Symposium on several complex variables, Park City, Utah, 1970 (Lecture notes in mathematics, 184) [184] 0387053700, 9780387053707

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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich

184

Symposium on Several Complex Variables, Park City, Utah, 1970

Edited by R. M. Brooks, University of Utah

Springer-Verlag Berlin· Heidelberg· New York 1971

ISBN Springer-Verlag Berlin . Heidelberg . New York ISBN 0-387-05370-0 Springer-Verlag New York . Heidelberg . Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number 76-153464. Printed in Germany.

Offsetdruck: Julius Beltz, Weinheim/Bergstr.

PREFACE This volume contains articles based on talks given at the Symposium on Several Complex Variables held at Park City, Utah, March 30 - April 3, 1970, under the sponsorship of the University of Utah.

The papers herein represent a broad spectrum of mathematical

research (e.g., function algebras, sheaf theory, differential operators, manifolds) but are related by the fact that they are all related to some degree to the area of

complex variables.

On behalf of the organizing committee, I wish to thank the contributors for their cooperation and Springer-Verlag for their willingness to publish these

We also thank the

University of Utah for their generous support of the project. I wish to acknowledge responsibility for any mistakes the reader may find in the manuscript, since the authors did not have the opportunity to proofread their contributions. Finally, I wish to thank Joyce Kiser for her remarkable typing of the manuscript.

R. M. Brooks University of Utah

ORGANIZING COMMITTEE R. M. Brooks, University of Utah

E. A. Pedersen, University of Utah

H. Rossi, Brandeis University

J. L. Taylor, University of Utah

INVITED SPEAKERS F. T. Birtel, Tulane University Eugenio Calabi, University of Pennsylvania D. D. Clayton, Louisiana State University Michael Freeman, Rice University T. W. Gamelin, University of California, Los Angeles Reese Harvey, Rice University Eva Kallin, Brown University J. J. Kohn, Princeton University R. O. Kujala, Tulane University Andrew Markoe, University of Wisconsin L. Nirenberg, Courant Institute, New York University R. Remmert, University of Maryland (visiting) (unable to attend the conference) Hugo Rossi, Brandeis University Bernard Shiffman, Massachusetts Institute of Technology Yum-Tong Siu, University of Notre Dame D. C. Spencer, Princeton University Wilhelm Stoll, University of Notre Dame J. L. Taylor, University of Utah R. O. Wells, Jr., Rice University John Wermer, Brown University (unable to attend the conference)

TABLE OF CONTENTS

SOME ANALYTIC FUNCTION ALGEBRAS.. •••• ••••••••

••

1

A LOCAL CHARACTERIZATION OF ANALYTIC STRUCTURE IN A COMMUTATIVE BANACH ALGEBRA. • • •• • • • • • • •• •• • •• •• •• •• •• •••• • • • • • • •• •• • • • • ••

10

A DIFFERENTIAL VERSION OF A THEOREM OF MERGELYAN •••••••••••••••••

37

POLYNOMIAL APPROXIMATION ON THIN SETS

50

ON AN EXAMPLE OF STOLZENBERG.....................................

79

FLAT DIFFERENTIAL OPERATORS......................................

85

F. Birtel and W. Zame

D. D. Clayton

Michael Freeman T. W. Gamelin

John Wermer

D. C. Spencer

FIBER INTEGRATION AND SOME APPLICATIONS •• • • • • • • • • • • • • • • • • • • • • • • • • 109

Wilhelm Stoll

PARAMETRIZING THE COMPACT SUBMANIFOLDS OF A PERIOD MATRIX DOMAIN BY A STEIN MANIFOLD•••••••••••••••••• , •••••••• 121

R. O. Wells, Jr.

GENERALIZATIONS OF GRAUERT'S DIRECT IMAGE THEOREM •• • • • • • • • • • • • • • • 151

Yum-Tong Siu

COHOMOLOGY OF ANALYTIC FAMILIES OF DIFFERENTIAL COMPLEXES•.•••••• 175

Andrew Markoe

FAMILIES OF STRONGLY PSEUDOCONVEX MANIFOLDS •••••••••••••••••••••• 182

Andrew Markoe and Hugo Rossi

EXTENDING ANALYTIC SUBVARIETIES •••••••••••••••••••••••••••••• • • • • 208

Bernard Shiffman

ON ALGEBRAIC DIVISORS IN

Robert O. Kujala

k C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . 223

PROBLEMS••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 231

SOME ANALYTIC FUNCTION ALGEBRAS by F. Birtel* and W. Zame

1.

Let

X be a compact subset of

tn.

By

P(X) , R(X) , and

respectively, we denote the uniform closures in nomials

POlY(X), the continuous rational functions

the holomorphic functions convex subset

Hol(X)

X is a rational (polynomial) set.

AR(X) = X(AP(X) = X)

where

of the polyRat(X) , and

Every rationally (polynomially)

(polynomially) convex set we mean a compact set that

C(X)

H(X),

By a rationally X with the property

A denotes the complex homo-

morphism space; by a rational (polynomial) set we mean a countable intersection of open rational (polynomial) polyhedra.

These facts

together with the Oka­Weil theorem imply: P(X)

= R(X)

iff

(1)

AP(X) = AR(X)

Less well­known but equally valid is the following equivalence: R(X)

= H(X)

iff

AR(X)

= AH(X)

(2)

.

Equivalences (1) and (2) suggest a general question, which is the primary motivation of the ensuing discussion. Question: P(X)

A

algebras

B

Let H(X)

A and

A and

B be function algebras on

and with

AA

AB.

X­ with

Does it follow that the

B coincide?

Before addressing ourselves directly to this question, it will be profitable to consider the function algebra * Partially supported by NSF GP 13064.

H(X)

in greater

2

detail, because its perversities, unlike the agreeable behavior of P(X)

and

R(X) • provide informative clues to the difficulties

surely to be encountered in the general question.

(See [1.2.5. 9,

10] . )

2.

The homomorphism space of

H(X)

is not always identifiable with Worse is true.

point evaluations on some subset of

X is holomorphically convex. the converse

every holomorphic set

Bjork provides this counterexample:

does not hold. X

(;2 :

[(z.w) 2 C

U [(z.w)

Although

:

Iwl

Iz I

, Iz I

=

0

0

• Iwl

Let

1) 1) U K

where K

with K

n

[ (z , w)

(;2

However. containing

X there are a unique smallest compact holo-

morphically convex set

b(X) • called the barrier of

unique smallest compact holomorphic set Nebenhuelle of

X.

few facts about

X. and a

n(X) , called the

For orientation and future references we list a

X, n(X)

and

b(X)

and about their associated

analytic function algebras. (a)

X

(b)

H(bX)

algebras in (c)

of

b(X) and

n(X) ; b(X) H(nX)

and

n(X)

and

X is.

are isometrically isomorphic to function

H(X) ; identifying, we have

int n(X)

are connected, if

int b(X)

H(n(X))

H(b(X))

H(X)

are Stein submanifolds of

(d) The closures of a relatively compact Stein submanifold n Cn is not always holomorphically convex, although n with smooth

3 boundary is; e.g., [(z,w)

1[:2:

0
a. = t+>a.

on

Wk n Wj

By "unique" we mean that V a. , To prove that t+> = t+>a. on Wk n Wj consider a.

the coordinate functions by part c); in

W.

consequently

Hence

in

Wk

into

t+>i

and

,

t+>i

.

We have that

Yo(t+>i) =

is zero in some neighborhood of zero

is identically zero on its domain of definition,

26 In other words, if

Hence k ;;; j

then

is an extension of

j

then

extends

k

4.4.

Definition.

to

Wk ' and conversely, if

to

aj

Let

be the set of all

n

a

in

from

property that there is a holomorphic mapping

with the into

V

satisfying Lemma 4.3. Observe that

aj

4.5.

If

Lemma.

mapping of n

= h

Wk

e ne

morphic on

V

into

f

and

h (f)

on

4.6.

Let

et k

*f ne

then

By Lemma 4.3,

y

is holo-

o (qJ n *f) =

.

be in

aj ,

and let

Y

y

=

=

tr

n U l3

We need

)

such that

eyqly

on

Wk

n U l3 , we have that each coordinate

qln agrees with one of the coordinate functions of either By using the coordinate functions of

we can extend

qly

holomorphic mapping tained in

s

and

n

or qls

c

is a directed subset of Q .

For each

In other words, since function of

nl3

V

At any rate there is a Y is in Q j By Lemma 4.6, qJn = nyqly and qlS

to show that Y is in

j.J

e , then

it follows that

=

(f) .

=




and that

S

Further­

are not distinct. fi(p)

m of the then

of

to

Fof, hence by

By using the same argument as above we may assume that and

r

f(p)

Cn .

orhood of zero in

is holomorphic in a ne

I- j , for

taking

be the translation on

t

i

em

into

en

=

0

f. ' s

are

is in

a..

l

such that

is holomorphic, hence by

32 definition

Yp«Fot)oa)

is in

Theorem. hp

there

a continuous mapping

W into qJ(O)

b)

qJ*f

c)

hp

A

qJ

is holomorphic on Yo(qJ*f)

=

.

h(f)

Yo(qJ*f)

f

A

W'

eh(f)

f(qJ(O))

=

Since

the local ring

Sp

maximal ideal of

from

in

f(p)

a

0

a ,

Sp'

eh p

a

of zero

f

in

in

Sp

A

from

A

(W)

into

by

W into

X

qJ

such that W'

for every

I f we apply the evaluation mapping

f

A.

in

ehpYp(f)

e

In particular

for all

f

Specifically we

A.

Hence

qJ(O)

=

Define the homomorphism

(fl,···,f n) by

1.3).

Observe that

(above).

in

f'{p ) = 0;

i f and only if

0

in

Ho(W)

from

function on

WI

is therefore a nontrivial homomorphism of

Hence

f(qJ(O))

say

into

(1) = 1

Ho(W)

into a field, its kernel must be the unique

=

(Definition eh Y

h

for all

ndo

into

is holomorphic on

qJ*f

for every

therefore,

a

Yp(f)

of zero in

and such that

ehpYp(l) = 1 .

an

Sp

W' for every

for every

Part b) is proved.

have that

W be as in Theorem 4.12.

By Theorem 4 .12 there is a continuous mapping

from some neighborhood

we get

Sp'

p

-,«:

.

is in

such

hpYp(f)

in

Yp(Fof)

from some neighborhood

Define the homomorphism h(f)

and

nontrivial homomorphism of

X

a)

Therefore

x,

B ,

Suppose

in

is

Sp

P

Fix h

a

hayo(F)

By Theorem W into

h

a.

is nontrivial since

2.8 there is a holomorphic mapping

Cn

We also have that

such that

and

h Y (F) a 0

h a. Y0 (rSi ) -- h PYP (N*rsi ) U

33 neighborhood of zero in

= hnyo(F)

hpYp(n*F)

we conclude that

W.

yo(F)

in

=

Since

n

yo(G*F)

hpYp(f)

The advantage

Hence for any

=

for every

we have was arbitrary

Yp(f)

in

Sp

of Theorem 5.3 over Theorem 4.12 is that under

reasonable algebraic conditions on the homomorphism mine topological properties of the mapping

w'

hp

we can deter-

For example Theorem

5.5 and Corollary 5.6. The following lemma essentially appears in [3] as Proposition 3.l. We shall state without proof a slight improvement of this proposition. However, the proof as it appears in [3] can be used verbatum (With, of course, notational adjustments) to prove the generalization.

5.4.

E

Lemma.

the maximal ideal space of

be a closed subset fixed point in

Y

Suppose

(in the relative

Let

be a commutative Banach

f

on

is

let

E

x

Y be a

mapping from some neighborhood

Y) of

such that: For each

a

E

ne ighb orhood

there is f(x)

function

such that

ga

defined in some

Yx(;!Y)

=

Yx(

f),

Gelfand transform of a in

5.5. Ho (W)

hp

Theorem.

If

h

is an injection

some neigh­

nontrivial homomorphism of

hp

such that

W

5.3.

W" W"

W'

is an injection

neighborhood retract of

X.

is a

GB

an isomorphism and

neighborhood morphism

into

Sp

W'

and If

be

f

p

of zero

b)

then

the derived mapping given by

and let a)

Y

Y •

of zero in onto

(p}

p

X,

there is a a homeo­

W such that

neighborhood of

into

Proof. n

V .

in Let

(a)

Let

A(D)

Recall that

D

W

be a closed polydisc centered at zero with

[10].

is

D

WI

of zero in

The proof that W into

X

D. cp

fixed point

x

that

and A(D)

is an injection of some neighborhood

5.4

of Lemma

Y = Wn D ,

E = A(D) , closed subspace f

zero, and continuous mapp

thesis of Lemma

D

U U

The maximal ideal space of

is a direct

applied to the Banach algebra

cp.

=

The main hypo-

5.4 follows from Theorem 5.3.c, and the hypothesis

is an epimorphism. ,

restricted to

denote the i-th coordinate function on

WI'

Choose

that

h Y (

each

i

,· .. ,f

continuous in some neighborhood of

n

p

such

P P

W'

This implies there is a neighborhood

WI ' =

D

be the Banach algebra of functions continuous on

holomorphic on the interior of

f

is a subvariety of some open set

such that

(f l , · · . ,

Hence

isomorphism) and that

is a retract of

.

is a

that

is the identity mapping on

We have that

homomorphism on

By Theorem 5.3,

B.

cp*a

a neighborhood of zero in locally-A function. cp*f*

W'

Sp

f-l(WI)

X.

W'

Let us denote the

Notice that Let

is holomorphic on

consequence of the definition of

.§:

In particular,

and

cp*f*

It is an immediate

W'

Yp(f:F)

V

be any element of

that for any

we have

W' , where

is a subset of

cp(WI)

Ho (W)

W

is a monomorphism (hence an

p

set in

G5

f

is the ident

V

[p}

h

domain of focp

by

of zero in

is the identity function on

Suppose, in addition, that

(b)

W'

F

holomorphic in

is the germ of a is in

is the identity homomorphism we have by Theorem

Sp

Since

5.3 that

A

Since

h p Y0

is an isomorphism we have

From our assumption that Lemma of

5.4 implies that

p.

Let

VII = V'

qJof

[p}

is a

G 5

is an injection on some neighborhood

n f-l(qJ -l(V') n W') , and let

V'

35

W"

w' n

=

cp-l(V")

Obviously W"

cp(W")

is a homeomorphism of

Then

is a subset of

is a little more involved.

W' , which implies that focp(f(x))

injection on

V'

injection on

V'

x

hence

is also contained in

V'

cpof

subset of Hence

cp

f ocp W'

V" Since

and

cp-l(V")

f(x)

f'{x )

W'

hence x

f(V")

focp(x)

cpof

,

and

(Observe that V'

we have that f

is an is an

f .)

for any

x

Hence

in

f(x) W".

W"

is the identity function on

is a homeomorphism of

W"

onto

Note

In

V"

Since

V"

n

cp -1 (V")

is a subset of

is the identity function on

we know that

Then

V'

cpof(x)

x

=

is a subset of

f(V" )

is an injection on

that we have also shown that other words

x

V'

cpof(x)

since

that

Let

cpof(x)

But

f(x)

=

V"

V" •

onto

is a W" .

V" .

The following corollary appears in [3], although the proof is somewhat different than the development given here. 5.6.

Corollary.

homomorphism of

Suppose onto

W onto cp(O)

=

b)

cp*f

is holomorphic on

Proof. some integer

radical.

Suppose that n.

Then

hYO(F)

o

p

=

in P

and

h

is a nontrivial

subvariety

W, con-

and a homeomorphism

cp

such that

W for every

=

Yp (f)n

Hence

o

hYo(F)

f

Yp(f) =

Yp (f

f

in

A

and that n)

fn

and

is zero in

is zero in some neighborhood of

Hence the kernel of

h

is equal to its

From the Nullstellensatz [7, p. 90] we have that

modulo the kernel of W.

G 5

P

some neighborhood of p , therefore

U

neighborhood of

a)

is a

Then there is

Sp

taining zero, of some open set from

(p}

In particular,

h

is isomorphic to

Ho(W)

for some subvariety

Sp

is isomorphic to

Ho(W)

The rest of the

proof is an obvious consequence of Theorems 5.3 and 5.5.b applied to

36

the isomorphism

h- l

from

Sp

onto

Ho(W) .

BIBLIOGRAPHY 1.

Arens, R., The problem of locally-A functions in a commutative Banach algebra, TranS7 Amer. Math. Soc., lor-(1962), 24-36.

2.

Arens, R. F., and Calderon, A. P., Analftic functions of several Banach algebra elements, Ann. Math. 2) 62,

3.

Clayton, D., Local analytic structure in Banach algebras (to appear) .

4.

Dugundji, J., Topology, Allyn and Bacon, Inc., Boston, Mass. (19 67 ) .

5.

Edwards, R., Functional Analysis, Holt, Rinehart and Winston, Inc., New York, New York (1965).

6.

Gleason, A., Finitely generated ideals in Math. Mech., (1) 13 (1964), 125-132.--

7.

Gunning, R., and Rossi, H., Analytic Functions of Several Complex Variables, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1965).

8.

Hormander, L., An Introduction to Complex Analysis in Several Variables, D-.-Van Nostrand Inc., Princeton,-Wew Jersey (19 66 ) .

9.

Narasimhan, R., Introduction to the Theory of Analytic Spaces, Lecture Notes in Math. 25,-Springer-Verlag, New York, Inc., New York, New York (1966).

10.

Rickart, C., General Theory of Banach Algebras, D. Van Nostrand Co., Inc., Princeton, New-Yersey (1960).

11.

RUdin, W., Function Theory in Polydiscs, W. A. Benjamin, Inc., New York, New York

12.

Shilov, G., On decomposition of a commutative normed ring in a direct sum of ideals, (N.S.) 32(74) (in Also in Amer. Math. Soc. Transl., (2) 1(1955), 37-48.

13.

Zariski, 0., and Samuel, P., Commutative Algebra I, D. Van Nostrand Co., Inc., Princeton, New Jersey (1958).

Louisiana State University Baton Rouge, Louisiana

algebras, J.

A DIFFERENTIAL VERSION OF A THEOREM OF MERGELYAN by Mi.cnae L Freeman*

INTRODUCTION

1.

Suppose D

f

is a real-valued continuous function on a closed disk

C , let

in

C(D)

valued functions on with identity of

be the Banach algebra of all continuous complexD

C(D)

and consider the uniformly closed subalgebra generated by

This subalgebra is denoted on

D

f

z.

and the identity function

[z,f], and comprises all uniform limits

of functions of the form

(1. 1)

with complex coefficients

a

kt

It is easy to see that if any one of the level sets La = (z of

f

V

of

separates

C - L

a

If:, then

D:

E

f(z) = a}

[z,f]'; C(D).

is clearly contained in

For any bounded component

D, and on

oV

each function

(1.1) becomes n L:

k=O a polynomial in

z

n (L:

t=O

a kt

t

a )z

k

,

with the indicated constant coefficients.

shows that any function

g

in

[z,f]

is a uniform limit on

* This work was supported by NSF Grant GP-8997.

This

oV

of a

sequence of polynomials in

z

By the maximum principle, this

sequence also converges on

V

i. e.

V

of a holomorphic function on elementary arguments that

p.

g

is the "boundary value"

It follows by any of a number of

C(D).

[z,f]

yield the same conclusion if any points

g I oV

must be holomorphic.

La

Similar reasoning will

has interior points, for at such

[5, Theorem 1.5,

On the other hand, in

27], Mergelyan proved the following result. Theorem 1.

If

f

is a continuous

such that for each

13,

is connected, then

[z,f]

L

S

function on points and

is

C(D)

Mergelyan's original result goes further to describe

L

the level sets

have interior points.

S

make use of his famous theorem mation by polynomials in

z

very much on the fact that

when

f

=

Z ,

C

takes real values. [z,f]

C(D)

They also depend Now it is of great when

f

f

is a

For example, a positive answer is obtained

by the Weierstrass approximation theorem.

general functions

when

All proofs of his result

on compact sets in f

[z,f]

[2, Theorem 4.3.1] on uniform approxi-

interest to determine conditions for complex-valued function.

D

For more

however, the problem is difficult.

It is the purpose of this lecture to announce and discuss Theorem 2, a result similar to Mergelyan's in which a differentiable function is permitted to take complex values in such a manner that its image is still appropriately thin in Theorem 2.

If

f

_i_s

_tw __ i_c_e

valued function on a neighborhood

2(a) where

C . differentiable complex-

U

origin in

C

such that

f(z)

Q

is

real valued quadratic form with non-zero eigenvalues of

opposite sign, and

39 f

2(b) then

o

;§l

has

D

exists a closed D c U

and

[z,fJ

=

in

U ,

of positive radius such that

C(D)

The condition 2 (b) re t'e r-s to

f

U c lR

as a map of

2

and simply means that the ordinary Jacobian determinant of vanishes on

U.

By Sard's theorem,

property of thinness in

C.

f(D)

2 lR

into f

will have measure zero, a

Using only this information about

f(D)

and some straightforward topological facts about the level sets of it is relatively easy to show

[3, Theorem 1.3J that

C(D)

exhausted by uniform limits of rational combinations of To show

r4 J

f,

is z

and

f.

the same for the;Jolynomial combinations (1. 1) requires a

considerably more detailed ane.Ly s Ls of

f(D) , as indicated below in

the sketch of the proof. Now Theorem 1 implies Theorem 2 when then

f

f

has real values, because

has a non-degenerate critical point at

p. 4 J •

the Lemma of Morse

coordinates

(S,11)

near

0

of index 1

0

[6,

[6, Lemma 2.2, p. 6], there are such that

f

=

2

S

-

2

11

This shows that

the hypotheses of Theorem 1 are satisfied on SUfficiently small disks D , from which

[z,fJ

=

C(D)

Of course a real-valued function auto-

matically satisfies the rank condition 2(b).

On the other hand, while

Theorem 2 does not imply Theorem 1, it is similar enough that it may be of interest that its proof in [4J is independent of Theorem 1. An immediate consequence of [z,fJ over

=

C(D)

is that the graph of

D M

=

[(z,f(z)): z

D} ,

is polynomially convex [2, p. 40J in

(:2.

This property is not in

general sufficient for

(let

f = 1 , for example), but

[z,fJ

C(D)

it turns out to be in certain situations applicable to a proof of Theorem 2.

The proof will employ the following result of J. Wermer.

f

40 Theorem 3. closed disk in borhood of

D

3(a)

M

3(b)

E

then

(Wermer If

C •

[8, Theorem in Appendix]) f

Let

D

be a

is continuously differentiable in

neigh-

and such that [(z,f(z)): zED}

is polynomially convex, and

o}

[ z , f] '" C(D)

has Lebesgue measure zero in

•

In showing that a function satisfying the hypotheses of Theorem 2 also satisfies those of Theorem 3, the hard part is to prove 3(a), that

M

is polynomially convex.

The property of polynomial convexity of a compact set in and more generally, a description of its polynomially c'Onvex hull, has great importance in function theory.

The quoted results are offered

in support of this contention, and there are many others.

The problem

has recently received attention [1] in the case of a small disk in a real submanifold ditions on disk.

M

of

C2•

The idea is to find differential con-

M to enable a computation of the polynomial hull of the

In this connection, the conditions of Theorem 2 are sufficient,

but not necessary.

See also Remark 3.2.

Results like Theorem 2 occupy a more central position in this problem than might at first be suspected.

This is because of what is

already known [9, Theorem 2] about the cases where first-order effects predominate, and also because of an elementary biholomorphic coordinate change due to Bishop [1, p.

5] which reduces all the other cases

to ones in which the manifold can be described locally as the graph of a smooth function

f

not constrained, and dition 2(b).

of the form 2(a). f

The eigenvalues of

Q are

is not necessarily subject to the rank con-

It is therefore these additional features of Theorem 2

which are special in the problem of describing the polynomially convex hull of a small disk in

M.

41 2.

SKETCH OF PROOF OF THEOREM 2 To use Theorem 3, i t must be verified that a function satisfying

the hypotheses of Theorem 2 has a polynomially convex graph over some small disk

D, and that the Bet

E

has measure zero.

The latter condition is easy to verify in general. Jacobian determinant

Jf

of

f

Note that the

is

Since this is assumed to vanish throughout a neighborhood it is clear that when of/oz

and

of/oz

D c U,

vanish.

E

U

of

is exactly the set on which both

has already been pointed out that

It

0

0

is a non-degenerate and therefore [6, Corollary 2.3] isolated critical point of Ref.

Thus, after shrinking

U

of/oz will vanish simultaneously on small disks

D,

E

nomially convex if

[OJ

i f necessary,

U only at

O.

of/oz

and

Therefore, for

and it remains to show that

M is poly-

D is small enough.

The complete proof of this in [4] is cluttered with technical details and will not be reproduced here. given for two special cases.

Instead, proofs will be

The first is very simple, providing an

opportunity to introduce the notations and machinery of the proof and exhibit its basic ideas.

The second case is designed to embody most

of the real difficulties which may be encountered in general, so that its proof will give a fairly accurate impression of the complete argument in [4]. Case 1.

f(z)

case, the graph convex. (2.1)

z

M of

f

=

x + iy

See Figure 1.

over any closed disk

It must be shown that i f Ip(o.,e)1

:§

( 0.,

sup Ip I M

S)

D

In this

is polynomially

is a point in

(:2

such that

42

Re w

f(D)

fl

o

z-plane Figure 1 The support of

must be contained in

M

n v;l(S)

.

for every polynomial

p

algebra homomorphism P(M)

of the Banach algebra

)..

extends to an onto

P(M)

,

C

where

is the uniform closure on M of all polynomials in two variables

(z,w).

It is well-known [2, p. 81] that there is a probability

measure

on

M which represents

(2.2)

)..(g)

for all

g

in

valent to

P(M)

(a"I3) C(M)

an isomorphism of

C(D)

and because in 2.2

g

.

This is because g .... g

(the map with

0

"1

[z,f]

,

=

with

which carries

C(M)

"l(z,w)

jections

"1

=

z

onto

,

P(M) ),

In the argument which follows,

M by examining the support of

on the

z-

and

will be and

w- planes by the coordinate proThese measures have standard

(z,w) .... w

and

defini tions:

(2.3)

J

for all Borel sets

E.

=

)),

J

j

1,2,

E

Letting

consequence is that

(2.4 )

is

[(a,S)}

narrowed down through its relations with the measures induced from

is equi-

C(D)

[z,f]

is a

u

can then be taken as a pointwise approximation

must have small support.

is placed in

in the sense that

( a" 13) E M as asserted, then

to the characteristic function of Thus

)..

=

Now if

P(M)

unit point mass at

(a"I3)

p .... p ( a" 13)

(2.1) implies that the map

The bound

( cr., 13 ) EM.

in two variables, then

support u c

-1 (support u·) J J

tr .

A similar calculation will show that

(2.5) From the well-known property that

, a particular

44

(2.6) for all bounded measurable functions side of (2.6) with P(f(D)) f(D)

j

2

=

of polynomials in

w.

Re w-axis.

be found a polynomial

p

in

f(D)

from which it appears that

A2(P)

=

p(a)

w

> sup

Ip(a)1

P(f(D))

Moreover,

as an interval in the

p

Thus

supported on

homomorphism

ee

of

clearly represented by

5

e, such representing measures on

f(D) , or else there can

would violate

f(D)

P(f(D))

is polynomially convex,

Ipl

(2.1).

Since

A

it follows from (2.2) and (2.6) that

for each polynomial

measure

is the uniform closure on

f(D)

5

of

such that

0

(a,a) ,

evaluates polynomials at

A2

defines an algebra homomorphism

C, where of course

onto

g, it follows that the left

p

in

w.

(2.5),

by

In other words the represents the evaluation

defined b y e .

But

ee

the unit point mass at f(D)

e

is also f(D).

Since

are unique (by the Weierstrass This

approximation theorem for example), it follows that fact and (2.4) show that (2.7)

support

(See Figure

c f(z,w):

f(z)

51

M

n

-l( e)

1.)

Now consider

by

(2.5)

and hence for any polynomial

q

and (2.7) it follows that

in

z

(2.8)

But from the definition of

f

it is clear that each of its level sets

La

is polynomially convex.

a

f(a) =

(a,s)

Thus

This and (2.8) imply that

L

a

M , completing the proof for Case 1.

S

, or

The experienced reader will be chafing under the suspic ion that this argument is too circuitous.

It is true that there are several

places in it where alternative and simpler routes may be taken.

In

Case 1, measures may be dispensed with entirely, if one notes instead that

M

= {(z,w): z

Re z2 = Re w,

D,

polynomial convexity follows easily. that

[z,f]

and

1m w

O}, from which

There are even direct proofs

= C(D) which do not first establish polynomial convexity.

The reader is invited to consider such alternatives and try them on the following example. Case 2. x

"§

Y

f(z) + i exp (

Here the imaginary part of

f

is a

the rank condition 2(b) is satisfied.

2-\ 2) s i nx(-sh ) ,

(x -y )

COO

x

>y

function of its real part, so Of course, condition 2(a) is

clear. The argument in Case 1 depends on the polynomial convexity of the image and level sets of

f.

In Case 2,

f(D)

no longer has this

property but i t turns out that one can successfully press an indirect argument involving representing measures by taking a careful account of the connectivity properties of the image and certain more general "level" sets of

f.

Figure 2 shows how the image of a circle f(D).

As a point

p

C

in Figure 2(a) traverses

in C

D exhausts in a counter-

clockwise direction beginning in the first quadrant of the z-plane at the point marked the left on the

0, its image

f(p)

in Figure 2(b) moves from

Re w-axis, arriving at point

1

as

preaches

0

to

46

(a)

(b) Figure 2

Showing how the image of correspond under

f

C

exhausts

f(D)

in Case 2.

Points which

carry the same number.

(a)

(b) Figure 3

A bounded complementary component oV

=

f(L l

U L 2)

•

V

of

f(D)

and its boundary

47 point

1

in Figure 2(a).

0

path, reaching

as

other three "b r anche s

The image point

p tl

f(D)

f(p)

The support of

then retraces its

0

arrives at the second point marked

of

similarly described by

f(p)

indicated in Figure 2 (b) are

as

p

continues around the circle.

is affected as before by the support of

but in a more complicated way, with each of the branches of participating.

The

f(D)

It will be shown that their influences are independent

enough that the same type of argument is still feasible. Of course,

f(D)

has many bounded complementary components, so

it is far from being polynomially convex. f(D)

However, the union

of

with its bounded complementary components is a compact set with

connected complement, hence polynomially convex.

E.

reasoning as before will show that showing that 13

E

e

f(D) .

Thus the same

The difficulty lies in

At this point, it is quite conceivable that

lies in some bounded complementary component

any case, it is clear that complementary component As before,

)..2

defined by that

S

V

of

f(D).

is in the closure of some such bounded

V, and it will be argued that

evaluates polynomials in f(D) c E.

supported on

In

w

Since

at

13

oV

and is

E , it follows

13

represents the evaluation homomorphism

13

e

P(E).

acting on

13

also acts on the (up to an obvious identification) larger

P(V),

and here a standard construction involving the ordinary

maximum principle and elementary functional analysis will provide a real measure acting on

supported on

(J

P(V) ,

supported on

hence on

f(D)

=

oE

P(E).

follows that

P(E) u2

=

(J

which also represents

Thus

r a [2, P • 207 J on

This proves that support

From the special form of

P(E)

[2, Theorem 3.4.14] (or what is the same,

is a Dirichlet

as in Case 1, that support

eS

is a real measure

which annihilates all functions in

By the Walsh-Lebesgue theorem the fact that

oV c f(D)

c

oV

c f-l(oV) f,

it can be concluded that

f ( D) )

it

and hence,

48 for this set does not separate

C, and the same com-

putation as (2.8) can be made with the supremum taken this time over f-l(oV). algebra Ll

Thus P(

\..Ll

represents evaluation at Moreover

( oV) )

(oV)

(l

acting on the

has exactly two components

and

L2, each coming from one of the two branches of contribute to oV, as indicated in Figure 3. Suppose that

L2

(l

.

is in the algebra

that

Since (by Runge's theorem, for example)

the characteristic function f-l(oV)

f(D)

X

relative to

of

P(L l U

)

,

Ll U

1

But

\..Ll

is a probability measure, so this shows that support Thus

\. L

must be supported on that part of

over"

L2 , by (2.4). Moreover, support \..L2 c f(L 2 ) by (2.5).

P

Therefore if

maps this set onto

is any polynomial in

f(L 2)

and this shows it contains

5

f(D)

representing

follows that

3.

(l

e

S

Thus

13 ' which implies that

f(L 2) , so

w,

Furthermore, as an arc in the w-plane a

M which "lies

is polynomially convex,

is another real measure on As before, it Q.E.D.

La .

REMARKS 3.1.

The function of Case 2 incorporates all the worst topo-

logical features that will be encountered in any function satisfies the hypotheses of Theorem 2.

f

which

A detailed proof of the

Theorem [4J is in its basic structure much the same as in Case 2, but is burdened frequently with digressions to verify the topological

properties of 3.2.

f.

The same techniques can be used to study the polynomially

convex hull of a manifold

M presented as a graph like 2(a) where

Q

is "elliptic", having non-zero eigenvalues of the same sign, and still retaining the assumption that rank that the polynomial hull of

M near

f

1 0

near

o.

It is found [4]

is a three-manifold with

boundary

M which is foliated by a one-parameter family of disjoint analytic disks in c 2 whose boundary curves exhaust M. This verifies in the situation here a conjecture made earlier by Bishop [1, p. 12].

REFERENCES

[1]

Bishop, E., Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1-22. --

[2]

Browder, A., Introduction to function algebras, W. A. Benjamin, Lnc , , New York (1969). --

[3]

Freeman, M., Local holomorphic convexity of two manifold in to appear in Rice University Studies, Summer 1970.

[4 ]

, The polynomial hull of

-.,.p-r-e-p-a-r-a"""'t'""'i..-o-n-.

(;2,

thin two-manifold, in

[5 ]

Mergelyan, S. N., Uniform approximation to functions of variable, Amer. Math. Soc. Translations 101 (1954).

[6]

Milnor, J., Morse N. J. (1963)-.----

[7 ]

Nirenberg, R., and R. O. Wells, Jr., Holomorphic approximation on real submanifolds of a complex manifold, Bull. Amer. Math. Soc.-,378-381--. -

[8]

Wermer, J., Polynomial1y convex disks, Math. Ann. 6-10.

[9]

Ann. of Math. Studies 51, Princeton,

__, Approximation on

Rice University Houston, Texas

complex

(1965) ,

disk, Math. Ann. 155 (1964),

POLYNOMIAL APPROXIMATION ON THIN SETS by

T. W. Gamelin

The first section is devoted to a brief survey of results concerning polynomial and rational approximation on compact planar sets and on thin subsets of

tn.

The remaining sections are devoted to an

exposition of the uniform-algebraic principles which lead to the recent approximation theorems of Alexander and Bjork.

Survey of Results Let closure in let

K

be a compact subset of C(K)

en

let

be the uniform

of the polynomials in the coordinates

be the uniform closure in

C(K)

which are analytic in a neighborhood of

zl •... 'zn;

of the rational functions

K

and let

be the

subalgebra of C(K) of functions which are analytic on the interior O K of K. The inclusions P(K)

A(K)

are all evident, and has no interior.

R(K)

A(K)

C(K)

coincides with

C(K)

if and only if

K

We are interested in questions of the following

nature.

Main problem: guarantee that

f

Give conditions on a function belongs to

Subsidiary problem: P(K)

=

A(K),

or that

P(K),

or to

Give conditions on R(K)

f

E

C(K)

which

R(K) K

which guarantee that

A(K) .

The classical Weierstrass theorem solves our problems whenever lies on the real axis in

VI.

In this case, every function in

C(K)

K

51

can be approximated uniformly on

K

by polynomials in

z

More

generally, the stone-Weierstrass theorem shows that these algebras all coincide with

C(K)

whenever

coordinate axes

K

is a compact subset of the real

in

In the special case that

1

K

, a great deal of progress has

been made on these problems during the past several decades.

Some of

the milestones in the development of the theory were the following theorems, in which

K

compact subset of

Hartogs-Rosenthal Theorem R(K)

(1931):

(1936):

Lavrentiev's Theorem

P(K)

no interior and the complement of Keldysh's Theorem (

has zero area, then

KG

is dense in

plement of

K

K

K, then

P(K)

z

K

=

=

R(K) , then

): R(K)

Vitushkin's Theorem R(K)

C(K)

(ii )

y(D)

y(D\K)

=

is connected,

if and only if the com-

K

0

0--0

on which f

C(K) f

is such that

is uniformly

R(K) . K

is a peak

The following are equivalent:

D

for every bounded open set 2

>

0

is the analytic capac ity, and with radius

f

If

C(K) .

lim sup y(i:l(z;O)\K)

z

K

If almost every point of

(1959):

( i)

centered at

A(K)

(1958):

orhood in

Bishop's Theorem (

y

0

for almost all i:l(z;o)

z

K

is the open disc

The "almost all" refers here, and in

Bishop's Theorem, to area measure. Vitushkin's Theorem

(i)

has

is connected.

has a ne

(iii )

K

A(K) •

approximable by rational functions, then

point for

i f and only i f

is connected.

P(K)

Bishop's Localization Theorem every

C(K)

=

If the complement of

):

Mergelyan's Theorem (1951):

Here

K

C(K) .

=

and

If

R(K)

=

(1966):

A(K)

The following are equivalent:

52 y(D\K) (iii)

Here

a

For each

z

for every bounded open set

bK • there exists

r

1

such that

denotes the continuous analytic capacity.

It turns out that Vitushkin's characterization of those which

D.

R(K)

=

A(K)

K

for

in terms of analytic capacity is reasonably tract-

able. and it provides a satisfactory answer to our original subsidiary problem.

It can be used to give purely geometric criteria for rational

approximation of a fairly general nature.

For instance. Koebe's 1/4-

theorem shows that the analytic capacity of a continuum is comparable to its diameter.

The resulting estimate then shows that condition

(iii) of the preceding theorem is valid at any

z

the closure of some single component of

We deduce that

A(K)

whenever each point of

ponent of

It\K

bK

bK

which lies in =

lies in the closure of a single com-

Even this simple corollary of Vitushkin's theorem a proof.

had resisted successfully earlier attempts at discove The inner boundary of a compact bK

R(K)

K

It

consists of the points of

which do not lie on the boundary of a component of

C\K.

One of

the outstanding problems in this area is to show that if the inner boundary of a compact set

K

is sufficiently small. then

R(K) = A(K).

We have already indicated that this is true when the inner boundary of K

is empty.

boundary of

It can be shown that K

compact set

11::.

Is K

=

A(K)

whenever the inner

lies on a countable union of twice continuously differ-

entiable arcs in Problem:

R(K)

R(K) C

One open question in this area is the following. A(K)

whenever the inner boundary of the

has zero continuous analytic capacity?

Solving problems of this sort via Vitushkin's theorem leads immediately to difficult questions concerning subadditivity properties of analytic capacity. Vitushkin also has provided us with a solution to our main

53 problem for the algebra

R(K) .

Vitushkin's Individual Theorem: a(5)

>0

satisfy

(i)

there exists

a(5)

IHf(Z)

0

as

r

1

5

Let O.

If either

\I

+ 00

whenever

g

(ii)

is

r

1

I S f(z)dz I bS(zO;5) S

and

a(6)

II

f

)Y(A(zO;r5)\K) 00

or if

A(ZO;5)

such that a(5)y(S(zO;r5)\K)

is a square of center

Conversely, if

II

-function supported by a disc

there exists

whenever

C(S2) , and let

such that 5a(5)(\\

dxdy ]

f

5 ,.

then

f

R(K).

R(K) , then (i) and (11) are true, with

r

1

and side

a fixed multiple of the modulus of continuity

w(f;6) of f.

It turns out that this characterization is not so tractable, for those sets

K for which

R(K) I A(K).

For instance, the theorem

does not seem to answer immediately the following open question which has been circulating recently in Scandinavia: that

f2

R(K),

then is

f

R(K)?

If

f

C(K)

is such

One of the principles of uni-

form algebra theory is that one should try to attribute the defect in C(X)

of a uniform algebra to the analyticity (somewhere) of the

functions in algebra.

Mergelyan1s theorem can be regarded as a state-

ment to this effect. The defect of parts of in

K

R(K)

in

C(K)

can be attributed to the Gleason

R(K) , which can be regarded as a form of analytic structure This leads us to the following question,

which would perhaps

provide a more satisfying answer to our main problem. Problem:

For

K a compact subset of

define a tractable

notion of analyticity for functions defined on the Gleason parts of R(K) , so that

R(K)

consists of precisely the functions in

Which are analytic on each Gleason part.

C(K)

54 One area in rational approximation theory which has seen some development recently, by A. M. Davie, John Garnett and the author, is the study of pointwise bounded approximation by the algebras and

R(K)

A(K) , and the problem of uniform approximation by the real parts

of the functions in these algebras.

The classical theorems related to

these problems are the Farrell-Rubel-Shields Theorem on pointwise bounded approximation by polynomials, and the Walsh-Lebesgue Theorem on uniform approximation by harmonic polynomials in

x

and

y.

A

theory parallel to Vitushkin's has been developed to handle pointwise bounded approximation by

A(K)

In this case, though, Vitushkin's

constructive techniques do not suffice at present to yield the end results, but the functional-analytic tools of uniform algebra theory also playa crucial role. functions in

R(K)

The pointwise bounded approximation by

is still not understood, and hopefully an answer

to the preceding problem would also help describe the pointwise bounded limits of the functions in

R(K)

Now we return to a compact subset convex hull of hull of

K

and

will be denoted by

will be denoted by

subsets of P(K)

K

K

K.

K

of

and the rational convex

The sets

=

K

and

K

are compact

which can be regarded as the maximal ideal spaces of R(K)

respectively.

It is always true that

The Oka-Weil approximation theorem shows that R(K)

The polynomial

K

K

if and only if

P(K) . C2

The situation even for polynomial approximation in drastically more complex than in

becomes

Three items which indicate

(1.

impending difficulties are the following. Bishop-Hoffman Theorem: doubly generated, so that in

(C2

If

R(K)

=

K

I:

(C

P(K')

is compact, then

R(K)

for some compact subset

In other words, polynomial approximation in

q;2

is K'

is at least

55 as complicated as rational approximation in Stolzenberg's Example [21J: such that

K

fl.

There is a compact subset

K , while the projection of

axis has no interior.

K of

A

K onto each coordinate A

In particular, the set

K\K

contains no

analytic structure.

of

Cole's Example: There is a compact polynomially convex subset 2 C such that P(K) f C(K) while every point of K is a peak

point for

P(K).

K

This shows that Bishop's Theorem fails in several

complex variables.

It also shows that if one wishes to find some

generalized form of analytic ity to account for the defect of

P(K)

in

C(K) , then one must look beyond the notion of Gleason parts. While the general picture in

en

is bleak, there are several

special cases which have been treated with varying degrees of success. We will be interested in one such case, in which striking progress has been made, the case in Which

K

is appropriately thin, say a finite

union of arcs with some smoothness assumptions.

The type of theorem

which has been proved is of the following form. Metatheorem 1: A

thin, then

K\K

variety of

Cn\K

If the compact set

K

It n

in

is sUfficiently

is a (possibly empty) one-dimensional analytic sub-

With a theorem of this sort in hand, one can derive results on uniform approximation by impos Theorem (Stolzenberg [22 J ): such that

R(K)

C(K)

analytic subvariety of The hypothesis that

additional hypotheses on

i{l(K,Z)

H (K,Z)

and

0

Cn\K , then "1

=

is a compact subset of

K

If

K

The hypothesis that

[;n

is a one-dimensional P(K)

=

C(K)

is equivalent to requiring that

0

R(K)

K\K

K , and

=

every continuous nonvanishing function on logarithm.

K.

=

K have a continuous

C(K)

is automatically satis-

fied for all classes of compact sets to which Metatheorem 1 applies. The proof of the theorem is Short, so we give some details. Sketch of the proof of theorem:

Let

Zo

Cn\K.

Using the

56

rational convexity of f(ZO) = 0 , while

K, one can find a polynomial

f

does not vanish on

tinuous logarithm on K\K. Zo

¢

obtain

This shows that P(K)

Then

f

such that has a con-

K, and hence off a compact subset of the variety

By the argument principle, K

K.

f

= R(K) =

=

K

f

K.

cannot vanish on

K\K, so that

From the Oka­Weil Theorem we

C(K)

The pioneering work in the direction of Metatheorem 1 was accomplished by Wermer [24], [25],

[26].

He proved that

dimensional analytic variety whenever Jordan curve in singular on

K

such that one of the coordinate functions is non­

K.

P(K)

=

Riemann surface

is a one­

is a simple closed analytic

Under these hypotheses Wermer moreover gave a

reasonably complete description of the algebra either

K\K

C(K) SUbS

or else

K

P(K).

He showed that

is obtained from a finite bordered

by identifying a finite number of pairs of

points and introducing a finite number of singularities, so that is the sUbalgebra of continuous functions on analytic on

SUbS

P(K)

which are

S, which identify the appropriate points, and which have

the appropriate singularities. L. A. Markusevic [17] pushed through Wermer's methods, showing that

K\K

is an analytic variety whenever

K

is a simple closed

Jordan curve such that the coordinate functions satisfy certain smooth­ ness conditions weaker than continuous differentiability, and such that one of the coordinate functions separates all but a finite number of pairs of points of

K.

Wermer's work was extended by Bishop

[5], [6] and Royden [18],

and later by Stolzenberg [21], so that by 1965 the state of knowledge was reflected by Stolzenberg's version of the theorem, which was apparently also known to Bishop: tinuously differentiable arcs in variety.

If

K

is a finite union of con­

, then

is a one­dimensional

However, if the arcs are required only to be smooth rather

than analytic, the variety

K\K

may be hooked up to

K

in a quite

57 complicated manner, and we do not have a satisfactory description of the algebra

P(K)

It should be mentioned that Bishop [5] and Royden [18] obtained independently results related to Wermer's Theorem which, together with an earlier paper of Bishop [4], solved the following problem: compact subset

K of an open Riemann surface

of functions on hood of

Given a

S, and an algebra

B

K, each of which extends analytically to a neighbor-

K, describe the uniform closure of

B

in

C(K)

Later

Bishop [6] attacked a more general problem of introducing analytic structure into the maximal ideal space of an algebra of analytic functions, reworking the theory so as to obtain the Cartan-Oka-Thullen Theorem in the case of analytic functions of several complex variables, and Wermer's theory in the case of one complex variable.

Expositions

of Bishop's techniques from [6] to handle these cases respectively are found in Gunning and Rossi [16] and Wermer [27]. For several years the matter lay dormant, until a recent (19681969) flurry of activity initiated by H. S. Shapiro.

He observed that

Wermer's maximality theorem could be used to prove the following: A

is a uniform algebra on the closed unit interval

I , and if there

is a function in

A which separates all but the two endpoints of

then

He asked, for a starter, whether the theorem was

A

=

C(I).

If

I,

true providing the function separates all but one pair of points, not necessarily the endpoints, of

I.

Intermediate results were obtained

by Bjork [7], and later by the author [13] and Shapiro and Shields [20].

Alexander [1], and independently Bjork [9], then came across a

device which had the effect of lifting certain smoothness restrictions on the curves in question, and which allowed them to obtain the following theorem. Alexander-Bjork Theorem:

If

K

is a (not necessarily disjoint) finite union of compact Jordan arcs in t n such that one of the coordinate functions separates all but a finite number of pairs of points

58

of

A

K\K

K, then

is a one-dimensional analytic subvariety of

rn\K.

Using the same device, and some technical preparatory work, Alexander [3] was able to prove the following theorem, which shows that the smoothness assumption in Stolzenberg's Theorem can be replaced by requiring only that the arcs be rectifiable. Alexander's Theorem:

Let

K

be a compact subset of

which

lies on a connected set of finite one-dimensional Hausdorff measure. A

K/K

Then,

is a one-dimensional analytic sUbvariety of

The proofs of these theorems utilize the abstract techniques of uniform algebra theory.

In fact,

this sequence of theorems represents

one of the most significant contributions which that discipline has to offer.

The remainder of this presentation will be devoted to giving

an account of the current state of the uniform algebra theory which leads to these approximation theorems.

We have singled out four main

principles which embody the meat of the theory. a sampling of applications.

To these are appended

This development is completely parallel

to the treatment given by Bishop [5], except that certain of his smoothness hypotheses have been lifted, and less elementary techniques Rossi's local maximum modulus principle) are used.

A more

recent exposition along these lines has been given by Bjork [8], [9]. The forthcoming monograph of Wermer [27] also includes an elegant exposition of this circle of ideas, concentrating primarily on analytic arcs.

Notation and Conventions By

A

we will always denote a uniform algebra on the compact

Hausdorff space C(X)

X, that is,

A

is a uniformly closed subalgebra of

which separates the points of

stants.

The maximal ideal space of

X, and which contains the conA

is denoted by

MA, and the

59 Shilov boundary of

A

by

Then

0A'

0A

is a closed subset of

X,

X can be regarded as a closed subset of the compact space

while

The functions in functions on

A

M A extend, via the Gelfand transform, to continuous

MA, and the extensions form a uniform algebra on

which we will identify with

M A, In particular, we will always regard

A.

each

as a continuous function on

set

will denote the set of

f(p)

E

so that if

U

X !)

iC ,

the

such that

U .

We will freely use theorems from the book Uniform Algebras [12],

v.8.2",

and theorems from that source will be referred to here as "UA, for instance. Not all of the theorems which are stated will be proved. only sketches of proofs are given.

Usually

When a sketch of a proof is

attempted, though, the steps which are omitted are always of a routine nature and are straightforward to complete.

The First Principle Let f-l(U)

f EA.

An open connected subset

of

iC

is f-regular if

is homeomorphic to a (possibly empty) one-dimensional analytic

variety, such that the functions in In view of the connectedness of an integer

m

,

f

U

A

,

become analytic on

f-l(U)

over

of the ramification points of

the variety

f-l(U) , then

A. E U\S , there is a

is a disjoint union of

disc m

mapped homeomorphically by

S

,

f U

f'

.

such that If

S

centered at

analytic discs in f

onto

one, then

f

If

is a homeomorphism of

is an

is the image

MA

U

If

A., such that

f-l(

1::,)

each of which is

1::,.

Note that every connected open set disjoint from regular, of mUltiplicity zero.

f

and the singUlar points of

is a discrete subset of c U\S

1::,

f-l(U) •

i t is easy to see that there is

called the multiplicity of

m-sheeted analytic cover of under

U

U

f(MA)

is f-

is f-regular of mUltiplicity

f-l(U)

and

U, and every function

60 in

A

is an analytic function of the coordinate

f

on

The first principle is Lemma 17 of Bishop (5J. is f-regular, then

f-l(V)

cannot meet

f-l(V) •

Note that if

V

Consequently the

statement of the principle makes sense. Theorem 1: V V ,

Let

is f-regular. then

f

A , and suppose the connected open set

If

V is the component of

containing

f-regular.

V

Before proceeding to the proof, we wish to clarify one of the ingredients.

A function

h

is A-holomorphic at

be approximated uniformly on some neighborhood of A.

For example, if

g

A , then

l/(g-AO)

MA if h can by functions in

x

is A-holomorphic at each

If the continuous function holomorphic at each point of Let

x

h

on

MA is Ais A-holomorphic.

MA, we say that h B be the uniform algebra on MA generated by

holomorphic function

h.

A and an A-

It follows from the local maximum modulus

principle (cf. VA, 111.8.2) that

0A'

The theorem we wish to use

is Richart's Theorem (cf. VA, 111.9.2) asserting that also

MA •

By throwing in one function at a time, and using transfinite we arrive at the follOWing version of the theorem, Which is convenient for our purposes. Ric kart's The orem: a uniform algebra

B

Let

A

Then there is -------

uniform algebra.

on

B

A ,

Moreover,

and every B-holomorphic function B

can be chosen so that the functions in

analytic structure in

Replacing

remain analytic

MA on which the functions in

Proof of Theorem 1: author (cf. Bjork (8]).

B

A

any analytic.

The proof we will give was discovered by the A similar proof was found by Wermer (27].

A by the larger algebra described in Rickart's

Theorem, we can and will assume that every A-holomorphic function in

61

C(MA)

belongs to Choose

AO

A. U , and a disc

60

centered at

AO' such that

f- l(6

is the union of m analytic discs in MA, and 0) local coordinates on each disc in Suppose g

f

gives

vanishes on f-l(AO)' Since g/(f- AO) is analytic on each disc in f- l(6 0) g/(f-AO) is locally approximable by polynomials in f on each disc in

On the other hand,

by polynomials in Consequently shows that

f

which vanish on

is a A-holomorphic, and

f-l(AO)'

A

,

MA\f-l(Ao)

g(f-AO)

A.

coincides with the ideal of functions in

finite codimension in For

is locally approximable

on a neighborhood of each point of

g/(f- AO) (f-AO)A

l/(f- AO)

A

In particular, the ideal

(f-Ao)A

This A has

A.

define a continuous linear operator

T

A

on

A

by

setting

The operators

T move continuously with A 0A ' then maximum modulus at X o

A.

If

h

A

attains its

so that

Consequently if

A

¢

, then the range of

is closed, and A By the stability theory for semi-Fredholm

T

T

has zero null space. A operators (cf. Gokhberg and Krein [15]) the codimension of the ranges of the

T

A

is constant (finite or infinite) on each component of Hence

(f-A)A

has finite codimension in

Now each maximal ideal generated, by

f - f(x)

span the ideal modulo

x

in

f-l(V)

A

for all

A

is algebraically finitely

and any other finite set of functions which [f - f(x)]A.

By Gleason's embedding theorem

V.

62 (UA, VI.6.1), every

x

f-l(V)

has a neighborhood which can be given

the structure of an analytic variety, such that each of the functions in

A becomes analytic on the variety.

MA is a linearly independent set, when regarded as a family of functionals oO'A. So if A

V , then there are a finite number of maximal ideals containing

(f-A)A. on

That is,

f-l(V).

dimensional.

f-l(V)

assumes each value

A

V

It follows that the variety near Since

and the variety near x

f

Now

¢

x x

0A'

x

only finitely often

x

is at most one-

cannot be an isolated point of

cannot have dimension zero.

Hence each

lies on a one-dimensional analytic variety.

together the analytic structures we find that one-dimensional analytic variety.

f-l(V)

MA,

By patching is indeed a

That completes the proof.

The Second Principle The second basic principle is a criterion for f-regularity, which corresponds to Bishop's Lemma 20 of [S] and the proof of Lemma 13 of [6] .

Theorem 2: dary

bU

Let

U be a domain in the complex plane whose boun-

is a simple closed Jordan

and let

E be a subset of

bU which has positive harmonic measure (With respect to f

A.

If

f-l(U)

MA\X , and if

f-l(A)

U).

Let

is a finite subset of

MA for .all A E , then U is f-regular. If, moreover, the number of points in each f-l(A), A E , does not exceed m, then the multiplicity of

f

over

U does not exceed

----

m.

The proof of the principle depends on Wermer's maximality theorem (UA, II.S.l), asserting that i f complex plane, then

P(6)

6

is the closed unit disc in the

is a maximal closed sUbalgebra of

C(b6)

As a mechanism for introducing analytic structure, Wermer's maximality theorem is used in the form of the following lemma, which could also be deduced from an earlier result of Rudin [19].

63 Lemma:

r

that

Let

be an open subset f-l(U)

is such that

A

------

to-one onto

r

U

------

and every

U

g

Let

B

be a closed disc contained in

U ,

.

and let

A

B on

q

which contains the

By the local maximum modulus principle, the functions in

as a subalgebra of B

U

of the functions in

attain their maximum modulus on

that

f-l(U)

on

is a closed subalgebra of

polynomials. B

f-l(U)

f

Suppose

f-l(U) one-

maps

f

homeomorphism of

is

be the uniform closure in Then

and

MA \X

an analytic function of

A

Proof:

the complex plane.

C(bc;'o)'

b C;,O ' so that

can be regarded

Wermer's maximality theorem then shows

, so that the functions in

=

B

B

are all analytic on

C;,O ' as required. Proof of Theorem 2: proof of Lemma

13 of [6] .

f(MA)

If

We follow the argument used by Bishop in the

does not meet

U, then we are done, since

regular of mUltiplicity zero. meets

U

Since

we see then that

First we replace The new

U

;;
" E which is suffic iently near to l f- ( >,, ) will consist of precisely one point in each Yj By

applying the case placed by onto

f-l(U)

Hence

V

f-l(V) n Y.

V , we find that

V, and that

f

Yj , with U reis mapped one-to-one by f

to the closure of

1

m

J

A on

is an analytic coordinate on

is f-regular of multiplicity

m.

f-l(V)

n Yj

.

That does it.

The Third Principle The third principle deals with extending the analytic structure lying above an f-regular component over

bU.

U

to points lying in the fibers

The third principle corresponds to Bishop's "triangle

lemma", Lemma 19 of [5].

It was at this point that smoothness assump-

tions had to be imposed, so that each point of from a triangle lying in

U

bU would be accessible

The main contribution of Alexander and

Bjork was to circumvent this restriction. Theorem 3.

f

A.

connected) subset regular. P

>"0

does

then there is

U be an open (not necessarily

, such that each component

Suppose

f- l ( >,, )

Let

bU

is

peak point for

R(f(MA)\U)

belong to the A-convex hull of open neighborhood of

A are analytic.

Moreover,

of the variety

f-

If

in

has the struc-

is not constant

functions in the branches

p

Note that when peak point for

f

is

h-l(>"O) n X ,

MA analytic variety, on which

ture of

p

U

U

is connected, then every point of

bU

R(f(MA)\U) , so that Theorem 3 applies to all

is a >"0

bU.

To prove Theorem 3 it will be necessary to first prove a lemma which will be improved upon considerably in the next section. Lemma:

Let

r

a Jordan arc in the complex

and suppose

66 that

U and

V are connected open neighbor-

hoods of that

respective sides

U

f-regular, while

not meet

V.

Suppose

W is

maps

V

is f-regular.

W.

V

Then

f

V U r , or if U U r , then

f

maps

W homeomorphically W n f-l(V) .

is an analytic coordinate on

f

is connected, the local maximum modulus principle shows that

is A-convex. on

If

Suppose for instance that

V Ur

Since

does

W homeomorphically onto

Proof:

such

open subset

W homeomorphically onto

f

onto

f(X)

f-l(r) n X

that maps

r

Applying the first principle to the uniform closure is a one-dimensional

MA\W, we find that

analytic variety.

Hence, so is

Proof of Theorem 3:

Our line of proof will follow the proof of

Theorems 3 and 4 of Bjork [9J. disc

Let

,U2'· ..

t.j

centered at

be the components of Uj

Aj

such that

Choose a small open

U

t.j c U.J

a union of a finite number of analytic discs in uniform algebra on

its action on Suppose. q

MB

'=

MA\U

is

MA. Let B be the A and the functions

MA\U f-l(t. j) generated by Since every homomorphism of

A,

, and f-l(t. j)

B

is determined by

(t. j ) .

lies in the B-convex hull of X. Then O) there is a representing measure for q supported by X. The measure

-1

f-l(A

will represent

Since

is carried by

R(f(MA)\U)

Since

the point mass at q

AO AO

,

AO

f(X)

on the algebra

,

it

R(f(MA)\ U t.j

will in fact represent

is a peak point for that algebra and

is carried by

belongs to the A-convex hull of

f-l(AO) n X

f-l(AO) n X

)

AO -1

on is

, that is,

I t follows that

p

67 does not lie in the satisfying

Ig(p)1

B-convex hull of

>

1

and




is g-regular with respect to the algebra

In particular,

g(p)

is g-regular with respect to

MB I , and hence in

has a neighborhood in

dimensional analytic variety.

B' , so that

MA , which is a one-

That does it.

Almost Separating Functions As a corollary of the three basic principles, we obtain: Theorem

4:

complex plane.

Let

f

E

A ,

Suppose that

let U

and

T

V

be

is

f-regular and

--

f-regular, and variety on Proof: each fiber

meet

f-l(A) n X

f-l(U U

in the

be connected open neighbor-

hoods of the respective sides

-

open Jordan

f(X).

If

finite for

r u V)\X

the functions

is A

U

is

one-dimensional analytic are analytic.

Theorem 3 there are only a finite number of points in (A), Since

A E

r , which do

f-l(A)

n

X

not lie in the A-convex hull of

is finite, it is already A-convex.

68 l f - (,) 1\

Hence

A

is finite for all

we can assume that

bV

an open subinterval.

r.

By shrinking

V

and

r,

is a closed Jordan curve which contains

Then

According to Theorem 2,

V

r

r

has positive harmonic measure for

is then f-regular.

as V

f-l(U U r u V)\X

That

is an analytic variety follows from Theorem 3. The Alexander-Bjork Theorem cited earlier is an immediate consequence of the preceding theorem, if we take for function

zl

in

4;n.

f

the coordinate

With a little more effort, we obtain the fol-

lowing version of the Alexander-Bjork Theorem, which stems from Theorem 1 of [2J or [lJ. Theorem 5:

Let

Y _b_e _t_he_ _ u_n_i_o_n _of_

number of copies of

B be a uniformly closed subalgebra

the closed

interval, and let

of

contains the constants (but

C(Y)

separate the

of

countable

E

hood of each

of

of

of

Y such that Y\E

f

Suppose there f

M:s \Y

Then

If, moreover,

B

B

and an at most

one-to-one in can

neighbor-

gi ven the structure the functions on

analytic variety, on

become analytic. B =

Y).

not necessarily

separates

points of

B

Y, then

C(Y) . Proof:

Here we define

MB\Y

to consist of the homomorphisms in

MB which are not evaluation homomorphisms at points of Y. We may as well take E to be the complement of the set of points Which have neighborhoods on which closed.

We can write

Y\E

f

is one-to-one.

Theorem,

l' J

IS

R(f(

follows that

such that f r1'12"'" cluster only on E as j

) ) = C(f(r j)) , R(f(Y) ) = C(f(Y))

Note that if

F

E

is

as a (not necessarily disjoint) union of

closed intervals and the

Then

j

« 1

is one-to-one on each 00

By Lavrentiev1s

and

R(f(E)) = C(f(E))

It

is any finite union of compact Jordan arcs in

the plane, then there is a totally disconnected subset

Fa

of

1j ,

F

69 such that the arcs passing through each in a neighborhood of noting that the

A

f(r j)

A

E

F\F O actually coincide

Applying this remark to the

f(r j) , and

cluster on the at most countable closed set

f(E) , we see that there is a subset

J

of

with the following

t

properties: J

( i) (ii )

in

f(Y)

A E f(Y)\J

If

such that

f-l(r)

that

is compact and totally disconnected.

is a Jordan arc passing through

T'

(regarded as a subset of

ber of disjoint intervals on phically by

f

onto

(iii)

A

If

B

, then A can be joined to

Y

f(Y)

E

identified by X, so that

B.

Y by identifying

We can regard the functions

B becomes a uniform algebra on (A) n X

X.

is finite for all

Using property (iii) and Theorem 3, we find that

f(X)\f(E).

every component of every point of

by a

as in (ii).

Moreover, assertion (ii) shows that A

00

in only a finite number of points, each of

f(Y)

as defined on

and such

Y ) consists of a finite num-

X be the compact space obtained from

any two points of

,

A

Y, each of which is mapped homeomor-

C\f(Y)

E

which lies on an arc of Now let

A

of

r.

curve which crosses

in

, then there is a neighborhood r

C\f(X)

f(X)

is f-regular.

is a peak point for

can be invoked again, to show that analytic variety.

Since

MB\X

R(f(X))

R(f(X)). =

MB\Y

=

C(f(X)) ,

Hence Theorem 3

is a one-dimensional

That takes care of the first assertion of the

theorem. The second assertion follows from the first assertion and the argument of Stolzenberg, cited earlier in the survey, as follows. Since Y is a union of disjoint intervals, the topological condition VI H (Y,Z) = 0 is satisfied, and the argument principle shows that Since of

R(f(Y))

C(f(Y))

A must be contained in a fiber

every maximal set of antisymmetry f-I(A)

Now maximal sets of

antisYmmetry containing more than one point must be uncountable.

70 Since the level sets of antisymmetry of

A

f

are at most countable, each maximal set of

reduces to a point, and

A = C(Y).

That completes

the proof. Corollary:

Let

Y

a.::..::..:.:.:::..::.::.. union of compact Jordan Let

closed curves on a Riemann subalgebra of

C(Y)

containing

separating the points of such that

f

Y).

extends to

B

be

and

uniformly closed

constants (but not necessarily Suppose there is

==::L.::-='::'

is not constant on any

function

in

ne ighborhood of

Y.

Then

f

B

Y, and

f

is a one-

dimensional analytic variety.

The three basic prine

s we have developed so far can also be

used to handle Stolzenberg's Theorem on smooth arcs and Alexander's Theorem on sets of finite

In each case, one must obtain first

more information on the behavior of the coordinate functions on the set in question.

In the case of smooth arcs in

one can use

Sard's Theorem to set the problem up so that the third principle will apply.

In the case of sets of finite length, one must perform a more

intricate analysis.

In that case, one does not cross over open seg-

ments on arcs, but one must use Theorem 2 to cross over a set of positive length (and hence of positive harmonic measure) on a rectifiable arc. To see how sharp the hypotheses in Alexander's Theorem are, let's

c 3 which are not

consider the arcs constructed by Wermer [23] in polynomially convex.

Wermer started with an arc

r

in the complex

plane, such that there exists a continuous nonconstant function S2

the Riemann sphere ized so that

f(oo)

the three functions points of

S2.

which is analytic off

0

and if

f,

zf,

The mapping

and

Zo

¢ r

r·

satisfies

[f-f(zO) ]/[z-zo]

If

f

f

on

is normal-

f(zo) I 0

,

then

separate the

71

z - (f(z),zf(z),[f(z)-f(zO)J/[Z-zOJ)

r

embeds

as an arc in

r.

vex hull of

iC 3

, and it embeds

r

In particular,

S2

in the rational con-

is neither rationally convex nor

polynomially convex.

r

Now suppose we take

to be an arc of positive area, and

f

to

be the convolution

Then

f

is continuous, and the modulus of continuity of W(f;5) = 0(5 log

t)

as

f

5 .... 0 •

This is sufficient to ensure that the embedded image of

r

has (2+ )-dimensional Hausdorff measure zero, for all

>

On the other hand, if

K is a compact subset of

zero 2-dimensional Hausdorff measure, then and

C(K).

R(K)

iC

n

K onto

which is constant on the

fibers over some fixed coordinate axis lies in R(K)

which has

The Hartogs-Rosenthal Theorem

thus shows that every function in C(K)

real-valued functions in

0 .

K is rationally convex,

This occurs because the projection of

each coordinate axis has zero area.

satisfies

R(K).

separate the points of

Hence the K, and the

stone-Weierstrass Theorem can be invoked. This leaves us with the following open questions, some of which have been around for some time. arc in (1)

For these,

K

a compact Jordan

a;n. If

K has finite two-dimensional Hausdorff dimension, is

K rationally convex? (2)

If

1

q ,

consider the diagram

where the natural projections (n - p)(p - q)

and

and

(p - q)(q + 1)

The hermitian metric on

V

Which defines a Kaehler metric on

T

have fiber dimension

respectively.

induces a hermitian metric on lE'(vq+l)

q+l V

which restricts to the

117 submanifold that

Gq(V)

W(V P+l) Theorem 3.

Let

w be its fundamental form normalized such

has total volume 1.

Representation Theorem of Bott and Chern T

IJ

p,

*

rr*(C

IJ-q

(Q) A q

p+l,···,n, where d(p-l,q-l) - fq-l)! ......• l! (q(p-q))! - p-q)! ... (p-l)!

This can be proved by chasing the diagram:

118 The fiber dimensions of

tt

P (n-p)p

are The map

6.

'Tp

tr

p

(n-q)q

T

q

q

q

q

tt

T

X

and 'i'

A.

(n-p)(p-q) (p-q)q

(p-q)(q+l)

is the inclusion.

Equidistribution Using fiber integration, other results of Bott and Chern can be

obtained and considerably extended.

Let

vector bundle of fiber dimension

over the m-dimensional Stein

manifold

M with

1

s

m.

s

Define

E

q

m - s.

vector space of global holomorphic sections dimension

n + 1 , such that

e:M X V = V

E

M

The zero set of Z(v) For how many

generates

is defined by V - [oJ

v =

V

MI v(x)

[x

v

V

[oJ

e(x,v)

V

x with

point

of

E

V

with finite

E, meaning, v(x)

then

be a

e

if is surjective.

=

is

O(x)} Z(v)

empty? v

Assume that

V - [oJ

and that

Z(v)

is

M is

Then value distribution theory can be used to answer this

question.

If

v

Let

is

pure q-dimensional for at least one connected.

M be a holomorphic

Let

K

be the kernel of

M , the fiber p

=

f(x)

n - s. Gp(V)

is holomorphic.

Kx

An exact sequence is define&

is a (p+l)-dimensional linear subspace of

Hence it is represented by one and only one such that

E(f(X))

Consider the diagram: FpO(V)

1 M

e.

f

>

T

Gp(V)

1T

) !P(V)

119 For each on

V.

SLP(V)

• define

Sa

Take a hermitian metric

=

It defines a Kaehler metric on n

w On

that

a

with fundamental form

1 M

let

>0

ddch

h on

is compact for each

be a non-negative function of class

COO

such

M and such that

>

r

O.

The characteristic is

the average deficit is 6. f

(r )

Theorem 4. almost all

- 0

If

for

r -

00

•

then

f-l(Sa) r

0

for

a

The connection to the vector bundle question is easily established. First:

Second:

If

v

[o} , then

V -

VM•

Cl(E), ... ,Cs(E)

E

of

K, and

V

E.

are defined.

back the exact sequence (5) over

0 -K-

[,

VM

t

f

VG (V)p

defines hermitian metrics

Hence the Chern forms

Gp(V)

E __ 0

l,

Qp -

and

Observe that

M:

o-

LP(v)

The hermitian metric on

along the fibers of

over

a

0

f:M - Gp(V)

CO(E) , pulls

to the exact sequence (6)

120 Hence

Therefore

f::,f(r) Theorem 4 implies Theorem 5.

If

S Cs_l(E)

A (ddch)q+l

Gr

then

Z(v) I

... 0

for almost

V E

for

V .

REFERENCES [lJ

Bott, R., and Chern, S.S., Hermitian vector bundles and the equidistribution of zeroes of their holomorphic sections, Acta Math. 114 (1905), 71-112.------

[2J

Stoll, W., Value distribution of holomorphic maps into compact complex manIfolds, Notes 135.

University of Notre Dame Notre Dame, Indiana

PARAMETRIZING THE COMPACT SUBMANIFOLDS OF A PERIOD MATRIX DOMAIN BY A STEIN MANIFOLD by R.

I.

o.

Wells, Jr.*

Introduction The periods of integrals on algebraic. manifolds of arbitrary

dimension has been given new impetus by the recent work of Griffiths (see the survey articles [2], [4]).

The basic purpose of studying

periods of integrals has been to study the moduli of the given algebraic manifold.

Given an algebraic manifold of fixed topological

type, one can compute the periods and one obtains a matrix of periods dependent on the algebraic structure.

All such periods are points in

a classifying space determined by the topological type of the given manifold.

Such classifying spaces are known as period matrix domains,

to use the terminology of Griffiths [31.

The variation in moduli is

represented as a variety in the period matrix domain.

Classical

examples of this situation are the representation of the moduli of elliptic curves in the classical upper half plane and the representation of the moduli of Riemann surfaces of genus of the Siegel upper half space of rank

g

g

as a subvariety

Griffiths' work [4] is a

direct generalization of this approach to the general moduli problem, in the spirit of Hodge's work on harmonic integrals. The interest in this paper is the geometry of the period matrix domains themselves, and in particular, those aspects of the geometry which are not present in the classical cases.

In general a period

matrix domain (see [3]) is the open orbit of a real Lie

group in a

* This research was supported by NSF GP 8997 at Rice University.

122

quadric submanifold of some projective space, defined in terms of subspaces of Euclidean space satisfying generalizations of Riemann's bilinear relations (due to Hodge).

These are complex flag manifolds,

and have been extensively studied in a more general setting by J. Wolf [9].

Let

D be a period matrix domain.

If

D

is Stein, then it

falls into the classical theory of bounded sYmmetric domains. in general,

D may contain compact complex submanifolds of (maximal)

positive dimension

q, and hence cannot be Stein.

period matrix domain is

In this case, the

(q+l) - complete in the sense of Andreotti-

Grauert [1], a generalization of llSteinness ", and the best possible in this situation [7].

In this case also, there is a fundamental result

due to W. Schmid [7], Which asserts: bundles

Lover

D

for certain homogeneous line.

(see also [4]) (D,L)

=

0,

r I q

(Where cohomology is taken with respect to the sheaf of sections of L).

If

q

=

0 , then this is a consequence of Theorem B of Cartan.

The situation is now the following: automorphic forms on

much of the function theory,

D, global functions representing subvarieties

(modular subvarieties, etc.), are now necessarily cohomology classes, since there are in general no sections of line bundles available for this purpose, due to the vanishing theorem.

Thus, if

crete subgroup of the automorphism group of

D

r

is a dis-

(which arise naturally

from the algebraic geometry, analogous to Siegel's modular group, see [3J), one would like to understand the vector spaces

Hf(D,L) , the

cohomology classes which are invariant under the action of proposal by Griffiths

(see [3, IJ) for a representation of

in general is the following:

let

VD

r

A Hq(D,L)

be the disjoint union of all

translates of a typical fibre (of complex dimension

q) in

D by

123 elements of the complexification of the natural (linear) automorphism group of

D

(see Sections III and IV for precise definitions).

should be a holomorphic fibre space with a parameter space in the following diagram (where

Then M, as

p(fibre) is the natural inclusion as

a submanifold):

M

and Griffiths conjectures that

M might be Stein and that the natural

mapping (using inverse and direct images of sheaves)

might be an isomorphism (cf. [2], [3.1]).

This would then give a

representation of cohomology as sections of a line bundle and more generally would afford a representation for automorphic cohomology. In this paper we show that, for a particular class of non-classical period matrix domains, namely those arising from the periods of 2-forms on a Kahler manifold, the parameter space

M is Stein (one additional

dimensional assumption is made in Section V for the complete result as it stands now).

This gives one part of the above representation pro-

gram of Griffiths.

The results of this paper have been applied by

M. Windham [8] to study the geometry of there is a complex structure on holomorphic,

11"

VD

being proper, and

VD

itself.

so that the maps p

In particular, 11"

and

having Stein fibres.

pare One

obtains from the Leray spectral sequence, Grauert's direct image theorem, and Cartan I s Theorem B the result that

for any coherent sheaf

F

over

124 It is unknown at present whether

but Windham has also shown in [8) that, for any coherent sheaf

'lJD

F

on

' one has r

However, it is false that

p* L

what the actual relation between

>

0

L , and hence it is still unclear

Hq(Vo, p*L)

and

Hq(D,L)

is at the

present time. An outline of the paper is as follows:

In Section II we des-

cribe the period matrix domains studied in this paper, the groups acting on them and their homogeneous structure.

The period matrix

domains studied are of the form D

SO(2m,r)/U(m) x SO(r)

(see Helgason [51 for the notation used), and maximal subgroup.

G/H H

is a compact non-

The maximal compact subgroup is

and the homogeneous space

K/H

K

is a compact submanifold of

admits a complex structure as a complex submanifold of III, the deformation space

Vx

SO(2m) x SO(r),

is defined (D

D.

D which In Section

being an open subset

of the compact projective algebraic manifold X), and it is shown that GC/Kc is a parameter space for (Gc denotes the complexification of the real Lie group

G).

In the case

represented as a specific closed SUbvariety and for

r

>1

the fibres of space

'lJD

V of

, the situation is more complicated.

'J x

r

1, t

n

(n

GC/K c

is

2m + 1) ,

In Section IV,

which are contained in

, whose parameter space

characterization of points in

D define the deformation M c GC/Kc is to be determined. A

M is given by Theorem 4.7, in terms of

orthogonality conditions with respect to the quadratic form involved in the definition of the domain

D.

In Section

Q

V this

125

characterization is applied in the case

r = 1 , to show that

an open Stein submanifold of the variety has the following specific result: D

V c en

of the group

G

en

=

c2 m+ l

SO(2m)/(U(m)

by elements

is given by the Stein manifold defined ,

(z

(A)

(B)

For example, one

SO(2m,1)/U(m),

= SO(2m,1,e)

by the equations in

is

In the case

the parameter space for the translates of C

M

x + iy)

-1

2 2 xl + ••. + x 2m

< x 22m+l

2 2 Yl + ••. + Y2m

or

which is the intersection of a closed subvariety of

en

2

< Y2m+l

with the

union of two tUbe domains. We hope to study the general parametrization result and its application to the representation of cohomology in a later paper.

The

results here serve as a model for what is presumably true in general, and it's already clear that many technical points do go over, but some non-trivial points of generalization arise also, and that is what must be studied first. I would like to express my thanks and indebtedness to various people with whom I have been associated while working on this problem. First to Phillip Griffitrnfor his encouragement and inspiration, to Wilfried Schmid, with whom I worked out most of the details of the

..

first example of this parametrization in Gottingen in the summer of

1969, to Aldo Andreotti for several helpful conversations on the subject, and to my student Michael Windham, who read the first draft and helped out in many ways.

II.

Period Matrix Domains In this section we want to discuss a particular class of period

126

matrix domains, which naturally arise as the classifying space for periods of 2-forms on compact Kahler manifolds (see Griffiths [3, IJ. Namely, let

be an integral matrix, where Setting n lR c

c" ,

2m + r , then

n

I Q

denotes the unit matrix of rank

U

acts naturally as a quadratic form on

by Q(u,v)

=

u t .Q.v ,

representing vectors in Euclidean space by column vectors. of

U

Q on

(j)n

Let

Gm, n

is the iV-linear extension of its action on -- Gm, n

be the Grassmannian manifold of all t-linear iV n

m-dimensional subs paces of [u l ' ••. , u} m

be a basis for

(2.1)

Q(S,S)

(2.2)

Q(S,

The action n lR

Suppose

S

Gm,n , and let

S, then we write

o

0-,

e

1,· ",m

i.e., is a positive definite matrix.

We then define

Then

D

x

(S e G m, n

D

(S s Gm,n : Q(S,S)

o}

Q(S,S) =

0,

>0

} •

is an open subset of the algebraic sUbvariety

X.

As is

shown in Griffiths [3, IJ, the real Lie group G

SO(Q,R)

Q}

is a transitive group of biholomorphic mappings of

D onto itself and

127

its complexification Gc

[g

SO(Q,C)

=

SL(n,C) : gtQg

=

Q}

is a transitive group of biholomorphic mappings of Thus

X is a compact projective algebraic manifold, and

open submanifold of fold.

Helgason [5J).

D.

D

G

D

is an

is itself a homogeneous complex manih

k

or

has two components when

is zero,

mr I 0

(see

The main problem under consideration in this paper

becomes trivial if In this case

X, and

D is not connected except when

Note that

since the group

and

X onto itself.

mr

=

0

and we assume from now on that

mr I 0 .

D will have two components, which we will denote by

DO

Dl , where DO is the orbit of the identity component of G in There is a natural isomorphism between DO and Dl given by the

linear mapping -1

o

o

o

1 _ n 2

o

o

o

-1

which we will have occasion to use later on. Let

H be the isotopy subgroup of

G at a point of

D.

Then

Griffiths computes that H and

D

=

U(m) X SO(r) ,

is thus represented as a homogeneous space

compact subgroup of

G containing

H

is denoted by

by K

=

SO(2m) x SO(r)

where the natural embeddings are given as follows:

G/H

The maximal

K and is given

128

:::J

Re A

A

U(m) .---[

1m A

(A,B) , SO(2m) x sot r) ...... [:

SO(2m)

SO(Q,R)

: } SO(2m,r)

We then have the exact commutative diagram of Lie groups 1-+ H _ G

T

i

T

11

-7

G/H _

1

T

K --. K/H -

1

1

T 1

and from this we see that the homogeneous space submanifold of the homogeneous space of the identity coset in

G/H

K/H

is embedded as a

It is, in fact, the orbit

D under the natural action of the subgroup

K. Y

We want to show that the homogeneous space

K/H , which is a

=

real-analytic submanifold of the homogeneous space

G/H, is also in a

natural manner a complex submanifold of the complex manifold

D.

In

order to do this we need to specify the relation between the coset definition of

G/H

Gm,n If g block form where of

n X r

matrix.

If

GL(n,lI::) and h

g

is an

2

Proposition 2.1.

T(gH)

G/H

1 2 (g ,g ,

,

be represented in

let

g

are

n x m matrices and

g3

matrix, then let

1

and more generally for arbitrary period matrix domains.

What is

needed for a "good parametrizatiop theory" (such as the above results for

and the results of Section IV of this paper) is a good representation for the parameter space GC/Kc for the fibre space ' the "deformation space" of the period matrix domain for

r

=1

, we have the following representation.

D.

Let

In particular, ( e l' · •. , e n }

denote the standard basic vectors in Theorem 3.5. mapping

Suppose

r

1 , then there

biholomorphic

of the homogeneous complex manifold

onto the non-

singular affine algebraic hypersurface

induced Qy the mapping

c

G - V given

co(g)

138 Remark 3.6. The variety on

V

cp(g)

= g3 , the last column of the matrix g. n represents the orbit of under the action of =

g.e

en

given by left matrix mUltiplication. This result generalizes, but the representation V of GC/Kc becomes a complex submanifold of a noncompact "Grassmannian type" generalization of Euclidean space Cn . This theory and the applications to the problem at hand will be developed in a later paper. The proof of Theorem 3.5 will follow easily from the following propositions, which we state in a more general form than is needed at present. Let

columns of the identity matrix in GC. For r = 1 , Let Mn,r Cn. r be the vector space of n x r we have E = en complex-valued matrices. Then there is a natural action of GC on

be the last

M

n,r

r

given by left matrix mUltiplication. V r

=

(Z

Let given by

M : zt QZ n, r

Let =

-I } r

Then, letting cp(g)

=

cp: Gc ... M n,r

gE , we have by

cp

is the variety

and the mapping factors through the natural projection

to a mapping

cp-

be

as in the diagram, and

is

V

r

eM

n,r

P

biholomorphic mapping.

139 Proof.

First, by the definition of

factors through the natural projection defined.

It follows from -I

gt Qg

(E) , it's obvious that P

and that

is well

Q , that

Q g E

r

(gE)t Q(

so

Z E V then one can extend Z to an r ordered orthonormal basis (with respect to the quadratic form Q), [v

E

Vr •

Moreover, if

.

l,···,v2 m,Z} C

element

g E G

Using those as columns in a matrix so that

=

Z

and hence

g

we obtain an

is surjective.

One

has to check that two different choices of orthonormal bases give rise to elements

g,

g

so

gg-l

E

GC(E)

and this is not hard.

q.e.d.

Lemma 3.8.

Proof.

the other hand, suppose that

h

1:1

It i ' trivial that [: h

E

E

c G (E) , then

E , if

A

hE

implies immediately

E

E

SO(2m).

On

is of the form

h

where

C

E

M 2

r, m

.

Since

h E GC , we have

htQh

=

Q , and

det h

1,

which implies immediately that (writing out components) det A

C = 0

Hence

A

E

SO(2m, t), and

h

E

•

=

1 . q.e.d.

140 Corollary 3.9.

In the case

r

=

Theorem 3.5

The

the above corollary.

1

easily

the two lemmas and

More generally we observe the

Proposition 3.10.

Vr

holomorphic with the

homogenous complex

isomorphic

SO( r,t) .

This

the commutative diagram

is the mapping in Lemma 3.7,

where

q

is the natural projection

homogeneous spaces, and III

=

q

0

- -1

This proposition then tells us that r r n , whereas closed

q.e.d.

r

>1

, we have the natural

representation as a base space for a holomorphic fibration, where the total space and the fibre are both Stein.

This

will play a

role in the parametrization theory which will be developed in a later paper.

IV.

Characterization Recall

Submanifolds of

D

Section III that We had the following diagram (3.1).

141

Let

be the set of those fibres in

(which was defined in Section II by

which are contained in

D

[S

=

X: Q(S,S) )O}).

E

What we want to do is show that fact a Stein manifold. those fibres of

not in that

D

,

and

Q(v,v)

cise.

S 0

Then

M is in

First we need a suitable characterization for

which are contained in

A submanifold

D

.

D

gYo of X is not in D if a point S E gYo is is not in D if there is a vector v E S such We need to make this type of information more pre-

First we have the following lemma. Lemma 4.1.

with

t




=

0,

I

J

(2.5)

where of

v

is the volume of

1I x M , and (Rl, .•. ,Rn) is the polyradius

(c,. Let

f

with coefficients in

•

We can expand

f

in a power series in

z,

187

(2.6)

f(z,w)

The series converges uniformly on compact subsets. compact subpolydisc of

and

K

Thus, if

a compact subset of

is a

M,

using the uniform convergence of the series and Fubini's theorem. Letting

-

K - M we conclude that (2.6) is square integrable

if and only if

(2.7)

Now the lemma follows easily from this observation. given by (2.6\ be square integrable and suppose

S

=

For, let

f,

(k+l, .•. ,n} •

Let

Clearly

is in O ponding series for the

f

L f

2

, using the criterion (2.7), since the corres-

is just part of (2.7). O f j , but for factors of the form

The same is true for

188

1

R7 J Since these numbers are always bounded, the series for so

fj

fj

is finite,

also.

Since the lemma is proven, we can obtain the exactness of the

L

2

section sequence of the Koszul complex. 2.8. where

Theorem.

Let

A is a polydisc in

1

[a} x M in

be the idealsheaf of en

and

A x M,

M is a bounded domain in

em.

Let

1

be the canonical (Koszul) resolution induced sequence of L2 sections

o

L2 (F n )

L2 (F 1 )

...

by free sheaves.

L2 (1)

Then the

0

is exact, where 2 L (e-)

2 L (e')

and

3.

in

SAXM1f(ldZ Ildwl < co}

(f E e'(AXM) ;

n

O H ( AxM,1)

is defined as the direct sum of

Existence of holomorphic functions Let rn ,

M be a complex manifold of dimension 0

D

Suppose that

mapping ( the differential

dt

t: M has rank

N

,

D a domain

and

D is a regular holomorphic n

Then the

everywhere) .

t-1(T) are submanifolds of M of dimension m = N - n M T Suppose that p is a non-negative Cco function defined on M with fibers

these properties: {x EM: t(x) E K,

(a) for each compact set p(x)

c]

KeD

is compact in

M,

and

c

R

(b)

for some

"o :

if

p(x)

Co ' then

(s. psh.) at that

M

x

I

dp(x)

When

0

and

p

is strictly plurisubharmonic

M admits such a function

p, we shall say

is a regular family of strongly pseudoconvex

folds (certainly each

M

is s. psc.).

T

section to show that if

K

functions holomorphic on

It is our purpose in this

is a compact subset of

M

psc.) mani-

M there are O' K; in fact

in a neighborhood of

sufficiently many to blow down the exceptional set of neighboring fibers.

Since

is an extension p rob Lemj

M and O has many holomorphic functions, this

M O the obstructions to such extension lie in

the first cohomology group of the ideal sheaf of objective is to show that this group is small. sUfficiently large subgroup is a finite Throughout this section

t: M ­ D

MO'

Thus our main

We shall show that a

, o­module. will be a regular family of

s. psc. manifolds, so defined by the function

p.

We shall adopt the

following notations M

M

For such

M

T,C

CIS.

T

T,C

M

T

n MC

is an s. psc. manifold.

From time to time the domain

smaller domains;

We shall refer only to

D will be replaced by

it is assumed that the family

t: M ­ D

also shrinks

accordingly. Since

M is locally (on

find, for each

x

B x 6 where 6 x x x N n C ­ We may cover

M) a product of

M with D, we can O ' a coordinate neighborhood of the form

M O is a subpolydisc of

D

and

B x

is a ball in

M p(x) ;! c l by finitely many such coor­ O; dinate neighborhoods, choosing the same polydisc 6 for each. By shrinking

6

[x

further, we may arrange that

[x

M; t(x)

D, p(x)

;§

c]

190 is now contained in the union of these coordinate neighborhoods. now replace the original family by this family, with some specific value greater than

Co

will range between

and

c

coordinate neighborhood, then

l• U

Co '

say

chosen as

In the future

cl

Notice that i f

n Mc

c

U

' for any such

We

c

is such a c , is a pseudo-

convex domain in the sense of Hormander (definition 2.2.2

[6]).

Now,

we would really have liked a covering by product domains so that each simplex in the nerve of the covering is a product domain, but that is not possible to arrange. Mc t (c t

> c)

This forces us to compare an

M c

with an

via two coverings, one of which is relatively compact in

the other so that for each simplex, a product domain can be fitted between the supports of the simplex in the two coverings.

For technical reasons this has

can be done by suitably shrinking to be done, not once, but

n

This clearly

times;

that is the context of the

following lemma.

3.1.

Lemma.

Given the regular family of

described above, let

ct

cl

> c > Co .

(ii ) (iii)

(i v)

each

V..

l,J

We can find coverings with these properties.

=

(I )

psc. manifolds as

is biholomorphic to a pseudoconvex domain.

UV l .

j

UV

j

,J

.=M

n, J

n MT

V. . l,J

(v)

c =:; =:;

V. +1 . l ,J

each simplex

n

M T

all

T

a

[V . . , ••. ,V . . }

l,lO

l,lq

such that Now, let and let

V

T:

M X

be the graph of

be the projection onto the second factor t:

191

v V

[(X,T); t(X)

is biholomorphic to

resolution since the

M. t

i

T) .

=0

Its idealsheaf

- T i

I(V)

are global generators.

Thus the sheaf

also has a resolution

=0

Let

b e th e ana 1 ogous f ree vM-s h eaves an d

maps

with

T

considered fixed.

morphic functions on each

has a global free

T

in

M

If

T

111

F i- l

i T : F

is the sheaf of holofor

, we then have this resolution of

T

th e

t::.:

(3.2) Since

(3.2) is the Koszul complex of

a volume element on

I(M ) , it is indeed exact. T

Fix

M which is the product of Lebesgue measure on

with any finite volume element on

MO'

t::.

Then, the following lemma

comes from the results of section 2.

3.3. then

Lemma.

If

D

is any product domain,

D

DO

x

t::. ,

in

M

induced sequence of square-integrable sections

is exact. Now we fix attention on a specific

n on

o

described in lemma 3.1. FP

with values in p

n

and define

o i

a.T Ca.']"

3.4.

i

q : E

m i

Define

i+l

a.T

i

·a. T

as the space of q-cochains

LP,q

=0

0) .

Let

by i ((5a p,p+i + (-1) *']"(a p+1,p+i+l))

(a) ]p,P+i+l

Lemma.

LP,q

endowed with the covering

which are square integrable

(otherwise Ei+l

M c

0

Now we have to observe, that if we drop the square-integrability assumption on all the cochains, the corresponding sequence

(3.5)

-+

computes the cohomology of covering.

•••

-

Clm-l

E' .... 0 m

since the covering

vrn

is a Leray

In particular there is induced a map on the cohomology of

the square-integrable sequence.

Our main purpose is to show that this

map is injective.

3.6.

Lemma.

i .... H (M

r

As Hi(M

induced from the sequence (3.5)

There

map from

ker a.i

to

r

is considered

r

Hi(M

r

map

,

cr')

i t is continuous, when

be endowed with the Frechet space topology

induced by uniform convergence on compact sets. Proof.

Let

Then

with values in

on

n

i-cochain on

Restricting to with values in

==

0

a O is an i-cochain

M , this induces an r

Since is a cocycle.

This represents

Since the L2-norm on analytic functions is stronger than any supremum norm on a compact set, the map

is also continuous.

Now, let us return to the machinery of lemma 3.l.

Mc.

>c ,

and we have coverings J l The same setup holds on Mc'; let

selected a

c'

We have the commutative diagram

of

Mc

There we have

,

and 1/n

of

193 • I

Hi'

T

Hi (M

T

I

(3.7)

R

3.8.

a'

T

R(ker

Let

a'

E

Hi(M

E

ap

is solvable.

Vl

restriction from

T,c

,ff) -r

,

Mc' LP,P+i

ker

i

1

.

(li(a') on

Let

0

T

The hypothesis that

N(tlJ)

is just the assumption that

T

We have to show that, if' to V.

J

, then Rn a'

Mc By assumption there is an

ex.

=

R.

J

i-l

(c)

is the where

on

C E

that

5yo

=

aoiM

•

T

We can extend

Yo(cr)

Dl ' and then

cO(cr)

Co E LO,i-l(l!2) ,

1021

0

cochain

YO

n MT )

on

trivially to a function 1

cO(cr)

102 / will be square integrable.

such that

such

and in so

passes through a product domain

5c

-5c O + R2aO vanishes on

on R 2aO

O M-r

M

D1,0 holomorphic on This defines a

In particular, on

T

For each i-simplex

cr,

1031 , and by lemma 3.3, there is a square integrable

D2,cr

section of

i-l

By lemma 3.1, we can restrict to

doing each (i-I)-simplex

vr2

on

represents a class in

a O1M-r,c'

o ,

-r

Ei

with

(aO,···,an)

a'

T

are the restriction maps.

T

Lemma.

Proof.

,(J' )

R,

T

R,

C

1

R

Hi where

T,

F

l

Restricting to

on

vr3 '

D2,0

call it

cl(o), so that

defines an element of

Ll,i, and

194

Applying

5, we find that

from

to

c

2

E

wT ( R3a l

+ 5R C l) 3

0.

=

Again, by passing

through product domains, simplex by simplex, we find

L2,i+l

such that

Continuing in this way we ultimately obtain the desired such that

u;-l(c)

c

E

Ei-l

Rn a'

Now, we have to prove that obtained by a

R

is also surjective.

This is

to the present context of Grauert's

bumping technique and is most easily accomplished by considering square-integrable Dolbeault, rather than Cech cohomology.

The trans-

ference is easily accomplished by observing that our coverings are acyclic relative to square-integrable cohomology;

this is precisely

Hormander's Lemma 2.2.3 [6]. More precisely, let defined on

Mc

integrable.

Let

LP,q d

with values in LP,P+i p=O d

Let

be the space of FP

such that

and define

(O,q) wand

forms

oW

W

are square

Edi .... Ei+l d

by

Hi,d

be the cohomology groups of the complex defined by the

3.9.

Lemma.

T

Proof. differentials germs of

Let

K

5, w ' r

Coo_(O,q)

denote the triple complex with

ED

and

3,

where

Fm,q

denotes the sheaf of

forms with values in the vector bundle

, and

195 CP2(tt,Fm,q)

where

denotes the Hilbert space of p-cochains on the

L

nerve of ,j with coefficients in

S

e;(a)

L

a(s(a))

and

support of each p-simplex.

Fm,q, such that for are square-integrable on the

The coboundary operator

is the usual Cech coboundary homomorphism. pmq W: K T

Kpm-l q

operator

a:

Kpmq

0,

The operator

is induced from the Koszul complex (3.2) by multi(_l)p-m+ q.

plying each homomorphism by a factor of operator

KP+l mq

5: Kpmq

Kpmq+l

Finally, the

is induced by multiplying the Dolbeault

by a factor of

(_l)p-m+q+l.

It is easily checked

that these operators are pairwise anti-commutative so that if we define

EB

p-m+q=i

pmq K

and

d = 5+\11

T

+a ,

is a

differential complex. We define two filtrations, EB EB

(d K.}

(K i},

on

i.

K as follows:

n . EB KJ+P,P,q

j!!;i,q,p=O

These filtrations are compatible with the grading of makes

K

K and each

into a regular differential complex with filtration (see

Godement [3] for definitions). The corresponding spectral sequences

Ei

and

dE.i,

have

terms:

(1) (2 )

and and

for

q

o

for

1 •

q

i§;

1 .

Lemma 3.9 then follows from theorem 4.4.1 in Godement [3].

We will

196 verify only (1) as (2) follows similarly. The

EO

differehtial

term of

(1) is

Ki/K i +l

=

dO being induced by

d

.

P q , the

q p=O

to give

K gives

that the total gradation of

n

$ $

E

Note also

iq

p q

o

Hence

El = H(EO) and the differential d l of El is computed from the connecting homomorphism of the exact sequence

where the differential of the middle term is

It is easily checked that the connecting homomorphism is induced by the map a iq $

d

it is clear that of all cochains of

E1i O consists

which are square-integrable and holomorphic Ei

on each simplex, Le.,

as defined previously.

This clearly

implies that Furthermore, if Thus, if Ker

S

, then

as.(o) = 0 J

S

for each

'T

N(V)

and

j

= 1, ... ,n

Since

101

is

pseudoconvex, we can apply Hormander's theorem 2.2.3 (6] to conclude Thus and

aT)

= S

Hence

E2i q = 0 for q 1 The proof of (2)

o

for

q

1.

This clearly implies that

follows in a similar fashion, the only subtle

part being the demonstration that

for

q

1 .

This,

197 however, is accomplished by noticing that FP,p+l is a fine sheaf on which L2 sections can be defined. A close observation of the proof that Cech cohomology of fine sheaves is acyclic in dimensions greater cohomology.

than zero shows this carries over to easily checked that follow

n

dEiq 1

q

v:,

Since it is

,p+i) , the proof will then

as in (1).

This completes lemma 3.9. Lemma.

3.10.

Proof.

R

is surjective.

lemma 3.6 holds there as well.

Refer to these groups by

still have the commutative diagram

Rd: is surjective. generalizing Grauert's technique.

where

B

Wl, ... ,W d

to take

c'

and

c

Hi,d,. T

We

We prove this by

be a collection of domains of the type

is a ball in

i

and

M

(3.7) and it suffices to prove that

the mapping

Let

c'

We can define the Dolbeaul t cohomology groups on

em

and such that

-

d U

W. i=l sufficiently close to accomplish this). M

c

C

(we have Let

(1) d

(2)

Let

B

o i

C

C

B

L: IT.

i=l i

be such that

supported in

Let

Xi

=

and Xi n Wi enough) . g

i-l Ed

(x

(c '-c) L: IT. ](x)

M; [p

Xi - Xi_l

a

C

Bo i XA

is a pseudoconvex domain (so long as O.

Let such that

f-

IT i is still

and

c'

< c]

Further each and

c

are close

We shall show that there is a

extends to

The proof is by

198 extension from each the case from

X

Xi to

o

to

for this we need only demonstrate

Xi+l

Xl.

For this purpose, let us suppress all fixed indices.

Let

o . In particular, there is an

n Xo

defined on

W l

2.2.3 (6].

Then

L

=

=

L 2(0,n+i-2)

Fn

Yn with values in by Hormande r 1 s theorem

form

f n = oYn

such that

af n _ l

There is thus an

2(0,n+i-l)

,

$Cay n )

so

-

n- 1

0 F n- l

with values in

form

such

Continuing in this way we obtain squareintegrable forms

Y

=

(Yo' •. ·,Y n)

'on

Wl

n Xo

o w is compactly supported on

If

X 6 , then

is in of

f

E

X

o

=

i

c

on

which is near Lemma.

Proof.

Let

,

with

i- l

since

Xl

3.1l.

Rf'

E

wy

=

fl

strongly pseudoconvex,

E

-

R

0 Now

1"

on

vanishes identically on the part

0.(c)

,

E i,

1

X and O'

is injective for i

n

p

and identically

g = c

Thus we take

Xl

i 91 1" f

f

on

n Xo

W l

such that

R

1"

i

and 1"

is an isomorphism

1 i( f) =

but since

0 (c'

By Lemma 3.10,

0

1"

!;

c

cO)

M

,

1"

so

is

199 = 0 •

'I

3.8, the class of

Then, by lemma i

so on cohomology

in

f

Hi

is zero,

'I

is injective.

Finally, we are in a position to apply the coherence result of Markoe. 3.12

Theorem.

Let

E =

i

be

mappings of

are

Then

0,

0,:

--.

=

Ei + l

i

=

the cohomology groups

(J b. -maps

germs of holomorphic

sheaf

Hi

defined by

i

1

of the sequence

i-l o,r

continuously,

...

are coherent Proof.

For each

injectively into

'I

i

H (M

'I, C

,

'I

maps

i/o

k er

0,,,

,if), which is finite dimensional for

i '" 1.

r

has closed kernel and closed

In addition this shows that each map range, so is a direct map.

The theorem now follows from Markoe's

result. 3.13. let

Ri(F,C)

Definition.

For any coherent sheaf

be the germ of

c)

3.14.

Theorem.

along

Ri(F,c') Proof.

in

Ri(F,C)

'1-

1

on

M and

Co

(0) :

lim

F

be a coherent sheaf on

M which

> c > Co ' the image generated « O-module.

Then for

is a finitely

Consider first the case

F =

(j'.

c'

We shall show that the

factors through is a finitely generated

c

U--o

global resolution by free sheaves. of

F

Hi , which

o

by the preceding theorem.

Then

200

is finitely generated. Let

be a ball centered at

V

W

Hi(Mc'

in

Mx

n

0

in

and let

We can identify

Now

w

can be represented as an i-cocycle on the

nerve of the covering tl'l x V purposes we may suppose V

of

nY

(Mc'xV)

For our present

•

Mc l over a slightly larger polySince, for each simplex a of N(V'i), lal x U is

disc

covers

l

Stein, there is an i-cochain

=F

in

O

=

fi

defined on

w on

which induces where

fi+l

Y.

x U with values

Then

vanishes on

Y, so

is an (i+l)-cochain with coefficients in

Continuing in this way, we obtain a sequence of

t

w can be viewed as an element of

In this case

(MC' xV,O"y)

Y of

M with the graph

(p+i)-cochains with values in

FP

fP,

0

p

n

such that

(3.15) If we fix

T,

the restriction of

(fi, .•• ,fn)

to

is square-

integrable (in fact, holomorphic through the boundary) and thus is in Further, as phically, thus

varies this element of varies holomor(fi, ... ,fn) r(u,g,i) . By (3.15) a,i((fi, ... ,fn)) =0. T

Thus the correspondence -

i

w - (fi, ... ,fn)

, which obviously factors the restriction map

(er,c') _

.

Now the case of a general length of the free resolution. 3.16.

Lemma.

coherent sheaves on im Ri(G,c')

induces a map

Let M.

F

is handled by induction on the

All we need to show is this. exact sequence of

0 - F - G- H - 0 =

If for every

c'

are finitely generated under

c

cO'

--,---,---,---,-.;.;..;;..;.c.

im Ri for

i

c ') ,

>0

the same is true for Proof.

Choose

c"

so that

c'

> c" > c

cO'

We have the

,

201

commutative diagram of cohomology, with exact rows.

Ri(G,C ')

1

al

Ri(H,c I)

tl g.

l''

p'

Ri(G,c tl)

-)

a

Since

Ri(H.c ');

[aptl wi}

j

=

ptlp'Ri(H.c ') Let

c'

[f

O'(Mc I .0):

g

then

A

is

is also finitely generated. say L, •••• k] • Also. aptl(Ri(G.c tl)) is

3:

with

(j'(U)

in

>c

be the restriction homomorphism. =

pI (Ri+l(F.c 1))

J

Theorem.

A

Ri+l(F.C)

Let

generates

3.17.

tl

[aptl wi : wi (G.c tl). i l •.•.• t} aptl(Ri(G.c tl)). Then it is clear that

finitely generated. generators for

1

P

p '5'(R i(H.C '))

finitely generated, {p'5'g.: g.

c'

Ri+l(F,c tl)

Ri(H.c)

is Noetherian and since

J

l

1ptl

Ri(G,c)

J

Ri+l(F,c ')

tl

Ri(H,c tl)

lptl

by

51

be U

c) .

Co

p:

(). ... (J"'M Mc '.0 c. 0

If neighborhood

of

U

M

c.O

in

M and

pf

is of finite codimension

Proof.

If

I

global resolution.

is the ideal sheaf of

M

O

in

M. then I

The following diagram is commutative with exact

rows: RO (if. c ' )

i

p RO(O'.c)

has a

rr

Tr

RO(O'/I.c')

,

1

p

O

R (e""/I. c )

Rl(I.c')

lP R1(I.c)

202 Now

A

Ker 5p.

Since

5p

there is an induced map

p5

which is injective.

By the previous result then,

finite

Since

is a

acts by evaluation at

boffo

0

on

, the result follows.

4.

Blowing down the exceptional set The purpose of this section is to use theorem 3.17 in order to

blow down all the exceptional sets in the fibers in a regular family of s. psc. manifolds.

This is accomplished locally first, by means of

the holomorphic convexity theorem cited in the introduction and then by patching.

In order to do the first we must show essentially that

the fiber over

0

is holomorphically convex with respect to the

finite codimensional algebra

4.l. 2 C

map

Lemma. p: X

Let

lR

->

be a subalgebra of is a

cl

,

Ix s X; "Iff

E

does not intersect Since closure Let

t.:

such

A on

I

Xc

A

finite codimension.

::?

max[ If (y)

X; p(x)

For

=

I;

p (y)

!!l!

c

> Co

there

(4.1)

cO}}

c l} .

is of finite codimension in

tJ(X) , its uniform

Xc is of finite codimension, say n t in Rn be a continuous linear map whose coordinates span

L: tJ(Xc) -> We can represent

support in

s . psh.

that

{x

A

is

P

ff( X)

A If (x)

s. psc. manifold defined by the proper

X

> c l > Co

c

A .

Choose

L

by an n-tuple of measures of compact c

> cl >

Co

such that

contains

the support of these measures, and no measure has positive mass at any point of

(there are only countably many such points).

Now let

203

Kl

[x

=

X; p(x)

C

,

l)

KO

uniform closure of in

Let If(p)1 L


R/r o N (r;T) v

N'Ii (rr 0 ;(l/r 0 )T)

ro

>0

A A(Bror)

by (5); so that since

IITII

A, Br o ' and R/r o are the desired constants is a fixed positive number for all T in the distin-

guished boundary of some open polydisc with center at the origin. Conversely, suppose that there are A,B,R,

and

rk

k + 3

positive constants

such that

Then for each unit vector

N'Ii (2r/r 0 ;(r 0 /2)s)

N'Ii (r;s)

...

by

S and each

N (2r/r ;rle 'Ii

0

... , r k e

it k

)dt 1 "'dt k

(5), (6) and iteration of the usual Poisson estimate for subQ.E.D.

harmonic functions. Remark. "almost all

The proof shows that "all T

T"

may be replaced by

(with respect to the product measure)".

228

Ck

Let us call a subset of

distinguished if and only if it is k C

the image of the distinguished boundary of the unit polydisc in under a C-linear automorphism of c k . Note that the class of

distinguished sets contains the class of all distinguished boundaries of open polydiscs centered at the origin as a proper subclass. that for a C-linear automorphism

T

on

Note

k C

\10Th"" \lIT(T) , so that as immediate consequences of (1), (7) and

(10) we have: Proposition 11.

f(O)

on

;i

If

A

regular, then

N

(r;T)

\If

r

>

R

and all

Proposition 12.

f(O) -I

on and -an-

--

T

If

R

and a distin-

A A(Br)

D •

A

premium, then an entire function

is

0

::§

A, Band

product of

f

function of finite A-type

function without zeros if and only

----

f

0 divides a function of

are positive constants guished subset D of c k that

for all

entire function

the divisor of

f

satisfies the condition of Proposition 11. In the case of classical growth we can say more. n (r;c) v

=

er 1 n (r;c)J t- dt v r

J

er 1 n (t;C)t- dt V r

::§

Since

Nv(er;c) ,

a consequence of (3), (8), (9), (11) and (12) is: Proposition 13.

f(O) or of ..:...;.....;;...,;..;;;.. stants

A

p

and

;i

Let

p

>

0

be given.

An entire function

f

0 divides an entire function of order less than

and finite

R

p

are positive con-

r

>R

and all

k C

D

and a distinguished subset

for all

on

in

D •

such that

Moreover, when

229

P is not

integer,

order less than

f

is the product of an entire function of

or of order

p

p

and finite type and an entire

function without zeros if and only

satisfies the above

condition.

(9), and (12) we have:

Similarly as a consequence of (2), Proposition 14.

A divisor

'J

'J(O)

on

divisor of

polynomial if and only if there

set

k

of

D

C

such that

A divisor

Proposition 15. divisor of pl'f

n)r;'f)

polynomial

constant for all

if and only for each ­­­­­

Proof.

Let ph

P

P(O)

n

L P q=l q

P

Pn('f)

P(O)

0

So

with

UeU

zeros of

so that

there is


M • ­­­­­

r

degree of

'J on

in

r

distinguished sub-

is bounded uniformly in

such that

P

is the

0

=

in

C

0

k

for

be the

Then

where

s2 = min(1/2, so/sl) • is a divisor on

rk

with

'J(O)

o

and

230 for which there are a constant C

k

r D

vh(z)

such that is in

Then

D •

=

M and a distinguished subset

whenever

0

n \) (r;'f)

z

= n \) (M;'f)

is in

with

C

for all

r

>M

Izi

>M

and all

Moreover, it follows from the argument principle that

is a continuous integer-valued function of nected.

ThuS,

is constant in

n \) (2M;'f)

D.

is bounded uniformly in

'f

polynomial

But the degree of

P

on

which is constant on

on

r

'f

By (14)

D

on on \)

pl'f

D of and r

in

n (2M; 'f) \)

D , which is conD so that

n (r;'f) \)

is the divisor of a is then

n (2M;'f) \)

Q.E.D.

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R. O. Kujala, Functions of finite A-type in several complex variables, Bull. Amer. Math. Soc. 75 (1969), 104-107.

2.

, manuscript (70 PP.) of same title, to appear. .. , lof, Sur les fonctions entieres d'ordre entier, Ann. Sci. ole Norm. Sup. (3) 22 (1905), 369-395.

3.

E.

4.

L. A. Rubel and B. A. Taylor, A fourier series method for meromorphic and entire functions, Bull. Soc. Math. France 96

53-96.

5.

W. Rudin, A geometric criterion for algebraic varieties, J. Math. Mech. 17 (1968), 671-683.

6.

W. Stoll, The growth of the area of a transcendental analytic set, I II, Math. Ann. 156 (1964), 47-48 and 144-170.

7.

About entire and "Entire Functions Pure Math. II, La Soc., Providence,

Tulane University New Orleans, Louisiana

meromorphic functions of exponential and Related Parts of Analysis" (Proc. Jolla, California 1966), 392-430, Amer. R. I., 1968.

PROBLEMS On the second evening of the Symposium a problem session was held during which participants discussed several problems in the general area of several complex variables.

I'Ve include these here

together with their contributors. 1.

Characterize all biholomorphic maps of

2.

Which polynomial maps of a polynomial map of

tn

a biholomorphic map?

3.

to

tn

to

A.

=

Let in

log.]

¢n

are biholomorphic?

J

?

A.

¢2

[Known:

are defined by two algebraic varieties if

(R.O. Kujala)

and

0

If

has constant Jacobian, is it

Dr be the disc of radius r in ¢ , n ¢n-l M an n-dimensional manifold.

df(O)

(R. O. Kujala)

(R.O. Kujala)

Which O-dimensional varieties in functions of finite

4.

¢n

",2

r