

be so small that the part

56. fl ' Zj 1 2 • t.

Re Zl.l. -

<15

of

5

is contained in

We have

U.

5

6

5

f",A'1

• 5

d

J fOlA a,., = 5al

d(f",1\ IJ)

~5J

fWA/'.

Thus

ldl 'il CadZ"

!

(0) ~,}5

!

f wA f'

+ 5

d

Let

A

=t j=l ~ Iz j l 2 < "

+ Ssn(Re zl <:

B

=~ ~

o'/4}.

j=l

Izjl Z <

sf f

-d

W 1\

1

-dI2} and

A and B are Runge domains in ~n. Moreoyer Re( zl + dlZ) is < 0 on A and > 0 on B. Therefore A \I B is ~ao a Runge domain in ;n. Let K be ~ oompact subset of Then

B

containing the origin in the interior and let

omorph1c function on

B.

holomorphic functions in f" 'of g as any

K co B g

uniformly on and

We can find a aequence n II: such that K.

f •.~ 0

J Sci V (S - 56) ~

we must have

g

uniformly on "Sci

(S - Sd) "A.

be any hol-

'if \.l f U

of

(S- Sd)

Therefore for

- t2~-

.8 . .8n dreo.Ui

(Zryjf £

=

o.

are zero.

Hence all

(n-lll Let us now cunsider the case local coordinates at

lIquation

h = 0

zo'

= n-l.

d

aD

near

Zo

and we may aseume that

By suitable choice of is given by a local Zo

ie at the origin

and that

where of

1

D.

'\jZi Zj If

E.

>0

{! I zjlZ =

>0

and where

h.> 0

corresponds to the side

is sUfficiently small then

6.$

{h

the closed half sphere

> o}

p:.

cI with 0 < ci <.~ contains = ci5 (\ fRe zl~ O. By the

for any Z

Izjl

lemma above, in the coordinate ball

W of radius

e,

centered

at the origin the forms 'Va represent linearly independent d cobomology c4a&ses of H (W,6'), not extendable over Z00 In the general case Zo

can be written as

d So n-l b = 0

tbe local equation of

aD

near

with

where

is positive definite and where the side of the side If

h

D corresponds to

> O.

W is a sufficiently small polycilindrical neighborhood of

Zo = 0

in those coordinates t then

W 11

1Zl

• ••• • Zd+l = 0 S 11

II •

f zo1

-123-

A. Andreott i

CODsider the torms ot the lemma in the coordinates zl ••••• Zd+l'

aa forms defined in W. These have singularities outside of D. Moreovsr their clasees again satiRfy property ii) of (y) .. this is true for their restrictions to the s)ace

f Zd+l

= ••• • Zn = o} •

(1).

This achievss the proof of statement

7.3 Finiteness theorems. In theorem (7.l.1) ws emphasized only the positive eigenvalues of ths Levi furm. With ths sams procedure one can prove the following more precise theorem, ct.rl), Theorem (7.3.1). &!i D be a relatively compaot open subset ot the complex manifold X with a smooth. strictly d-Lsvi-convex boundar:c. Then tor any locally tree shear '3" ~ X we have diall: Hr(D,S) < ~

it

r tI d.

The group Hd(D,~) not coneidered in ths ~heorsm, is actually int1n1te - dimsnsional and moreover has a natural structure of a Fr'chet space. Theor.. (7.3.1) can be considered as ths "intersection" ot the tollowing general theorems of finiteness. A complex manifold X is called q~pseudoconvex if there exists on X a C" function X ~ lR and a compact set K .uch that

+,

i)

the sets

Be ·fx. X are relatively compact in X

I

c1

f(X)<

for every

on X-K the Levi-form L(.) n-q positive eigenvalue•• If we can~oose K. ~ the manitold

of

11)

Theore.

!h!!! Of'

(7.3.l). ~

X

diall:

11 X

c ' lR />

has at least

X is callsd

g-complete.

ie g-pseudoconvex then for any coherent

we have Hr (X,'1) <


it

Moreover if X is a-complete then for

r

> q. r> q

Hr(X, '3")

• o.

-124-

A. Andreotti Analosously we have Theore. (7.3.3). l! X is g-pseudoconcave, for any locallY free eheaf '1' 2!!. X we have 1.f

r < n-q-l

Due to the maximua principle there cannot be an analogue of "q-coaplete ll manifolds for the pseudoconcave case.

Theorea (7.3.3) is only true for locally free sheaves. Actually the real analogue of theorem (7.3.2) is that "for a g-pssudoconcave manifold and any coherent sheaf ~ 2.!!. X we have

.!2!: !h.!!:!.

ark

r)

q+l •

denotes cohomology nth compact support•• It

Theorem (7.3.3) is obtained from the one quoted above by dUality and thus, if stated for any coherent sheaf. the number n • d1ac X has to be replaced by the depth 0 f the sheaf '.t • As reference ses [2 J. [7].

7.4. a)

Applications to projective all5ebraic manifolds (c f. We revert to the case d = 0 to begin with.

[9]).

+ Let 0 be a complex manifold. Let CO(D) denote the space of positive O-d1.mensional cycles i.e. the free abelian mono1.d generated by the points of D. We have

Pi' oj = 0

n o(l) n o(2)v

where D(k) denotes the k-fold symmetric product of O. For each k ~ 1 D(k) is the quotient of the cartesian product D" ••• xO··(k-times) by the action of the symmetric group on k letters permuting the factors of this c>rtesian product in all possible ways. As such o(k), and therefore C~(O). carries a natural structure of a complex space.

- 125-

A. Andreotti

Moreover we have a natural map given by for Note that the image of Po is a set of holomorph1c functions on C~(D) which are additive with respect to the monoid

structure of

C~(D).

Proposition (7.1,.1).

C~(D)

The space

(holomorphicallY complete)

iff

D

is holomorphically convex

is holomorphically convex

(holomorphically complete).

~.

C~(D)

If

is holornorphically convex (complete) then

C~(D)

as a connected component of If

D

! cv~ c.

fact let c

=l

niPi

C~(D)

+ CO(D)

is holomorphically convex then also +

CO(D)

deg(c)

be a divergent sequence.

= 2n i •

is.

In

Let for every

Then :leg is a holomorphic function on

telling us on which symmetric product

cycles stay.

D

has the same property.

~

If deg (c y )

$

O(k)

the considered

there is nothing to prove.

If

1 CiS

deg (c v ) s: k, then we c an extract a d1 vergent sequer.c e contained in a given product D(l) (with 1 S k). v l Now D = D
1

D.

fT(svl)

= C \If.

The sequence

such that

su~ I g(s .) I =~.

,

~

polynomial in t 'if {.I (t - g(yx» l The coefficients

at (x)

t

l'

.

must be diver-

+ ~ (x) t

1'-1

Consicler the

+ ... + ~! (x).

are holomorphic functions which

ll-lnvariant thus for each i function f. < IiCO(l)) with ~

f Bv,S

Therefore there eXists a holomorphic function

there eXists a

rr • f 1

: : : ai'

~e

holo~orphic

1 So i

~

I!.

On the

-12'6-

A. A n dI:.e.o1.t i

sequence

at least one of these functions

BV.

~t

must h.-ive

unbounded atsolute value. Therefore sup /f,(s 1)/ =~. ~ v If D is Stein then also C~(D) is Stein. This follows at once from the fact that on a Stein manifold we can prescribe the

values of a holomorphic function on any finite set. b)

Suppose now that

D

is an open subset of a prcjective

algebraic manifold X of dill~ X = n. Given em integer d, + we can consider the space Cd(D) of positive compact d-dimensional cycles of D i.e. the free ~belian monoid generated by irreducible compact analytic subsets of D of dimensio~

05 d S n,

1 D1

I{

E-

almost all zero,

Ai

compact

irreducible analytic subset of D It is known that

of

dim~

Ai = d

"I

C;(D) carries a complex structure of a weakly

normal complex space.

Examples. 1.'

D = X ='Pn(~)' C+ n-l

2.

d = n-l.

(I' n(~» =

X = Pn 0:)

C:_l(6)

=

I

J;, (~)

11 lPn +2

D = ~(~) - lr oint }

a:n

L!

a:(n~2)_1

(~)

( 2 )-1

U P + n

(~) L!

( 33 )-1

...

then

a:(n~3)-1 u •••

L!

d

d

Let us consider the cohomology group H (D, 0. ). Every cohomology class E, on i t can be represented by a a-closed differentia:! form 4> d,d of type (d, d) modulo a of forms of type (d, d-l). + Given c • Cd(D) we can consider (by virtue of a theorem of Lelong) the integral E,(c) = ~ d,d. c

f

One verifies that this integral is independsnt of the choics of the representative ~ 0 f the class I).

-127-

A. Andreotti

As analogue of proposition (7.4.1) Theorem

(7.4.2).

a)

The function

<,(c)

for

c

we then have the follawiQg

C~(D)

a variable on

k..;L

holomorphic function so that we get a linear map

b) l! D is strictly d-Levi-convex then for any divergent seguence 1C) c C~(D) there eXists a class 0; f Hd(D. n d ) such that =

In particular

C~(D)

dJ.

is holomorphically convex.

l! D in addition is d-complete then given c l ' c 2

c)

C~(D)!!!!!. c l I- c 2 there eXists a class such that I;(c l ) I- ~(c2). In particular

+ Cd(D)

s"

~

Hd(D • .Q.d)

is holomorDhically complete. (cf. 19]).

We limit ourself to a very brief skotch of the proof

First one shows that the :unction ~(c) is a continuous function of c (this is the most difficult part). (C) Since C~(D) is weakly normal it is enough to show that (d)

~(c)

is holomorphic at

non-singul~'

p0ints of

essentially done via Morera's theorem.

is an analytic rr

I-dimensional disc in

+

Cd(D).

This is

Loosely speaking, if!:J +

Cd(D).

if

-y

= dil and if

+

~ Cd(D) is the fibered space of d-dimensional cycles over + their parameter space Cd(D). we have

F

t
J !c(t»)( ~ f1 dt. :=n-1(y/(

~·d,d A dt

~

1 h,~'dt = 0 1/'- (A)

=

1 11-(11)

f d(

1\

dt) •

-128-

A. Andreotti where

t

b

denotes the variable on

c(t) = ~-l(t).

and

n -leA)

last equality sign is for reason of degree as

d+l

The

is a

dimensional space. If

(1')

the class

e,

as we did in

(8)

is strictly d-Levi convex then one constructs

D

using the local non-extendable cohomology classes (Lev1.-problem)d

The separation of points on

+

Cd(D)

when

D

is

d-complete follows by an exact sequence argument from the d+l ~ d J -closed (d,d) H (D.'L) and the eXistence of a

vanishing of

form whieh has 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 0)

Consider. as an application, the following situation. Let

f : X

~

Y

be

~

proper holomorphic map between the projective

algebraic manifold Let

X).

X

onto the projective algebraic variety

A be a compact irreducible analytic subset of X

a compact irreducible analytic subset of f-l(B) = A Set

a

and

= dimll:

f: X - A

A,

b

In fact

f

The cycles

'" n~lnB

Y. B

such that

~ Y - B is an isomorphism.

= dimll:

B

and assume that

Then there eXists a neighborhood is holomorphically

Y

and

W of

>

a

C~(;d

b.

C~(X) which

in

co~vex.

induces a map f B. 2B, 3B, •••

b

+ + (Cb(X) - Cb(A»

+

are isol~ted points of

has a holomorphically convex neighborhood

One then verifies th
+

(Cb(Y) - Cb(B».

-of

C~(Y) U

in

W = C~U') U f-~(U) has the reqUired

property. Now the eXistence of a neighborhooJ

W of

th~lt

garantied by the eXistence of a neighborhood

which 1s strictly

sort is certainly

N(A)

of

A

in

b-convex, by Virtue of the previous theorem.

X

-129-

A. Andreotti Chapter

VIII.

Duality theorems on complex manifolds.

8.1 Preliminaries. a)

Ii

each homology vdth value in a cosheaf.

As in the case of poincar~ duality on topological manifolds, it is better unJerstood as a duality between cohomology and homology, so in the case of complex manifolds duality shoul~ be a pairing between cohomology and homology • For this reason we develo9 here • each homology theory as a preparation for duality theorems (c f. [7 J which we follow in this exposi tion). Precosheaves (cf. [18]). A precosheaf on s topological space X is a covariant functor from the category of open sets U C X to abelian groups, i.e. f9r every open set U c X an abelian group 2J (U) is given and i f V c U are open a homomorphism i

V U

is given, such that if

.0

(V) -.

We V c. U

2J

(U)

are open subsets of

X

we

have,

A precosheaf 1s called a coshea! and every open covering sequence is exact.

where

00

7-(

:=

1ui 1 i , I

if for every open set

of Q,

J). C

the following

is defined by

and

Example.

Set

support in

U.

elusions

.§(U) = continuous functions on De fine the "extension maps"

S(V) c

also a coshea!.

~(U).

U with compact

as the natural in-

We obtain in this way a precosheat and

X

-130-

A, Andreotti

v

Given a prec.osheaf, ~

Cach homology.

U. = t Uil i'I

an open covering

0

f

X,

= ~ 0(U), i VU S and one ~efines the groups

cS (U1

•• '1)

0-·· q

and the homomorphisms

by

for We have

aq-l

0

=0

q

g

={". i I ~o··· q~

for all

q;>. 1

"-

C

q

(U,S).'

.

and we put <J -1

= o.

We thus get a complex with differential operator of degree

-1

and an augmentation eX : C ( o

We define

Hq(U, $)

U, 3) ..,.

the q-th homology group of this complex,

(11, $) =

H

q

If

U =~

V j"l,jOJ

<9(X).

Ker O 1m

9- 1

aq

is a refinement of

the re finement function

,..: J ..., I

U,

to each choice of

there is ;l.Ssociated a

homomorphism

t. : Cq<"V;

S)

~ C

q (1.(,;;J)

for every

which is compatible with the differential operator.

q..>. O. Thus

~.

induces a homomorphism 1"

V

u:

Hq(V', ,,)') -r H q
which, as one verifies, is independent from the choice of the refinement function.

One can then define ~

H (X, CJ) = q

lim ~

U

- J 31-

A. Andreotti

Remark.

We may not wish to use all open subsets of

11(.

the open sets of a particular class vided

7Jr A i \

J

cosheaf :'

J (u)

= ...

rJ

the obvious way.

oS" l ~ (U),

U

iV

is called flabby

r k(U~ L) e5 (U)

and set

X.

On every open set

of sections of =rk(U,

r).

r.,

Define

compacti\

We thus obtain a flabby cosheaf on

i Vu}..

if

U.

be a Boft sheaf on

consider the group

ly supported in

u?

(U),

e5 (X)

I

For example let

in

X,

For every cosheaf of this sort and for

every locally finite covering

U.

Hq(U,S)=0 ~.

This can be done pro-

X.

is injective for all open sets

Uc X

but only

is stable by finite intersections and contains arbi-

trarily fine coverings of b)

X

Define the sheaf 2.

u

= ~ U i""f 101 if

of

X,

we have

q2.l.

by 10·~·1q

= as the sheaf which has for a point

x

the direct sum of the fibers of the sheaf

at the same point. We have an exact sequence of sheaves

... ~ r q "~q where

;E q-l

"'1"0 L.... " q-l -;-. •• ~ 2"0

0

-132-

A. Andreotti

The exactness follows homology of that complex e ffic1ent 1n of x. Now the sheaves 1. q are supports in the sequence functor r k 1s exact on the sequence ••• .., Cq (U,

.s ) ..,

Cq-l ('U

from the fact that at each point x the 1s the homology of a simplex with cosoft.

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

, .$) ~ •••

-Y Co ('l(,

.$ )

-'J'

$ (X) -;>'

0

and this proves our contention.

One can prove that any flabby cosheaf 1s of the sort described above. We have for cosheaves the corresponding statement to the Leray theorem: Let

-U

= fUi ~

be a locally finite covering of

i'1

X

~

!.!! <:S

be a cosheaf on X with the following property for every open covering if = 1Vj 5j "J we have

H (U "

q

1

V,$)

=

Hq(U i

(I

Uj

n

LJ-,~)

= ••• = 0

for every q> 0

then the natural homomorphism

Hq(1{,J)~ Hq(X,S) 1s an isomorphism. A covering 'LA of this sort will be called a Leray covering for the cosheaf $. As a consequence of the Leray theorem, we mention the following fact:

~

.3', 3 ,e5"

21 = Z Ui~

i~I

be cosheaves on X be a Leray covering for J'

d~

let an d

,S- ".

homomornhisms U -- Ui ••• i q O k U are given such that the sequence kU hU oJ "(u) 7' 0 J(U) ~ 0 -:)S'(U) '7

Sunpose that for each open set hu '

and

- 133-

A. Andreotti

is exact, and compatible with the extension maps. an exact homology sequence .y

HI (X,

S')

h.

HI (X,

--?

k.,

S) _

h.

H1 (X,

S II)

Then one has

d

~

k.

Ho(X, $!) -;;:> HO(X,:5) - 7 Ho(X, c0") ~O Note that exact sequences do not commute in general 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 .ani1'olds. The following lemma is a consequence of the Hahn - Banach theorem (cf. [47J). -7

~

(8.2,1),

~

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

tu __

B'

tv

--

C'

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

If

v

~.

t Ker u Im tv

.-:;>

Hom cont

(~ Im u

is a topological homomorphism then 0 Ker t u t Let (3 "" f3 = b' + v(C!) Im tv

£) •

is an isomorphism. with

b'

0

u

= O.

For every k E. ~;r ~ k = b + u(A) with v(b) = 0, we define p(k) = b'(b). This does not depend on the choice of the representative b' and b and thus it defines a linear map Ker v -;> £. For every £ > 0 the set t b Eo BI lb' (b) I <£ 1 d((3) YiiiU is open in B and u(A) saturated. Therefore 6(~) is continuous. Conversely given A ~ Hom cont (~ Im u' an i t lifts to a continuous linear map ), I Ker v ~ a:. By Hahn - Banach we can /\ B ~ E. Since extend ..l.. I to a continuous linear map A I A' lm u = 0 then.2 I lm u = 0 therefore J.' € Ker t u ,

I

I

-134-

A. Andreotti

Ker t u

and thus it defines an element of "

extension "

,v

Al - A'

0

f .A.. ,

to

B

1m

then)..

I

defines a linear map of

topolobical, then the map of

X

-

I

"

.N

and)..

.AI

This shows that

set

X

U c.c X

-

define the same element

I

One then verifies that Let

I

0

and T

a:.

into ~

into

1.

on Ker v.

0

,;.

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

is another

I

tv

v(B)

v(B)

-v

If A

i:

y

If

Then v

is

is. continuous and : C '~ D: I linear

= tY(}J). Therefore rU.) in Ker t u 1m tv I

are each others inverse.

be a complex manifold.

We consider only those open

which are open sets of holomorphy.

This collection

~ of sets if stable by finite intersections and contains

arbitrarily fine coverings. Let

b

be the structure sheaf of

X

and let

'}=' be any

coherent sheaf. For every U" /1(, we can find a surjection f}P ~ T -'f 0(1) from which we derive (by theorem B of H. Cartan and J.P. Serre) a surjection £(U,e P ) ~* r(U,e:f')~ The space f (U, f) p) Schwartz.

has the structure of a .pace of Frechet-

One verifies that there is a unique structure of a

space of Freehet-Schwartz on [( u, continuous.

of the presentation [28]).

e P ~ Y~

:(.(U)

for

VC

which makes the map

0

we have chosen.

c(

*

is independent (See also

U.; 7?( we set

For any

As transposed

If')

Moreover this structure on l' (u, T)

s

= Hom

cont

(£(U,j(), a:)

of the restriction maps

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

r Uv :[(V,j1" f(V,j='), we get extension

maps

(1)

This fact can be proved but it is not obvious

It is obvious for sufficiently small of the definition of coherence.

" ] ). (cf. ('l22

U's,' indeed it is part

A

Andreottj

and therefore Proposit1on cosheaf.

(8.2.2).

For any coherent sheaf

Moreover for any n,,??(

U = tUi} iEI

finite covar1ng

o ....

!!!!!. Uc: ll(,

of (l,

Hq('2(~"1.) = 0 !££

f!22!.

:I,
"""

~

and any countable locally

q

we have

> O.

v

The augmented eech conplex

r(fl, 1=') -+ CO (1(, '1')

-7

Cl("2(,~) -+ •••

is a complex of Frechet spaces (as

V

is countable) and con-

tinuous maps. By theorem B this complex is acyclic, i.e. the sequence ie exact. By the lemma (8.2.1) the dual sequence is exact. complex.

But that is the sequence of the augmented homology

Co("ZI,".t.)<'- Cl ('?(, 'F.)~ ... In particular any countable locally finite covering?( c: /:?( o ~ ".:(£1.) -

a Leray covering for the dual cosheav8s

T. 8.3.

g.

of coherent sheaves

Duality between cohomology and homology.

This duality results by comparison of the two sequ.nces Cq- l

(I)

(V,'1')

S

q-}

Cq(V,T)

&

--J

cq+lCZ'f,:f)

'2(

7?(

( II)

where

-:r

is a coherent sheaf and

finite covering of In

(I)

e

is a countable

X.

the spaces are spaces of freChet-Schwartz and the

mapa are continuous, in (II) the spaces are strong duals of sp aces 0 f rrechet-Schwartz and th e maps as transposeds 0 f 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.

is

-1"3"6-

A. Andreotti

By appl1.cat10n of the duality lemma (8.2.1)

we obtain

Theorem (8.3.1)

II S q

(a)

is a topological homomorphism then

=Hom

H (X,~.) q

Moreover

Hq+l(X , 'T) T

!!

(b)

dq_l

Hq(X,~) Moreover

H _ (X, q l

cont (Hq(X,~),

_and

Hq (X

or) ,.:r.

t).

are separate d •

is a topological homomorphism then

= Hom

cont

(Hq(X,

"f.),

'1'.)

11:).

are separated.

Note that the assumption in (a) is equivalent to the separation of Hq+l(X,;t) and that the assumption in (b) is equivalent to the sep,u-ation of certainly .satisfied if H _ (X, q 1

8.4. a)

sheaf

:t.)

Hq_l(X,'f.).-

These conditions· are

Hq+l(X,"T)

or, respectively,

are finite-dimensional.

v

eech homology and the functor Let

T , fi--

110m C>'

EXT.

be sheaves of CY -modules

("1 ,f)

OIl

X.

Then the

is de fined as the sheaf associated to

the presheaf

u~ and

y

Hom

e)u

(~ f U

,j I u)

are coherent so is the she af

'Jf om Ct

(1, f) .

is a family of supnort&, we set

r

A shea!),1 HOM(X; • ,

tf)

,11 m <9 c:r,~ )·

HOM 4> (U,:r, C,J = 4'( U of ty-moiules on X

0

is called injective 11'

is an exact functor on sheaves 0 l' {J -modules

for any exact sequence of sheaves of ~ -modules

O-i'gl~:r ~'J.1l-r0 0...,;> HOM (X;T",'~) ....,..HOH

is exact.

(x::r,::)~ Hal'! (X;~11,-j)......y 0

1. e.

A. Andreotti We have the following facts: i) 11)

an injective sheaf is flabby if

{j

T

is injectivs. for any sheaf

&-

of

-modules

%,om t9 ("f.:J) is flabby i11) for any sheaf}- of 11 -modules one can tind an injective resolution

07';'

(.)

;;Co ~;;l -> ::;27'

-?

sequence 1s exact and every

Y- 1

1

t

...

l.s.

the

0, 1s an

i

injective sheaf. Thsss facts follow directly from the definitions and the possibility to imbed every module in an jnjective module.

(r.

;Yom C1

Applying the functor

<.)

to the sequencs

we get a complex of sheaves and homomorphisms

o ..,,;7(om t9' (7,f.o~ .... J/om The

q-th

p ( T ' ; l ) .... .J/omtP-

cohomology group of tilis complex is denoted by

This is a sheaf of

&- -modules

and one verifies it is independent

of the choice of the resolution Moreover 1 f

(T'7 2 )-;'

7

a coherent sheaf.

(.).

t::-

are coh eren t then Ext This can be seen as follows:

and

q

t1

(1'!-)

is

Let

o

"""

be a resolution of f

(•• )

be locally free sheaves(l) •

Applying

(1) On any open set of holomorphy

tion, as any coherent sheaf on

U c.e.-X

X

we have such a resolu-

1s the quotient sheaf on

of a free' sheaf by the theorem of Coen

(cr.

[~2J).

U

However since

here we need only this resolution locally one can invoque the theorem of syzygie.s of Hilbert to derive the eXistence in a sufficiently small neighborhood

0

f a given point

finite free resolution: 07 & Pd .... t9 Pd-l."..•• ,>t)po ~.-J _> 0 where

x

€.

X

of a

d s:. dimxX (cf. f@.

-138-

A Andreo.1ti the functor 0 .... /fom

whose

;fom

&

c9

•• ~)

(

we get anothtr complex

r

cio'j.) .,. ;t(om 6'

(.L

1'9),.. . "-:f

q-th cohomology group is again

om6'

::. xt'6

(L

2';)

(1' .;:. ).

follows by a standard spectral sequence ar-gument.

as it

By this

construction GxtJ(jt.~) is a coherent shea~. In particular if :J. is locally free we get [. xt q if

q> 0

as we can take); 0

=T

,ell

=.J;

2

-7 •••

.

(1'.;.) = = •••19= o.

Note that in any case

0

because

the sequence

.J

is exact as

o -4fom t9 (T.~ ) .., 11' om
Applying to ths resolution

we obtain a complex whose

(0)

q-th

the functor

HOM ~ (X;

'1'. . )

cohomology group is denoted by

'J.p.

For

EXT~ (X; we have analogously to the urevious case

q = 0

EXT

~(X;

1.;.)

= HOM ,/,(X;"f.j..).

The spectral sequence of the double complex KP,q =

and

d

L4>

K = [KP,q, d}, where

(X; 1fom 6' (,(;p'f q)

is induced by the ma's of the resolution

(0)

and

( .. ).

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

t

EXTn (X;.},;)

( n = p+q)

where

b)

The connection of homology ani the functor

EXT

is

established by the following ~

n.

(8.4.1).

Let £l

n

~

X

be a complex manifold of pure dimension

be the sheaf of germs of holomorphic

n-forms on _X.

-139-

A. Andreotti For any open set of holomorphY ~

sheaf

and for any coherent

2.!!. X one has

El{~

1.(u) =. the suffix ~

U cc:.. X

k

n

(U;-:(, "~n)

El{T~ u;'J',o't ) =

°

if

q I< n,

denoting the family of compact supporte.

..

Let '3" be a locally free sheat on X and let U be an open set of holomorphy in X. The slieaf;r' can be consider(<<)

ed as the sheaf of germs of holomorphic sections of a holomorphic

vector bundle E, .~ = (E). Let E* denote the dual bundle of E; if E is defined by the transition functions -llI • f~j} then E* is defined by the transition functions)< t gij

JZ

Let r,a(E) denote the sheaf of germs of Cd' forms of type (r,s) with values in E and let ;XU,V (E*) denote the sheaf of germs of forms with distribution coefficients, of type (u, v), with value in E*. Since

U

1s an open Bet of holomorphy, the sequence

O-.>[(U, 19(E»

~ (U,AO,o(E»

l

(U,JlO,l(E»

~ •••

~ [(U,J1o,n(E» ~

°

1s an exact sequ.nce of spaces of Frechet-Schwartz and contin-

uous maps.

By the duality lemma

also exact.

But this 1s the sequence

°~

(U,e (E»'

(8.2.1)

:-i k(u,;kn,n(E*»!

the dual sequence is

k(U,;7(n,n-l(E*))!

... ~rk(U,Xn,o(E*» Now for any vector bundle of germs of holomorphic

denoting .by J
E*,

n-forms with values in

E*,

~

0.

the sheaf we have

in the exact sequence of sheaves

A soft resolution of ~n(E·).

Therefore, since

holomorphic vector bundle and hence j\n(E*) sheaf, we get:

E*

can be any

any locally free

-149-

A. Andreotti

f

for 81lY locally free eheaf

o

H~{U';) = {

El{~

and

(U;

'1.~)

k

EXT"

5:

T

Let

($)

(U;

k

n = J2 {E*) = trom 11{'1, if

r tl n

Hom cont (f{U,<9{E». ~) i f

1f

= 0

P

if

tl

n

T,; ) C"t q (T.f))

#zq = H~{U;

tl 0

q

as ~

Ext~ (7.J.)

r = n.

be locally free then we have

BlCT:{U; J',~)

In fact the epectral sequence converging to the term

Jl. n )

1s locally free.

(p+q=s).

Thus

has

#z q = 0

Moreover

s;.)

= :fomtJ {'}, 1s also locally free, hence by EP O = 0 i f p tl n and Z

«()

we get

n o E Z

= ~ (U; p{om&

(Y,; ».

degenerated so that we have

But the epectral sequence 1s

EX~(U; T

J)

= #ZO •

This proves our contention.

(r)

Suppose now

th~t

'T

is coherent and

1-

locally free.

The we have

EX~

(u;T,j)

=0

if

p
and an exact sequence

o

-?

EXT~(U;T'J)

->

H~(U; /10m{) (J: 0';) ..,. ~(U; .:rom Cl (.£'1'; KP , q = f k(U, ~om~ (h

In fact consider the double complex

p

has for cohomology groups the groups For any ~ the sequence of sheaves

o -:>.1( omIi (001; p' /I) ~

;tom is (.,6 p'

EXT*k(U;

».

.fq»

7'1)'

/0) ..,..?1 om <9 r); p .ft'l)

7' •••

is. exact as ~ P. is lO~allY free and provides an injective resolution .of /(ome: (); p,;),

Tak1ng cohomology with respect

to the differential coming from the resolution

(*)

we get

-141-

A. Andreotti

#i q = H~(U. ;(om~
as

is locally free.

EX~ (U;T.f) = 0

It follows that

q.,l n

if

if

P < n

and

EXTn~1 (U;~'Ji) is the l-th cohomology group of the complex: ~(U;

(0)

71 0m ty (et o';))

~H~(U;.;fom&(06I'f)-? ~ ~(u;.!omli(;: Z·.1) ,. •••

In particular we get the exact sequence (1)

0

"7"EXT~(U;'1,;).., H~(u;;fom&1 (.t

o.f)..,

~(u;c77"omcJ

iLl';'))

Consider the exact sequence of spaces of Frechet-

(0)

Schwartz obtained by applying the functor

r

to

( •• )

." .,>F(U.obl ).,. f(U.et'o)-' f(U;1)-"> 0 Exactness follows from the assumption that of holomorphy.

By the duality lemma and

U

is an open set

(~)

we get an exact

sequenee

H~(u.;;fom.9(,jfo,jl.n»~H~(u.ffom&(Jfl'!\n)

o"? '1'.(U)-">

(2)

Comparing

(0).

"f.(U) =

Given

T

U "7 EXT~(U;

and

(1)

EXT~(U; T. Jl. n)

and,¥.

'i.;)

Cxt~(·"J'.~).

U.;?1(.

with

E-~·q = Hp(U • Ext~( c

7J(

0 if

q';' n.

We denote this precosheaf u

We do not need to verify it is a cosheaf.

(8. 4.2).

2{

EXT~(U; Y,nn) =

and

coherent sheave., define a precosheaf by

for

Lemma

where

we obtain

(2)

There eXists a spectral seguence

1. ,;;:)).

is a loc ally fint te covering

0

f

X.

- 142-

A. Andreotti

Consider the double complex

~.

K-P,q = Cp(~(.1Tomk(-j(.;1q))'

If we take cohomology with

respect to the differential coming from

(.)

we get

= Cp (2(, Cxt~C:r'. ~)) and then taking homology with \I _ respect to the Cech differentiil we get I

E-i' q

2 = H~
q IE-P

,ffomk (T•.1q)

Now we remark that Thus we have in is soft.

~

Jif- q

rl!om k (T,9q) being injective.

(U) =fk(U;;;fom

("f,fq))'

a flabby cosheaf as '1/om

(f.I-OJ

Therefore taking first homology

\I

with respect to the Cech differential we get

"E-i' q

=0

if

P

~

0

Taking now the cohomology with respect to the differential coming from

(4)

we get that the spectral sequence degenerates

having as total cohomology the groups

EXT~(X;'J',; ). we apply this lemma to ,; = Jl.n. q ,J. n

and

(8.4.1).

Then

[xt~( 1, S\, n) = 0

if

Ext~( T, Jl, n) = 'J'. as it follows from lemma Therefore we get the following

Theerem (8.4.3). ~ X be a complex manifold of pure dimension n and let ~ be any coherent sheaf on X. !h!!O we have

Hp(X,

T.)

~

EX~-P(X;":t, .n,n).

The combination of this theorem (8.4.3) with theorem (8,3.1) is what is usually called the "duality theorem"; in this form i t is due to Malgrange and Serre (cf. r39J. [lW], [44]).

-143-

A. Andreotti Remarks.

1.'

"Y'

If

is locally free

i.e.

the sheaf of gsrms of

holomorphic sections of a holomorphic vector bundle E,

"'f

Ext q

CJ

2

(E),

thsn it

('(,nn) cO

q" 0

and c

Eo

being the dual bundle of EI(~-P (X;61(E),Jl. n )

so that theorem (8.4.3)

c

E.

E xt~ ('1,.I\.n) 1(o,m&(T,f/, n)

2

c

.ll.n(E.) ,

Therefore

~-p(X, Jl.n(E.)

gives

H (X,19 (E).) ~ ~-p(X, Jl,n(Eo».

p

In particular if

E

c

E- l

18 locally free and if

X

is a line bundle,

Eo

and we get in

this case

~(x,9 (E).) ~ ~-p(X, j1.n(E-l » •. 2.

If

T

2

tJ

(E)

manifold then the theorem

(8.3.1)

is a compact

can be applied without

restrictions and. we get HP (X,6'(E»

• Hom (~-p(X, .j1.n(E*», I)

as the cohomology groupe being finite-dimensional any linear map into C is automatically continuous.

8.5_ Divisors and Riemann-Roth theorem. a) Let X be a connected complex maoifold of complex ' dimension n. The sheaf 61 of germs of never vanishing holomorphic functions as a sheaf of multiplicative groups, can be considered as a 8ubsheaf of the following sheaves:

.

the sheaf~·

of germs of non identically zero meromorphic

tunc tions, the sheaf f) o·

functions.'

of germs of non identically zero homolorphio

-144-

A. Andreotti While the sheaf /h?* /'l is a sheaf of Multiplicative groups. the sheaf is a sheaf 0 f multiplicative aonoids. We thus get two exact sequences ot sheaves

&:

(1)

°~ tJ· --?J(. -) fJ ~ °

(2)

o~tJ·~tJ·~f) ~o +

o

where the sheaves ~ and ~ + are defined by the exactness of the sequences. The sheaf j) is called· the sheaf of gems of (meromorphic) diVisors. The sheaf ,;<9+ is called the ~ of germs of holomorphic or positive divieors. The elements of HO(X.)?) are called meromornhic divisors on X and the elements of HO(X,)}+) are called holomorphic or positive divisors on X. An element D e HO(x.0) (reap. D. HO(X,J)+» is given on a sufficiently fine open covering "Z( = {Uiti.I of X by a collection ffil of meromorphic (resp. holomorphic) functions. not identically zero. on each Ui • and such that i, j

6

I

is holomorphic and never zero. The functions fg ij ! are transition functions of a holomorpbic -" • line bundle I DJ and represent tbe element ,,(D) E H1 (X,cO' ). where 8 is the connecting homomorphism of the exact cobomology sequences associated to (1) or (2). In every meromorphic divisor D we can distinguish tbe

O-part DO' and the polar part divisore and we usually write

D~,

these are holomorphic

Correspondingly we have

The study of gensral

(i.e. meromorphic) divisors can thus be

reduced to the study of holomorphic divisors.

-145_

A. Andreotti

Consider the space HO(X.t9(D)) where <9(D) is the sheaf of germs of holomorphic sections of the bundle [DJ. To every s HO(X.6!(D») there corresponds a holomorphic divisor, this in its turn determines s up to mUltiplication by a global never vanishing holomorphic function. If X is compact (or pseunoconcave) ther. the divisor of a section s £ HO{X,6!(D» determines the section up to multiplication by a non-zero constant.

The set of positive divisors

corresponding to elemsnts of HO(X. _'(D» is called the linear system of D and is denoted by \DI. Its elements correspond one-to-one to the point of the projective space

(HO(x.19 (D» - lot> / a:* .. We attribute to

I D\ dim

the dimension of this space

IDI = dima:

The problems we want to

X

a divisor

D

HO(X.{J(D» - 1 •

~nsider

IDI.

compute dim

°

Thus

is the following:

given on

Note that if we are able to

eatablish that dim 1~2 then we have proved that the holonorphic line bundle i D ~ admits a holomorphic section different from the O-section. b) Let us suppose now that X is comp~ct and dim~ = 1. Then every divisor D is given by a finit. sum D 1 niPi with n E :rl: and Pi being the divisors ass. -.iated to points i of X. A divisor is positive iff all n are;~. The integer i En is called the degree of the divisor D. i Assume first that D = ~ niPi is a positive divisor. In this case the bundle 1 D} has certainly a holomorphic sec tion 6 ~ 0. the section corresponding to the divisor itself.

Given

s

we

have a sheaf homomorphism c; ~ () (D) given by mul tipl1c ation 6. The homomorphism is certainly injective and the quotient

by

sheaf is concentrated in the points Pi

by the vector space

V(n , Pi) i

ing the Taylor expansion at truncated at the order

n -1. i

Pi

Pi

and given at each point

of dimension

n

represent1 of holomorphic fun;tions,

-146_

A. Andreotti

In other words we have an exact sequence of sheaves:

This gives the exact cohomology sequence

° ~ HO(X,S) -> HO(X,{l (D))->

E'i.ni

-> Hl(X,D) ~ a l (X,<9 (D)) "'>0 1

as

H (X, LL v(n ,Pi)) i

support.

'=

°

the sheaf having

O-dimensional

Therefore we get, in particular:

dima: aO(X,tJ(D)) - dimE Hl(X, 6'(D)) = deg(D) + dimE HO(X,O) - dimE Hl(X,D ).

No,,: HO(X,e ) = E

~(X,.9 )

as

X

dimE HO(X,Cl) = 1.

is compact thus

1 is finite-dimensional and, by duality, :::: HO(X, _'1 ),

the space of holomorphic differentials on is called·: the genus

~(x, D(O))

g(X)

of

X.

X.

i f finite-dimensional and, by duality

~ aO(X,.n.l(_D)) = HO(X, e(K-D))

"here

bundle associated to the sheaf Jl, 1 • called the canonical bundle,

~ K~ is the line

The bundle

We thus have the following formula dim

IDI

e

D.

i(D)

D = DO - D", even

the meaning of

(K-D».

Indeed let HO(X, e(D~)).

s

denote the section, corresponding to

We havs an exact sequence of sheaves

0-) 61(D) ... e(D

D~

is

(Riemann - Roch theorem)

This formula can be extended to any divisor

where

K l,

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

if not positiv, maintaining for dimE HO(X,

f

dimE HO(x,D(K-D)) = 1(D)

and

is called ths speciality index of the divisor

(1)

Its dimension

o )-'

= Z niP i •

JL V(Pi,n i )

~

°

Dd

ot

-147 _

A. Andreotti

Therefore

dim~ HO(X,19(D O» - dimll: H (X,t'J (Do) = dim~ HO(X,e (D» Thus using for dsg

DO

dim~ Hl(X,O(D»

+ deg(D,.)

the formula already established we get

I D\

(Do) - g(X) = dim

-i(D) + deg (D,,).

From this ths assertion follows as we have

deg(D)

= deg

(DO) -

- deg

(D~).

c) we add a few remarks to the Riemann-Roch theorem in dimension one. The genus bl(X) (1)

21

g(X)

21

X

equals

t

of the first Betti number

X :

2g(X)

= bl(X).

First we show that

b (X) .>. 2g(X). l

From the exact sequence

o~ ~-i>ed->.5l1....,0 Since

HO(X, 11:) ~ 11:,

HO(X,

CJ)

N

11:,

we get an injective map

0-7 HO(X,Jl. 1) -> Hl(X, 11:) • This map associates to every holomorphic

class as a closed form. Now HO(X, Jl.l) and HO(X, ji 1)

l-form its cohomology

can be considered as subspaces

Hl(X, ~)., Their intersection is reduced to [O~ In fact "HO(,Jl l ), 13. HO(X, ji·.l) and i f « = 13 + dg where

of if


g

is a


C ~ function on X, by reason of bidegree we get ;; a g = i.e. g is (plur1-)harmonic. By the

°

Thus

max1aum principle Therefore

I

must va constant hence

dim~ Hl(X, 11:) ~ 2 dimll: HO(X, .51,1)

We show now that

bl(X)

~

2g(X).

Q

= O.

i.e.

bl(X) ~ 2g(X).

From the exact sequence

0~1I:-?6J& e~,I/~o we get an injection

°7

l Hl(X, 11:) -, H (X,19 ) It Hl(X,

0).

-148-

A. Andreotti

°

° ,. ,

In fact H (X, 1:) ~ 1:, H (X, C/) maximum principle). Therefore

~

dime H1(X, '11:) ::. 2 diml: H1(X, ~)

1:,

l.e.

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

=

IKI

dim

deg

(K)

=

=°

we get

g(X) - 1

If we apply the eame theorem to (3)

D

D

K we get

2g(X) - 2.

To do this we have to know that i K f comee from a divisor. Now if g(X) ~ 1 this follows from (2). I f g(X) = then for any positive divisor D we have dim 101 = deg (D) + i(D). But thsn necessarily i(D) = 0.

°

There eXists on

first order.

X

a rational function with a

sing~e

pole of

This function extablishes an isomorphism of

X

onto the Riemann sphere and on this manifold one verifies immed-

iatsly that K -Zp, p being a point of X. In pClI'ticular for a divisor D With deg(D) > 2g-2 we must havs

= 0.

i(D)

As an exercise one can show now that every compact manifold X

ot complex dimension one admits a porjective imbedding

is projective algebraic.

Indeed if

.0' ••• ' St

HO(X,e (D)) the map X ... Ft (!:) defined by x ~ (so(x), ••• , St(x)) is holomorphic everywhere. (D) > 2g

i.8.

is a basis of

If

then one verifies by means of the Riemann-Roch theorem

that the map 1s one-to-one and biholomorphic. d)

Let of

Let

D

X

be a connected compact manifold of

be a holomorph1c divisor on

"multiplicity"

X

dimll: X

= 2.

which will be supposed

one 3.t each point and non-singular.

We have now an exact sequence

° where

B

-;0

<9~&(D)-?

1s a section of

1 DJ

O(D)/ 0

-7

°

correS90nding to the divisor

D.

- 149-

A. Andreotti

We get an exact cohomology sequence:

°

-7

HO(X,ll )

-,7

& (D» ~

HO(X,

HO(D,

~ ~(X, e );'" ~(X, 8

(D»

-- Hl(D,

__ ~(X,

(D»

..,. 0.

e) ~

H2 (X,

c9

·8 (D)

I D)

~

19

I D)

""'

(D)

We have

dim(CI)HOC D ,19(D)/ D) -

dima:~(D, D (D)I D) = deg

tD tiD

- genus of dima:H2(x,O)

= dimE

= Pg(X) = geometric

2 HO(X,Jt )

(D) + 1

genus of

X

dima:Hl(X,e) = hO,l dima:H2(x, 19 (D» = dima: HO(X, e(K-D»

where

t

Kt

denotes the

bundle corres~onding to the sheaf of holomorphic 2-forms 2 S1. = () (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

.>- deg

1 DJ ID

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

This is Castelnuovo's theorem. right side is

dima: Hl(X,

e (D»

The jifference of the left and which is called the

tlsuperabundance" •

If

X

is Kahler then

q = hO,l = hl,O = dima: HO(X,S1 1 )

of linearly independent holomorp!'1ic p g(X) - q = Pa(X)

number

I-forms, and

is called the arithmetic genus.

The inequality can be extablished without the restrictive

assumptions we have made on ID

of

D

D.

In particular for a multiple

we get the inequality

dim liDI:z. 1

2

degiD11J) -

(l(~-l)

deglDqo + 1 genus (D) -

_ 1+1) + (p (X) _hO,l) - i(lD) + 1. g

If

D

is positive and

1

large enough and positive, then illDl = 0.

-)~O-

A. Andreotti

Therefore if

deg~oJlo > 0,

dim

IIDI

2 1 •

grows like

Therefore

If X contains a divisor 0 trans. degree

1<. (X)

with deg

i 0110

= 2. ·One could show that

X

) 0

~

is in this

case a projective algebraic variety as it is always the case

for a complex surface

degree

2.

(cf.

X

[21J).

with ~ (X)

of transcendence

- 151-

A. Andreotti

Chapter

IX.

The

H. Lewy

problem.

9.1.

Preliminaries. To simplify the exposition we restrict ourselves to the space ~n although the results will hold on any complex manifold with only formal changes of notation. Let U be an open ~
1Z

E.

U J P (z)

2.

0:5

U Ip(z),;

os

U

=fZe

S

={z«U/p(z)=o}

°

and we will assume d p " on S, so that S is a smooth hypersurface. On U we consider the Dolbeault complex C* (U) = {CO,O(U) where

CO,s(U)

.2.

CO,l(U)

2

CO,2(U) ~ •••

denotes the space of

Cd

5

forms on

U of type

e

(O,s) and where is the exterior differentiation with rsspect to antiholomorphic coordinates. Analogously we define the spacss

Co,s(~t,

resp.

CO,s(U-),

as

the spaces of thoss forms of type (O,s) on a+, resp a-, having C" coefficients with all partial deriviatives continuous up to the boundary S. In this way we obtain two similar complexes, c*(u+) and C*(U"'). Define ""O,s(U) =

H, ..

Co,s(U), 4> =

13 We have

Ii

t

tj0s(U) c ;:10,s+l(U),

pdE.

+ ~

p/l(3 , " "

Co,s(U),

Co,s-l(U)l.

therefore

.,;;*(U) = (/O,s(U) is a sUbcomplex of C*(U) and indeed a differnetial ideal. in a similar way on. defines the sUbcomplexes tJ * (U!) of C* (U!).

-152-

A. Andreotti

Finally one defines the quotient complex O*(S}

= [QO,O(S}

by the exact sequence

o -- J* (U)

-i'

c* ( U) ~

Q*( S) ~

o.

The quotient complex is denoted by Q*(S} as each one of its spaces is concentrated on S. The operator is by definition induced by the operator on C*(U} and c1*(U}. One could define the quotient complex also for the inclusions /'1* + c C• (U-) + CJ (u-) but one obtains' in this way the same complex Q*(S} as only the values on S of the coefficients of the forms considered are of importance.

as

a

We thus can consider four types of cohomology groups H*(U}

the cohomology of

C*(U}

H*(U:}

the cohomology of

c* (U:)

H*(S}

the cohomology of

Q* (S).

Note that while the cohomology H*(U} is the cohomology of U with values in the sheaf tJ the same is not true for H* (U!). Remark. We have

-d O,O(U}

::~t>o,o : U~

a:

I

q, 0,°/ 5 = ° 5 thus QO'O(S)

represents the space of C fi' functions on S. For u to: QO'o(S} the condition 8S u = is a necessary condition for u to be tV 00 + oJ 00the trdce on S of a function u € C ' (U ) (or -u ~ C ' (U }) is a which is holomorphic in g+ (or g-). Indeed if to U- (ar U+) we have C IS' extension of ~ U :: on U+ (or U-) and therefore 2i U =rei. for some d e CO,l(U), i.e. as u = o.

°

°

u

u

In general calling the im~ge of a form of

CO,s(U)

on

QO,s(S) the trace of that form on S, we can state that for u E QO,s the condition u = is a necessary condition for

as

°

-153-

A Andreotti

u to be the trace oli S of a form (1 " Co,e(u+) with -J u = N Indeed let u be any form of CO,s(U+) having u as trace on Then , N p f\ f3 for some and r; in C"(U+). u = U + pd.+ ~

a

°

s.

"

Thus

"U.:.
H 0 8+1 + + ap(a+~f') e CJ' (U)

i.e.

Same argu.. ent for

u-.

Example. Take on

~2.

as co ordinates Let

p Then lR 3

S

4 -

5 X

(x

2

=~

2

+ x )

(Z2-- Z2) -

I zl 12

x = xi

is the product of the paraboloid 4 are coordinates, by the xl' x ' x

where

2

4

-

+ x~ in x -axiS.

3

1-

At each pollit . dZ 1 and ap = - 21 dZ 2 - ~ d~ for the (O,l)-forms. Thus we have QO,O(S)

= C""(S)

QO,l(S)

N

C""(S)

QO,2(S)

=

o.

Thus

dZ

1\

as'

~

S

1

II

= fc,,"<S) To compute

= cO' functions on

form a basis

C (S)

1\

dz1 ....

° 1".

by its definition, we have to do the following ~

- given u ~ C~(S) extend (in any way) to a C ¢ function u on ;2. Due to the shape of S we m.ay assume u injependent of the variable x ; 4 ~

- compute

au;

-154_

A. Andreotti

-

-'"

au;

compute

'V

-~

i1U

=

~

=

~

dZ

zl

.It

+

l

dz

Zz

'V

N

-

dZi.

aZl

Z

II

(2171 p + ZiZi.dz )

l

i1 Z z

'" ou Ziz l ~)dzl

'" = (ll zl

'" 3 P• ,

21 0l...!!.

Zz

Zz

- "restrict" the form thus obtained to 0,1 (I: Z)) to get

t1

;iSu

= (~ aZ

i~

l

as

'"u

il u) ax 3

is independent of

x

dZ

ean be taken as coordinates on In conclusion the complex on

lR 3

~

where

= xl

~·(S)

+ 1%Z'

a

-

and

iz

zl

The operator

9.2.

L

3

can be taken

x

-L

l oX

3

are coordinates

-"0

.

3

is the operator of

H. Lewy,

07J.

of

Mayer-V1etro1s sequence.

a)

A

C '" function on

vanishes on

S

0

S}.

is called

flat on

S

with all of its partial dsrivatives.

'1 0 ,s(0) = l ~ "CO.s(O) I on

x

x Z'

S.

C"'(lR3)~ C (lR 3)

= 0-

~.

is Lsomorphic to the complex

where L

compute modulo

1.e.

l

and

4

S

all coefficients of 4>

i f it

Set are flat

-1~5-

A. Andreotti

a1 o ,8(U)

We have

'j O,8.1(U)

G

~.

·u

(U).

. .

C• (U)

i8 another subcomplex of

S

coefficients of the

=

-:i- '1 0 ,e(U)

and in fact a eubcomplex of

The quotient complex

concentrated on

"J' (U)

therefore

..

= C (U)/:¥ (U)

C (S)

i8

and is obtained by restricting to C'" forms on

S

the

U.

We have CO,s(S) =

t

!:

aet

a ( •••"et 1 s

..

1-· ·($8

Co,o(S)

0-7 C..(U) -;>

for all

c· (U+) • C· (u-)

-7

(x) dZa: ••• dZ a I" s

(a ••• Ci ) l s

c' (S) ~

°

is an exact esquence

from which we get a cohomology sequence

° ~ HO,o(U)

-p HO,o(U+) • HO,O(U-) -> HO,o(C' (S»-7

(1)

To connect the cohomology groups HO,s(S)

= HO'S(Q'(S)

J-l*

..

HO,s(C'(S»

with the groups

we may use the following exact sequencs ..

•

0-3> <J (U)/')' (U) .... C (U)/-:J (U)

•

-7

Q (S)

-;0

°

i.e.

(2~

o."t;"(U)/':/- ·(U) ~C·(S) --;> Q'(S)

~

(9.2.1).

For any choice of

0.

-i'

U

and

S

the seguence

1s exact.

Proof. (Q)

Let

u'" {jo,o(U)

thus

Assume th,.lot the coefficients of

u = fa

au

l

are flat on

S.

Then

-156-

A.. Andr:entti This means that ex 1 = u

2

=P

for some

pet 2

Q'

EO

CO,OCU)

2

thus

'

But then

(;(2'

;p (1'215 = 0. thus

= pa 3

et 2

for some q

3

Eo

Co,oCU),

Continuing in this w4Y we see that

we have exactness at. C~)

d 0,° / ,;,°,°.

u = (,3dy

thus

must be flat on

u

5

i.e.

To treat the general case we will make use of the

following fact Given on

S

f ' f , 121 l O C '" function F 2.!!. U

~ltF

"pit

I

2.!

a sequence

there eXiets a

=

for

fit

5

k = 0, 1, 2.

Cd' functions,

such that • ••

This can be derived as a particular case from the Whitney extension theorem. Also direct proofs are available. Cl ) Let

f € {jo,sCU),

f3.

and

Then

Ili l

5

for some

f!,

1\

1t ap

=13/ 5 for

=0

It

a p' Ill)

f = Cf -

as the coefficient of

:ip

f -

A

satisfies the conditions Co).

A

B

l

with (31

III

+ ~p"

.u

'\ =

vanish on

Let now

assume that the coefficients of

31'

= 1, 2, 3, •..

5

Thus we can write

(1)

ap ~

such that

a ltp

u = perl +

f = pa +

Using the above remark, we can find a form

I'> 1 ~ CO,s-lCU)

Co)

s 21.

5

1"'\

while (31

u. :jo,sCU)

are flat on

satisfying

+ .. fA 13 1

5.

and Write

Co).

In particular it follows from this lemma that the space

CO,O(5)

can be identified with the space

power series in

p with

C

~

G (5)

coefficients on

S.

) e}

of formal

- 157-

A Andren.lli

Then u - ;jCff'l) = pca l

Set

y 1 = ai

-

ij~l'

- ~~).

By the assumption we get ~

2

satisfying Co).

2 P CO' 2 Set

'Y 2 = a 2

'Y 2 =pa3

-

H

(l 2'

P3

+a1'I'

13

3

II

° thus

Y

I

=

=

°

thus

1 S

Then

to- f32)'

By the assumption

with

~P

;;1''' y 21

satisfyingC").

S

Then

Proceeding in this way we construct a formal power series in c) = r.I. th

~

m

satisfying

u _ JC II • l P

1.

k

It

13

k

i1

~ II 13

) _

11

and r.I. th the property that

C")

- P

m+l.". r

m+l

m+l

., CO,sCU).

Using the remark made of the beginning, we can find f e CO,s-lCU) such that

fl s Set

v_pf

k~

= then

Lf il

P

=0 j.{

I'f~-CI

has flat coefficients on o,sCU)/~O'sCU).

for

°' s-l.CU) S.

k = 0, 1, 2,

and we have that

u - ()v

This proves exactness at

j

This lemma tells us that the cohomology of

/j" CU)/T" CU)

is

zero in any dimension and therefore the cohomology sequence of (2)

gives a set of isomorphism ffO,s(c'(S»

~ Ho,sCQ'CS)

= HO,sCS).

-158-

A. Andreotti

Introducing this result in the sequence

(1)

we obtain an exact

sequence

which is called the Mayer-Vietoris sequence for

U and

S.

The previous considerations can be repeated replacing the C OJ functions on U t U': with the space of those C fUnctions with compact support on U, U! respectively. This b)

space of

leads to an exact sequenee

~ HOkl(U) .... HOkl(u+) • HOil(U-)

-;>

H°itl(S) ....

(Mayer-Vietoris sequence with compact supports).

9.3. Let

Bochner theorem. X be any (n-2)-complete connected manifold of complex

dimension n ~ 2, for instance a Stein manifold. Let S be any connected closed C ~ hY1Jersurface in X such that X-S = ~- u ~+ is the union of two connected open sets i- and ~+, 0 f which say ~- is relatively compact. Theorem f

~

S

(9.3.1).

C

0>

function

aSf = 0 is which is holomornhic

satisfYing the compatibility condition

the trace on

in

Under the above assumptions, any

S

of a

C ~ function on

X-

~-.

~.

The Mayer-Vietor1s sequence with

com~act

supports gives

-159-

A.- Andrect1i

HOkO(X)

Now

= HOkO(X+) = °

as

X and

X+

are connected and

and not compact.

Moreover

HOkO(X-)

= HOkO(X-)

as

X-

is compact. By the dUality

theorem we get

HOkl(X) ~ Hom cant by the assumption that Since

Hn-l(X,

the maps

Hence

8

J( n)

(Hn-l(X, n

and

~

2

and

Hn(X, J\, n)

n),;)

=°

X is

(n-2)-complete.

are f1n1te-dimensio-nal, i)'

are topological homomorphisme.

Hn_l(X,Jl~) ~ Hom cant

(Hn-l(X,.n.n),II:)

And Here E• is the dual bundle of the canonical bundle Hence the restriction map HO,O(X-) ~ HO,O(S)

E.

ie surjective.

X = ~n

Fbr

(cf. Note

9.4.

this theorem is due to Bochner t Fichera, Martinelli [16,1 [24.] (41J).

th~t

no assumption is ma.e on the shape or convexity of

S.

Riemann-Hilbert and Cauchy problem.

These problems are generalizations of a problem concerning holomorphic functions in several variables first considered and

solved by Hans Lewy (cf. Assume that

[37.4

[381).

U is an open set of holomorphy in

~n.

Then

-the Mayer-Vietoris sequence (with closed supoorts) splits in the short exact sequences

°~ HO,OeU) .->'HO,O(U+) •

HO,O(U-) --. HO,OeS)

7

°

-1.60 -

A. Andreotti

This shows th "t (a~

Every cohomology class on

S

can be written as a jump

of a cohomology class on

U+

and a cohomology class on

U-

i.e. the Bo-called Riemann-Hilbert problem 1s always

solvable for an open set of holomorphy. Moreover if

s on (b)

=°

s >·0

the solution is unique while for

it is determined up to addition to the functions U· and U of a global holomorphic function on U.

Let us agree to say that the Cauchy-problem is solvable from the side of U· if HO,s(U·) ~ HO,s(S) is surjective.

Then we realize that if

U is an open set of

holomorphy the solvability of Cauchy-problem for

s >

°

is equivalent to the vanishing theorem

HO,s(U-)

=

°

and in this esse the solution is unique. If s = the Cauchy-problem is solvable if and only if

°

HO,O(U) ~ HO,O(U-) is surjective

1.e.,

loosely speaking,

iff

contained in the envelope of holomorphy of solution will be unique if for instance,

U·

U

U

is The

is

connected.

Example. Let us consider the situation of the example given at the end of 2 section 9.1. Here U• = < 1 x xl. x 22 ~ is an elementary con-

4 vex set while U is the closure of the complement in ~2. The boundary S of U· is strongly Levi-convex. Let a E S and let Jl. bs any nsighborhood of a which is also a domain of holomorphy. Writing for fl, 1t·,.Il- and SJ\, = nnS the MayerVietoris saquence, we get the exact sequence

-161-

A. Andreotti

We have

i)

HO,l(JL) = 0.

HO,2 Ut)

0,

as Jl,

is a domain of

holomorphy.

11l

HO,lCn.+) = 0.

This fact is a consequence of the regularity theorem of

Kohn and Nirenberg (cf. next section). It would be desirable to obtain a direct proof in this special case. i11)

Since

U-

has a pseudoconcave boundary one realizes that

tor every point

a

Eo

S

of neighborhoods A of morphy and such that

we can find a fundamental sequence

a,

which are domains of holo..

HO,O(.,!\) .... HO,O(J\,-)

is surjective (and an isomorphism). Making use of this information the sequence of Mayer-Vietoris gives us the following isomorphisM

HO,lCn.-)

'!'". Ho,l(SJ\~

The first tells us that given any

cO> functions on

=°

Sft

satisfying the compatibility condition asu ~t~h~e~r~e~e~x~i~s~t~s~a~ C