deforming cohomology classes - American Mathematical Society

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

Volume 181, July 1973

DEFORMING COHOMOLOGYCLASSES BY

JOHN J. WAVRIK ABSTRACT.

Let 77: X

>S be a flat proper morphism of analytic

spaces.

77 may be thought of as providing a family of compact analytic spaces, Xs, parametrized by the space S. Let if be a coherent sheaf on X flat over S. ? may be thought of as a family of coherent sheaves, ifs, on the family of spaces Xs. Let o eS be a fixed point, S be a flat

on X flat

In this

to be simple

to an algebraic

the deformation

morphism

of compact

not be locally

[3] and the theory

proper

, A ).

generalizes

in the more restricted

to reduce

herent

is a family £ Hq(X

to a cohomology

Reduction

theorem

X/S S, ç"

77 is not required

of manifolds) of Griffiths

(so

over

of o e S.

The morphism

n: X —» S be a flat

spaces,

o e S, A co-

4.11.1]

that

fo e HqÍXQ, 3M to some neighbor-

Primary 32G05; Secondary deformation

theory,

32C35.

families

of analytic

of moduli. Copyright © 1973, American Mathematical Society

341

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

342

J.J.WAVRIK

hood

of o is equivalent

We recall

the following

Definition

pactum

1.1 17].

to Hq{X , A).

definition:

A subset

K of a complex

space

X is called

a Stein

com-

if {i)

K is compact,

(ii)

K is semianalytic,

(iii) Stein

to extending

K has

a fundamental

system

of open

neighborhoods

in X which

are

spaces. The main

properties

Proposition

1.2.

A a coherent

(i)

sheaf

r(K,

of Stein

Let

compacta

X be an analytic

which

we need

space,

are the following:

K C X a Stein

compactum,

on X

0)

is a noetherian

C-algebra,

(ii) ViK, A) is a finite T(K, 6)-module,

(iii) HqiK, A) = 0 if q > 1. Proof.

See [7].

Now let

X o n K..z

\K A be a finite

Then

K. = \K.\z

theorem [2, Corollary we have

cover

isa

of X

finite

cover

by Stein

compacta

of X o by7 Stein

in X.

Let

compacta. r

K. =

Leray'sj

to Theorem 5.2.4] shows that Hq{XQ, 3) = HqQ\, A).

Thus

a complex

o - c°(X, J) —-,

cm{K,CJ)— o

of flat A-modules (A = 0.S , o ) and HqiXo , A) = HqiC "(K, ?)). Let

T be an analytic

space,

Let X ' = X xs T, it':X'-*T, is flat sheaf.

and proper

o £ T, and g: ÍT, o)

and represents

We may identify

in X' [7, (2.20)].

►fo, o) a finite morphism.

and A ' the pullback of A to X '. Then X '/T the family

X ' = 7T'~

Let B = l\

(o) with

induced X .

by g.

K. is still

Q. ß is a finite A-algebra.

A

is the induced

a Stein

compactum

We have VfC, ?')

Tfo., Ï) ®A B [7, (2.21)] and'so CqiK, A ') = Cq{K,J) ®AB. Proposition Proof.

1.3. Hq{C '(K, J")) is a finite A-module.

We have

already

observed

that

Hq{cAK,A))= Hq{Xo,A) = lim Hqin~Hu),A) = (R^fofoV By the direct

(R^foif)) *

o

image

Now we show

of finite

theorem

L3], R977^(i)

is a coherent

sheaf

on X, thus

is a finite A-module. that

the complex

C" = C'{}\,

A) can be replaced

free A-modules.


by a complex

=

343

DEFORMING COHOMOLOGY CLASSES Proposition

1.4.

Let

A be a noetherian

Cq = 0 if q > m, H9iCA D ' of finite

that

free

the induced

Prool. begin

Dq = 0 if q > m, and a morphism

^m

a commutative

II.9].)

can

The proof

^

qb ': D ' —> C ' such

is by descending

free A-module

be ufted

together

^m . fím _^ çm_

of A-modules,

Then there is a complex

HqiD A —» HqÍC ') are isomorphisms.

Dm a finite

•)

C" a complex

morphisms

(Cf. [5, 0m.

Hm(c

ring,

A-module for all q.

A-modules,

by choosing

Dm ^

a finite

induction

with ^^

on q.

an epimorphism suppose

that

We i/7m:

we have

diagram: dp + l

DP+l

D

Dm -*0

; Dp +2

kP + I

c*>_* cp+1-►P+l

with the Dl finite (i) (p'

free A-modules

induces

We have

such that

an isomorphism

(ii) (bp + ] induces produced

H'(D ') —> HJiC),

an epimorphism

this

situation

Cm — 0

ci+2

j > p + 2,

Ker (dPD+A —>Hp + lÍC ").

for p = m ~ 1 above.

Let Zq = Ker df, Bq = Im df, Z'q = Ker dqD. 6p+l Z'z? + 1 _,WP + I(c-),

let

Kp+] denote

the kernel.

induces

an epimorphism

We have

O—» zV/í,+1—» Z'p +1 — tfí+1(C*) —>0

0-► commutative

finite

F"~*Kp Let

with

exact

A-modules.

+ l.

Since

latter

morphism

F"-^Cp

and

¿£

may verify

that

with

a finite

with

Z p+

A-module

these

to a morphism

choices

By induction

Let

and

F

Kp+

F ' — Zp,

F*-*CP.

F —>Zp/Bp~ hence,

= HPÍCA).

by composition,

by p'^

we obtain

(i) and (ii) are satisfied

Then

We with

p

D ', çb ': D ' —> C ' as required.

A be a noetherian

Cq = 0, q > m, ó':

we

Cp,

F 'x F " — F " — Kp + l — Dp + I. conditions

are

and an epimorphism

F" — Kp + I — Bp to a morphism with an epimorphism

the composition

1.5. Dq,

free

Let Dp = F ' x F ", cí>p the map induced

by p - 1.

Proposition

I

A is noetherian,

we may find

free A-module

F ' — Cp.

modules

1

We lift the composition

obtain

replaced

rows.

Thus

/7 ' be a finite

We lift this

1

ßp — Zp +l —»//*+1(C*)—»0

ring,

D' —> C

isomorphisms

HqiD ') —» HqÍC ').

for every

isomorphisms

Hq(D " ®^ Ai) — W9(C ' ®^ M).

D ', C ' complexes

a morphism

of flat A-

of complexes

A-module


inducing

M, cp ' ® 1 induces

344

J.J.WAVRIK Proof.

We define

a complex

L' by

Lq= Cq~l © Dq,

We easily HqiC

check that

[— l\) = Hq~

*

\

dj

0

\

We have

an exact

so an exact

sequence

cohomology

C '[- l] is a complex of complexes

plex

Hq+ ÍC'[-

by ef>'.

Thus

constructed

exact

lY) with HqiCA,

ep ' induces

in this

the map 8 is identified

isomorphisms

iff

for all

q.

with the map

Hq{L ') = 0 for all

q.

induction,

By hypothesis, is true

The result

sheaf

22re have

Thus

the induced

mor-

is flat

0 —> Zq~

Since

for all

—» Lq~

Lm + 2 = 0, Zm + 1 = Lm + 1 is flat.

q.

—> Zq —» 0 is exact

0 —>Zq~l

for all q. By flatness,

®^ M —> Lq~l

®^ M —> Zq

®A M —>

follows. we have

1.6.

on X flat

modules,

Zq

for the sequence

To summarize, Theorem

L " ®. M.

Let

The com-

iff 0 —>Z?" ' 2>A M —> Lq~ ' %A M — Zq ®A M —►0 is

Now L« is flat for all q.

By descending

0.

way for ç6 ' ®A I is

are isomorphisms

the same

l]

sequence

Zq = Ker dqL, HqiL ") = 0 iff 0 — Zq~ l — L«" ' — Z« — 0 is exact. phisms

with

0 —» C'[-

Hqic'[- I]) -» Hq(LA -, Hq(D) -L, F/S+Hc't- 1]) -» • • • .

If we identify induced

(~dC

¿L o dL = 0. If Cq[- l] = Cq~\

(C')•

—> L ' —►D ' —> 0 and

-,

d, =

Let

n: X —> S be a flat

over

S, A = Cfo

.

Dq = 0 for

q > m, such

that for each

a natural

Then

proper

there

morphism,

o £ S, A a coherent

is a complex

finite

D ' of finite

morphism

free

A-

g: {T, o) —' fo, o)

isomorphism

Hq{D ®AB) — lim HHn'HU), A'), where

A

is the pullback

neighborhoods

2.

= X x 4(n) —>A("_n and so obtain

345

—» 0

of complexes

0 — D' ®a m"/m*+1 -♦ d' ®^ Am —>D* ®^ A("~n—*0 and therefore

an exact

->HqiD Thus its

cohomology

sequence

% AM)-*HqiD

ç"R_ j £HqiD-

image

®A A(n_1))

(zj(fn_j)

the obstruction

®4 A("~n)-W/«+1(D'

eH9 +Hü'

may be lifted to rj"*(D ' ®A A(n)) if and only if

®^ m"/m"

to extending

®a m"/m" +1)— ••-.

Ç

+ 1) vanishes.

Thus

«(£

) represents

..

Now Hq + 1ÍD ■®A m"/m" + 1) is naturally isomorphic to Hq*KXQ, c.fq®c m"/m" + 1) which

is a direct

Proposition

Hq + l(X , £ o

o

sum of copies

2.1.

The nth obstruction

®m"/m"

Corollary.

of Hq + 1ÍX , Î

).

We have

to extending

£

proved

£ HqÍX

, 3" ) lies

in

+ !).

// Hq + 1ÍX , J

) = 0,

'¿ere

are rzo obstructions

to extending

¿;

£

HqiXo , Ï o ). We will now show tions

exist

to extending

Proposition

,.

many obstruc-

A ).

X, S, A, q as above,

M

A

Since

space ¿

over

C.

AL is finite

This

means,

can be extended Formal

no obstructions,

by-step

9, only finitely

there

is a positive

72

is a finite

A(r!)-module,

If 72— > m, the morphism r

integer

n

íF("o)) then it can be

extension

the power

series

will

that

show

dimensional,

however,

that

to AI72 for all deformations it can

give

if ff

there has

dimen-

of AL0 and we have

is an 7zo such £ ALV which

o

a finite

that

hence

A1n A

Mn = A1„o for

can be extended

to Al„ o

tz.

be extended

is analogous

case

there anyJ ç

hence

M7Z —►M77Z is A-linear,

Let M72 = Im (/VI AITZ ' is a subspace 72 —>AL). 0 r

n > n . — o

3.

00 ,

A and

£ HqÍXo, 5F ) can be extended to HqÍX(n°\

HqiD ' ®. A(n))=

vector

C-linear. AI '

With

X/S,

to HqÍX0 £ HqÍX

"

2.2.

such that if tf extended

that

actual

deformations.

to ¿f

£ HqÍX^n\

to determining

is no guarantee

no obstructions

a power

If ¿j A^n))

series

of convergence. (i.e.

£ HqÍXa,

tot all

can be formally


A ) has

tz.

This

step-

term-by-term.

In

In this

we

section

extended)

then

it

346

J. J. WAVRIK

also

can be actually

extended

to a fo

Hqin

1ÍU), A) fot some neighborhood

U

3 o.

Let bra,

A be an analytic

cp induces

is given

by a matrix

is a morphism, its

image

eft: Am —» A".

then

r/>ß is given

by the matrix

K: Ker cpß —> Ker epc.

fo(aO).

and

cp

If K: B —> C by k

Ker cp .

K(ß)

of analytic

= |zs e Ker çS„| k = a\.

A-algebras

Proof.

a = fo j, • • • • , tz^) e Ker eV C Cm.

Let

Let

;

functor

of sets.

K is representable.

A is an analytic

a1, to Cjxi-

K is a covariant

to the category

3.1.

Since

A-alge-

morphism,

If k £ Ker c6ß we denote

Proposition

(a.).

If B is an analytic

If a: A —* B is the structure

a e Ker ç6_ and set

from the category

Lift

(aO

K induces

in

Let

C-algebra,

epß: Bm —>Bn.

C-algebra,

we have

R = C,x; z .,■■■, 1

z

777

Let

cp be given

A = Cjx .,••••,

!/(/.;

S a'fo.

' Z'

7

Z

by the matrix

x A/if

+ a.)).

, • ■ ■ , f ).

R is an analytic

Z

7

A-algebra. We have cSRfoT)=S a'fo. + a.) = 0 where fo (fo • •• , fm) e Rm, rf = z. + «¿. Thus that

ik ,,•■■, 1

z

z

^ß

K.

Let

fo- zz, so fo: K(r).

B be any analytic

A-algebra,

k m ) £ Bm.

A\z,,..., 1 Cjx;

f e Ker cSR. Moreover,

(R, ¿f ) represents

k z. - a z. £ m B so there is a unique ~i S —> B which carries z . to k. - a . (here

772

morphism r

ZZZ^

, ■ ■• , z

(/(/•))•

Since

k £ Ker ci„

this

We will now show k £ K{b),

k =

of A-algebras &

A{z ,,•■•, 1

z 777S =

a unique

morphism

induces

/: R

such that R(/)fo) = ¿. Let

D ' be a complex of finite free A-modules,

let tf

6 1Y9(D' ®A C).

We

set

FiB) = KetiDq ®A B —>Dq +1 ®A B), F, G ate

covariant

are ß-modules, a surjective Gip„)iK)

functors

and we have ß-module

on the category From

Proof.

—* F(C)

of F(C). GçoiA{n))/

r¡ \, F^

A-algebras.

3-1 it follows

is a finite

so S = u~ KC)fo is A-linear 7777

hence

ß.

and

F7)q

hence 72

A-algebras;

F(ß),

u: F—> G such

that

Let map.

G^

G^Q(ß) If rj

u: F^

G(ß)

uiB)

is

= \K £ G'(ß)| £ u{C)~

are also

u induces that

{¿¡o) we set

covariant

functors

—* G¿fQ.

is representable

for any

r¡o.

for all n, then Gç ÍA) / 0.

dimensional

Let S = l n S. S 0,

for each

3.2. // G^ iA{n)) / 0

Fíe)

of analytic

transformation

B —> C is the natural

FÍpR)ic¡)=

of analytic

G(C) is C-linear, FiA{n))

pß:

Proposition

Proposition

a natural

homomorphism

= ¿; \ where

F-n (ß) = fo eF(ß)|

on the category

G(ß) = Hq{D ®A B).

vector

space

) is an affine C-linear,

so

over

variety /

C and ztaC): Fie)

in F(C).

FipA(„)):

= lm{FÍpAin)))

is an affine variety and S D S J

7777

is a subspace

..By

+ 1

'

hypothesis,

Sn / 0- From [9, Lemma 2.1 ] it follows that D Sn / 0-


>

DEFORMING COHOMOLOGY CLASSES Thus we can find 77 euiO~lig 'o Lemma algebras

3.3.

Lez

) suchthat

F be a covariant

to the category

is surjective

o

of sets.

Fv'¡o ÍA(n)) i 0

functor

Let

347 for all n.

from the category

ÍR, rf ) be such

that

of analytic

Horn.

for all B. Then FÍB^n^) / 0 for all n implies

We have A-

ÍR, B) —> FÍB)

FÍlí) 4 0-

Proof. If FÍB(n))4 0 then Hom^R, B(n)) 4 0- Let f{n) eHomAÍR, B{n)). By a generalization

of a result

of M. Artin

we can find an / £ HomAÍR, ß). Now with

sentable. is what

r¡

we wanted

Hom^R,

GA

This

of Proposition

representable?

appears

3.2 can

A useful

us with

kernel,

2.4],

B) / 0 so FÍB) /- 0-

F v ÍA*-"') 4 0

fot all

I.e.,

to be the case

is there

in many

n.

But

a universal

examples.

F^

is repre-

family

If true,

of exten-

the proof

be simplified.

criterion

for no obstructions.

We have

a restriction

Hq(n~ ill), A) —>HqÍX , A ) fot any open neighborhood vides

in [9, Theorem

to prove.

Is

of ç ?

4.

Thus

above,

we proved

Hence, by the lemma, F ^ ÍA) 4 0- But then also G¿¡ ÍA) 4 0i which

Question.

sions

as chosen

which

a restriction

mapr

map

U of o in S. This pro-

Rqn *ÂA) o —►Hq(X o , A o )■ m • RqnAA) *

o

is in the

so we obtain

tq: RqnAcJ) /m . Rqnf'A) -* HqÍXo , f o ). O * O * o If tqo is surjective. '

reduced, appears

then

any' class

in

Hq{\ o , J o ) is unobstructed. If C 3, e o is we obtain a useful criterion for tqo to be bijective. This criterion first ' ' in the paper of Grauert L3J-»a complete proof is given by Riemenschneider

in [71, Theorem A a coherent If dimr

tq~ o

HqÍX

4.1.

Let

n: X —►S be a flat

sheaf

on

X flat

over

, A ) is independent

proper

morphism

S, o £ S a fixed of s in some

point,

of analytic and

neighborhood

spaces,

let C „

be reduced.

of o then

¡qo and

are bijective. '

Proof.

See [7, (3.5)]-

Corollary

1.

// dim„

HqiX

, A ) is independent

of s

in a neighborhood

of

o, anyz ¿^ o £ HqÍX o , A o ) is unobstructed.

Corollary

2.

// dimr

Hq + lÍX

, J

) is independent

of s in a neighborhood

of' o, any* B £ HqÍX o , A o ) is unobstructed. Bo Remarks.

1.

Hq(X{n\

'fin))

borhood

U of o.

"t?3

tot all

o

is unobstructed" n or, equivalently,

means

that

to £ € IIqiv"

c;o mayJ be extended l ((■'). if)


for some

to $~ n £ neigh-

348

J. J. WAVRIK 2.

The proof

of Theorem

4.1 is independent

of the "formal-to-actual"

machine-

ry of § 3. The converses section

show.

identifying

groups

In Theorem Using

of the two corollaries

We will

containing

the notation

Tq{M)

an A-module

(a)

in the next is useful

in

A = Cfo -modules. S, o

modifications,

since

we introduce

we

the co-

by Tq{M) = Hq{D ' ®. M). homomorphism r

with tqQ defined

4.2 ([5, III. 7.3.1])-

free

functor)

of A-modules

and we have

—>T*{M). If M = C, z