On Uniformization of Complex Manifolds: The Role of Connections (MN-22) 9781400869305

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ON UNIFORMIZATION OF COMPLEX MANIFOLDS:

THE ROLE OF CONNECTIONS

by

R. C. Gunning

Princeton University Press and University

of Tokyo Press

Princeton, New Jersey 1978

Copyright © 1978 by Princeton Universtty Press All Rights Reserved

Published in Japan Exclusively by University of Tokyo Press in other parts of the world by Princeton University Press

Printed in the United States of America by Princeton University Press, Princeton, New Jersey

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book

PKEFACE

These are notes based on a course of lectures given at Princeton University during the Fall tem of 197^, incorporating some material from lecture courses given during the year 1963-6^ as well. The topic of the lectures is the study of complex analytic pseudogroup structures on complex manifolds, viewed as an extension of the theory of uniformization of Eiemann surfaces. The particular pseudogroup structures considered, and the questions asked about them, are determined by this point of view; and this point of view also lies behind the choice of the role of connections as a unifying and limiting principal theme. A more detailed overview of the topics covered and the point of view taken is given in the introductory chapter. There remain many fascinating open questions and likely avenues to explore; and I hope these notes will provide a background for further investigations. I should like to express my thanks here to the students and colleagues who attended these lectures, for their interest and their many helpful comments and suggestions, and to Mary Smith, for the splendid typing of these notes. E. C. Gunning

Princeton, New Jersey

-!ί-

Ο OHTENTS Page §1.

Introduction

Part I: §2. §3· §1+. §5·

Description of the pseudogroups

The group of k-jets and its Lie algebra The pseudogroups defined by partial differential equations The classification of tangentially transitive pseudogroups: algebraic aspects The classification of tangentially transitive pseudogroups: analytic aspects

Part II: §6. §7· §8. §9·

1

16 21 39

Description of the connections

Pseudogroup structures and their associated connections Complex analytic affine connections Complex analytic projective connections Complex analytic canonical connections

Part III:

7

53 68 79 95

Complex analytic surfaces

§10. Complex flat canonical structures on surfaces §11. Complex affine structures on surfaces §12. Complex projective structures on surfaces

101 109 123

Bibliography

137

§1.

Introduction The general uniformization theorem for Eiemann surfaces is one

of the most remarkable results in complex analysis, and is at the center of a circle of problems which are still very actively being investigated. An interest in extending this theorem to complex manifolds of higher dimen­ sions has long been manifest, and indeed there have been several extensions of one or another aspect of the general uniformization theorem.

As has

been observed in other cases, some theorems in classical complex analysis appear as the accidental concurrence in the one-dimensional special case of rather separate phenomena in the general case; so a major difficulty is deciding just what to attempt to extend.

For compact Eiemann surfaces per­

haps the principal use of the general uniformization theorem lies in the possibility of representing these surfaces as quotients of the unit disc or the complex plane modulo a properly discontinuous group of complex analytic automorphisms.

Becent works (surveyed in [2]) have demonstrated the existence

and importance of a considerable array of different representations of compact Eiemann surfaces as quotients of various subdomains of the sphere modulo appropriate groups of automorphisms; but the detailed results seem to rest very heavily on purely one-dimensional tools.

On the other hand any such

representation has a local form, in the sense that the representation can be viewed as inducing a complex projective structure on the Eiemann surface, a rather finer structure than the complex analytic structure [2o].

The set of

all projective structures on a compact Riemann surface, being somewhat more local in nature, can be handled much more readily than the set of uniformiza-

tions of the surface and with tools that are less restricted to the onedimensional case; and these structures include, in addition to those induced by the classical and contemporary uniformizations, those associated to the more exotic representations investigated by Thurston [Ul], in which the groups of automorphisms are not discontinuous.

It is the extension to manifolds

of higher dimensions of this somewhat local additional structure on Biemann surfaces that I propose to discuss here; if the phrase did not already have a different generally accepted meaning, this could perhaps be called the local uniformization of complex manifolds.

There are many papers in the literature in which such structures on manifolds have been investigated, although not often have complex analytic manifolds been of primary interest; for this is really just a special case of the general problem of the investigation of pseudogroup structures on manifolds, an active area of research in differential geometry.

However the

model presented by the uniformization of Eiemann surfaces suggests restricting attention to a very special class of pseudogroup structures, those defined by families of partial differential equations having constant coefficients; for the defining differential equations can play the role in the general case that the Schwarzian derivative plays in the one-dimensional case, and that suggests the tenor of the treatment of the general case on the model of the one-dimensional case. The principal difference between the one-dimensional case and the higher-dimensional cases is then merely the presence of nontrivial integrability conditions in the higher-dimensional cases. That in turn suggests considering the connections associated to the structures rather than the

structures themselves; and the formal treatment in the general case is then precisely parallel to that in the one-dimensional case.

Considering

the connections rather than the structures really has the effect of linearizing the entire problem, and thus trivializing the questions of deform­ ation of structures and of moduli of structures.

The nonlinearity does

appear in the investigation of integrability conditions, although even there it is frequently possible to avoid the apparent nonlinearities; and the moduli can be introduced at this stage in a rather simpler and more explicit manner. Actually for some purposes it appears that the connections are all that is really needed of the structures, as will be evident during the course of the discussion; so the emphasis here will be primarily on the connections.

Even among the restricted class of pseudogroups mentioned above there is a great variety of possible pseudogroups; and any analysis detailed enough to be nontrivial seems to require somewhat separate treatment of basically different pseudogroups.

Therefore to limit the present discussion as much

as reasonably possible only those pseudogroups defined by partial differential equations with constant coefficients and having unrestricted Jacobian matrices will be considered here; the latter condition can be rephrased as the condition that the pseudogroup be transitive on tangent directions.

This subclass of

pseudogroups is still broad enough to include all the one-dimensional pseudo­ groups and some of the classical pseudogroups of differential geometry, the affine and projective pseudogroups; so this is perhaps the restriction leaving the general discussion closest to that of the one-dimensional case. There are enough complex manifolds admitting pseudogroup structures of this subclass to

lead to an interesting discussion.

However this restriction does leave

out a great many interesting and important pseudogroup structures, such as general G-structures, contact structures, and foliated structures, which must eventually be included in any complete treatment of uniformization of complex manifolds.

Some of these structures are well treated in other places

though [8], [lk] ; and the subject is anyway not sufficiently developed to warrant any attempt at a complete treatment.

In a discussion such as this it is a matter of choice whether merely to list the pseudogroups being considered, together with their defining equations and relevant properties, or rather to derive the defining equations and their properties from a classification of the possible pseudogroups of the limited class under consideration.

I have chosen the second alternative,

but to avoid requiring an unwilling reader to wade through the classification it has been included in a separate first part, from which the remainder of the discussion is essentially independent; so the unwilling reader need only glance at the list of pseudogroups contained in Theorem 1 at the end of §5, and refer to the properties of the defining equations as needed.

The general

study of pseudogroups of transformations was begun and carried very far indeed by E. Caxtan in a series of fundamental papers, [J] ; and the extension and completion of the classification of pseudogroups has been taken up recently by several differential geometers in a number of major papers, of which it may suffice here merely to mention [16], [29], and [39]·

However the classifi­

cation of the restricted set of pseudogroups being considered here can be carried out quite simply and completely, without use of the extensive machinery required in the general case; indeed the classification can be reduced to an

algebraic investigation of the subgroups or subalgebras of an easy and quite explicit finite Lie group or algebra, and some very classical analysis. The advantage of carrying out the classification in detail in this case is that it clarifies the relevant notion of equivalence and exhibits the possible alternative forms for these pseudogroups, while it also demonstrates the role of the defining equations and the parts played by their properties.

It may

also appeal to others, as it does to me, to see why such peculiar operators as the Schwarzian derivative must have the forms and properties that they do.

The second part contains a general discussion of pseudogroup structures on complex manifolds for the special class of pseudogroups being considered. here, with particular attention to the role played by connections.

The purely

formal aspects, which hold for all these pseudogroups simultaneously, are treated in §6, while the remaining three sections discuss some more detailed properties of connections for the individual pseudogroups.

The properties

treated are: integrability conditions, alternative characterizations of the pseudogroups (except for the projective pseudogroup, where this seems less interesting)j the differentiation operators associated to the connections, and the topological restrictions imposed by the existence of complex analytic connections.

To provide some illustrative examples the third part contains a

discussion of some aspects of these pseudogroup structures on two-dimensional compact complex manifolds, and is devoted primarily to the topics: which compact surfaces satisfy the topological restrictions the existence of complex analytic connections imposes; and then which of these surfaces actually admit complex analytic connections; and finally briefly which of these connections are integrable.

-6-

The group of k-jets and its Lie algebra. Consider the set of all germs of complex analytic mappings from the origin to t h e origin in the space k-jet of such a germ terms of order

f,

of

denoted b y

n

complex variables.

The

is defined to consist of the

in the Taylor expansion of the germ

f ; but since all

these germs are assumed to take the origin to the origin the conventional usage w i l l be slightly modified in that the constant terms in the Taylor expansion, the terms of order k-jet.

= 0,

w i l l not b e considered as part of the

Upon identifying a k-jet with its Taylor coefficients the set of all such k-jets can be viewed as a finite-dimensional complex

vector space; indeed

can be viewed as the direct sum

(1) where

is the complex vector space of dimension

consisting of the Taylor coefficients of order choosing any germs of complex analytic mappings

then such that

define

(2) noting that the k-jet of the composite mapping k-jets of the individual mappings operation (2) the set

depends only on the

It is readily verified that under the

has the structure of a semigroup with an identity

element, though not generally an abelian semigroup; the identity is the germ of the identity mapping.

The subset

form the group of invertible elements in

of germs of local homeomorphisms this group

will

-7-

b e called the general k-fold group or the group of k - j e t s , the special case being of course the general linear group. consists of all the jets nonsingular space

n x n

The group

such that the terms of order

matrix; thus

= 1

form a

is a dense open subset of the vector

and with the natural manifold structure inherited from that

vector space it is evident that

is a complex Lie group.

It is a quite simple matter to write the group operation in explicitly in terms of the natural global coordinates provided b y the encompassing vector space purposes.

,

or at least explicitly enough for the present

To do so it is necessary to be a bit more precise about the

coordinatization of the space

,

since there are various possibilities.

It seems most convenient for the present purposes to view the subspace of the (p+l)-fold tensor product

as consisting of

those tensors which are fully symmetric in the last

p

indices; the first

index w i l l be written as a superscript and the last

p

indices as subscripts,

so an element

is a tensor

(3) which is symmetric in the set of

k

p

lower indices.

A n element

is then the

tensors

w If in

f

is the germ of a complex analytic mapping from the origin to the origin and is given b y the

the k-jet

n

coordinate functions

then

w i l l be taken to be t h e element (!(•) with components (3)

-8-

given b y

(5)

This means that the k-jet is actually viewed as a set of derivatives of the coordinate functions rather than as a set of Taylor coefficients, just a difference of some combinatorial numerical coefficients; but the group operation (2) can then be obtained merely b y repeated chain rule for differentiation.

applications of the

In particular if

and

it follows readily that

(6)

(7)

(8)

and so on.

Formula (6) is just the usual m a t r i x product; and while the

ensuing formulas are somewhat more complicated, their general pattern is quite transparent. of the form

I n d e e d i s

a sum of

p

t e r m s , the q-th of which is

w h e r e d e n o t e s

nomial function of the components of the tensors

r) .

some poly-

That polynomial is in

-9-

turn a sum of terms of the f o r m w h e r e of the indices the indices

such tnai;

are various subsets '

is a permutation of

; all possible sizes of subsets

appear, since

all such differentiations appear upon iterating the chain rule, and the sum must be formally symmetric in the indices

.

Thus in general

(9) Here

denotes a sum over all sets of integers

such that and

so on,

consisting of

of the indices

a sum over some set of permutations of the indices

" .

denotes Actually

consists of the minimal sum needed to ensure the formal symmetry of in the lower indices, taking into account the symmetries of the tensors and

r) ;

be proved.

but that is a finer point than is really needed h e r e , so w i l l not Indeed the general formula is not really needed, and it is an easy

matter to verify any particular case of the formula. p =

(10)

the next case after (8), the formula is

For example in the case

-10-

where and

is a sum over is a sum over

6 3

permutations, permutations; for

symmetric in the indices and

(since

is a sum over

said

U

permutations,

the expression is already

and is also symmetric in the indices

is symmetric), so the summation is only extended

over a set of permutations i n the symmetric group on

^

letters which

represent cosets of the subgroup describing this symmetry, and similarly in the other cases.

The structure of the Lie group

can b e described in general

terms rather easily, without making m u c h use of the preceding detailed form of the product operation; describing subgroups of

but more details w i l l be needed later in .

Note that for any integers

possible to consider the

of a k-jet

it is , this defines a

mapping

which is evidently a surjective group homomorphism.

In terms of the represen-

tation (It) of course

For the special case

the kernel of this group homomorphism can be

identified with the vector space is clearly the point set where

T

k

J

indeed the kernel of this homomorphism in the decomposition (l),

is the identity matrix (the identity element in

being

and it follows easily from (9) that in this subgroup the group operation amounts to addition in the vector space

.

There thus arises

-11-

an exact sequence or groups

(11) for any index group

showing that and

is an extension of the vector-space

is as already observed the general linear

group.

Having obtained an explicit form for the group operation in it is a straightforward matter t o derive a correspondingly explicit form for the bracket operation in the Lie algebra Since

is a dense open subset of

of that Lie group. it is clear that as a vector space

can be identified with G^,

is a one-parameter subgroup of

expressed in terms of the global coordinates just introduced, the

corresponding element of the Lie algebra is the vector Furthermore the associated right-invariant differential operator on the manifold

is

where

f

is any differentiable function in an open neighborhood of the point .22] ; and writing

and recalling that the product

for the global coordinates (3) for short, is linear in the first factor, it

follows that

If

Y

is another vector in the Lie algebra

is the element of

such that

then the bracket

[X,Y]

-12-

hence (12) If

then the left-hand, side of (11) is

where the unwritten terms involve

; so to compute

it suffices merely to calculate the coefficient of the multinomial

on the right-hand side of (12).

For this purpose, consider-

ing initially only the first part of the right-hand side of (12), the only terms in

which need be considered are those which involve

multinomials in the tensor components

and the only terms in

which need be considered are those which involve the products of multimomials in

so writing

with

* ,

with the only

terms in the first part of the right-hand side of (12) which need be considered are

where indices

and

denotes a sum over some set of permutations of the .

When

the only nontrivial terms are those for

-13-

which

,

while when

the only nontrivial terms are those

for which

so this expression simplifies to

A

where

v

indicates that the v-th term in the product is omitted.

This can

be rewritten as

and the contribution from the second part of the right-hand side of (12) is of the same foim 5 but with. X

and

Y

interchanged, and a negative sign»

Con~

sequently

(13)

where

denotes a sum over some set of permutations of the indices

In particular, for some small values of explicit form

(Ik)

p

the bracket operation has the

and so o n , changing

where the unwritten terms in (15) and (16) are obtained b y interX

and

Y

in the first terms; the symmetrizations

are both summations over three terms.

These Lie algebras can be identified

with the initial parts of the Lie algebras of derivations of the rings of formal or convergent power series over

,

either directly from the defini-

tion or b y using the explicit forms just derived; thus this can be viewed as a rather complicated derivation of the Lie algebras which are basic to the customary development of the classification theory of Lie pseudogroups, as in [17] for instance. For some purposes, however, the explicit forms obtained here are quite convenient; and this approach is rather more primitive, hence perhaps more comprehensible to those not wishing to get involved in the traditional differential-geometric m a c h i n e r y , than some others.

The structure of the Lie algebra parallels the structure of the Lie group

in general terms of course . The Lie group homomorphisms

induce surjective Lie algebra homomorphisms

of the same form whenever

For the special case

the

kernel of this Lie algebra homomorphism can be identified with the vector space

viewed as an abelian Lie algebra, that is, as a Lie algebra with

identically vanishing bracket product.

Indeed the kernel of this homomorphism

-15-

is clearly t h e point set

in the decomposition (l); and

it follows easily from (13) that the "bracket operation in this subalgebra is trivial.

There thus arises an exact sequence of Lie algebras

(IT) for any index Lie algebra linear group.

showing that ; and

is an extension of the abelian is the Lie algebra of the general

§3-

The pseudogroups defined by partial differential equations. The definition and classification of the pseudogroups defined by

families of partial differential equations are rather straightforward matters once the preceding general machinery has been developed. of partial differential equations of order morphisms from !En subvariety

to

Kn

A C Gk(n,iE).

k

An analytic family

in the analytic local homeo-

can be thought of merely as being an analytic

Of course this is a somewhat restrictive definition,

since such families of partial differential equations do not involve the actual values of the mappings but only the derivatives of orders 1 through

k

of the component functions of the mappings, and the coefficients are constants; but for the purposes at hand this restriction is not unreasonable, indeed is rather natural.

The solutions of such a family of partial differential

equations, the set of those analytic mappings from subdomains of Oin

!En

into

such that the k-jets of those mappings at each point of the domains of

definition are contained in the subvariety whenever group

A

is a subgroup of

A,

ΰ^(η,ίΕ) ; and a closed subgroup of the Lie

G^(n,£t) is necessarily a Lie subgroup.

Lie pseudogroup of order

k

are closed under composition

of mappings in !En

A complex analytic restricted is defined to be the set of

Ο^ί(ζ) e A

in the domain of definition of f,

is a Lie subgroup of

ζ

into

JEn

f

for all points

from subdomains of

Bln

all complex analytic mappings

such that where

A

G-^n,!!) called a defining group for the pseudogroup;

the pseudogroup defined by a subgroup All the mappings in a pseudogroup

A C G^(n,(E)

will be denoted by

ψ{κ).

ψ(A) are complex analytic local homeo-

morphisms; the inverse of any mapping in

T(A) also belongs to

whenever well defined; and the composition of any two mappings in

ψ(A) If (A) also

-17-

f(a)

belongs to

whenever w e l l defined.

F o r the classification of these pseudogroups it is not necessary to consider all subgroups subgroups of pseudogroup.

since distinct subgroups, even

for distinct values of For any subgroup

minimal subgroup

defining the same pseudogroup :

integrable if

A subgroup

; equivalently a subgroup

if for any element

for all

U

k

will be called is integrable

;

of the origin in

z s U

pseudogroups of a fixed order

values of

consists of all

there exists a complex analytic homeomorphism

from some open neighborhood

subgroups of

m a y w e l l define the same

there is a naturally associated

the k-jets of all elements of

that

k,

k

f

such

and

Thus when examining Lie

it suffices merely to consider integrable

but integrable subgroups of

m a y still define the same pseudogroup.

for distinct The general problems

involved in an analysis of integrability or of the minimal order of a pseudogroup are nontrivial and quite interesting, but there are so few pseudogroups of fairly general form that a detailed treatment of these problems is not needed here; indeed for present purposes a rather simple necessary integrability condition, which can be described directly in terms of the Lie algebras, is all that is really needed.

To describe this condition, for any index

introduce the linear mapping

which associates to an element the element

with with

-18-

(18) and then to any linear subspace

associate the subspace

defined b y

(19) so that

and

Lemma 1.

be a Lie subgroup with associated Lie algebra where

If

A

is integrable then

is integrable and

then

and

define the same Lie pseudogroup.

Proof.

In an open neighborhood

submanifold

of the identity in

the

can be defined b y an analytic mapping

the sense that of the mapping submanifold the mapping

V

in

is the set of common zeros of the component functions 9 ;

and

is the tangent space to the

at the identity, so can be defined b y the differential of 9

in the sense that

(20 ) where before.

denote the natural global coordinates If

A

is integrable then for any fixed point

a complex analytic homeomorphism origin in

f

from some open neighborhood

such that

as there exists U

of the and

-19-

and

is near enough to the identity then for all

near

z = 0 ;

and setting

z

near

0,

and consequently

and upon differentiating this identity with respect to z = 0

it follows that

(21) This last identity holds in particular at all points subgroup

of

A,

for

t

this identity with respect to

of any one-parameter

sufficiently small; and upon differentiating t

and setting

so

and that

algebra

corresponding to the subgroup

t = 0,

recalling that

is the element of the Lie it follows that

(22) Upon comparing (20) and (22) it follows that hence that

whenever

which demonstrates the first part of the lemma.

For the proof of the second part of the lemma, the set of all k-jets of all elements

form an integrable subgroup

such that

and since necessarily

and

Now the elements

tangent vector

f

of which these are the k-jets

b y definition of the subgroup X

to the subgroup

B

to the subgroup If it is assumed that

B ; hence as above any

at the identity satisfies (22).-

comparing (20) and (22) and recalling that the tangent space

is integrable

near enough to the identity

evidently satisfy (2l), since the mappings satisfy

A

B

j ^ ^B = j ^ -jA

Upon

it follows that

at the identity satisfies then

and

-20-

consequently ment it follows that desired.

; but then in view of the previously obtained containB = A,

and hence

That suffices to complete the proof of the lemma.

as

-21-

The classification of tangentially transitive pseudogroups: algebraic aspects The detailed classification of pseudogroups w i l l only be attempted here for the special case of the tangentially transitive Lie pseudogroups, those for which all the defining groups

have the property that

; these are the pseudogroups for which there are no restrictions imposed on the values of the Jacobian matrices of the mappings.

The classifica-

tion apparently involves determining all the integrable subgroups with

and then determining which of these

subgroups describe the same pseudogroups; but it is actually a considerably simpler matter than might be expected. is an integrable subgroup for some

then

is an integrable subgroup; and the exact sequence (ll) induces an exact sequence

(23) where

can be viewed as a linear subspace

of the tensor space

The kernel

is a normal subgroup of

whenever

A,

so

the product and writing

(9) that

it follows readily from where

(2»0

Thus when

is viewed as a linear subspace nd any matrix

then for any tensor the tensor

given b y (2^)

-22-

must also be contained in

Now the expression (2^), when viewed as

as a function

of the matrix

describes a representation

of the group

transformations on the vector space

and the tensor as a group of linear

T ^ ; for (2k) is clearly linear in

and it is easily seen that two matrices

.

giving

r^,

for any

Indeed the representation

is one of the classical

symmetry representations of the general linear group, the representation in the notation of [Vf]. that

b e a normal subgroup of

A

thus amounts to the condition that

be invariant under the representation when

The condition

of the general linear group on

is viewed as a subspace

.

The same conclusion can of

course be obtained b y considering the Lie algebras of the groups involved. JUL

is the Lie algebra of the group

A

then corresponding to the exact

sequence of groups (23) there is the exact sequence of Lie algebras

(25) where

;

space of the tensor space The kernel

,

can also be viewed as a sub-

and then coincides with the subspace

is an ideal in the Lie algebra and

product

so whenever the bracket

; and it follows readily from (13) that where

(26)

Thus when

is viewed as a linear subspace

then for any

If

-23-

tensor

and any m a t r i x

the tensor

given h y (26) must also be contained in

.

describes a Lie algebra representation

The expression (26) which is indeed

merely the differential of the representation

,

as follows immediately

upon differentiating the expression (2k)-, for when considering a one-parameter subgroup

necessarily

of order in the bracket algebra of formulas.)

(The reversal

reflects the identification of the Lie

with right-invariant vector fields to simplify the The invariant subspaces of

under the group representation

coincide with the invariant subspaces of

under the Lie algebra

representation When

n = 1

the space

is one-dimensional for any

only possibilities for the kernel When

the representation

the direct sum

k; so the

are either for any

is decomposable into

of two irreducible representations, as is

demonstrated for instance in [36].

Thus there is a direct sum decomposition

where the subspaces

are invariant and irreducible

under

and the only possibilities for

the k e r n e l :

are either This decomposition can be described conveniently and explicitly b y

a projection operator b y a linear mapping a u s e d , for

for any

Q

commuting with the representation such that

that is,

and

and any nontrivial such mapping can be

-2k-

is then a nontrivial decomposition of under the representation

into subspaces which are invariant

so must coincide with the above decomposition.

It is a straightforward calculation to verify that the linear mapping

2

defined b y

(27) where

,

has the desired properties; so let

(28)

As a brief digression, but for use at a later p o i n t , an interesting alternative description of this decomposition of the representation should be noted here. subspace

In addition to the tensor space

of the k-fold tensor product

fully symmetric tensors; thus an element

introduce the consisting of the

is a tensor

(29) which is symmetric in the

k

indices

.

there can then be introduced the linear representation linear group which associates to any element the vector

(30)

having components

On this vector space of the general and any vector

-25-

where as usual

this is again one of the classical symmetry-

representations of the general linear group, the representation in the notation of [^7]-

For any index

there is a natural linear

mapping

(3D the contraction mapping, which associates to any tensor the tensor

defined "by

(32)

It follows readily from the definition (32) of the linear mapping

P

the descriptions (2k) and (30) of the representations

and that

(33) for any element linear mapping

and any tensor P

The image of the

is therefore a linear subspace of

invariant subspace of the representation nontrivial while

which is an

; and since this image is

is known to be an irreducible representation it follows

that the image of the linear mapping more the kernel of the linear mapping

P

is the entire space P

is a linear subspace of

is an invariant subspace of the representation is clearly a nontrivial proper subspace of

.

Furtherwhich

; and since this kernel and is contained in

as is evident upon comparing (27) and (32), it follows that the kernel of the linear mapping

P

coincides with the subspace

,

Thus (31) can be

-2k-

extended to the exact sequence of linear mappings (3b) which commute with the appropriate linear representations; and this also exhibits the decomposition of the representation tuents.

Furthermore this argument shows that

into irreducible constiis isomorphic to

in such a manner that

(35) an observation which w i l l eventually be useful but which w i l l not be needed immediately. Having thus determined the possible kernels in the exact sequences (23) and (25), it is a relatively straightforward matter to describe the possible subgroups

b y listing the corresponding subalgebras

Consider first a subalgebra with kernel

and

in the extension (25).

of the form

where

There must be an element is the identity matrix; here

is determined uniquely up to the addition of an arbitrary element of For any element

the bracket product

and it follows readily from (13), indeed from the special cases (l^) and (15), that

where

the Lie algebra representation ,

(30)

or equivalently

and being given explicitly b y (26).

Thus

-27-

This last equation is a linear equation in the variables and describes a linear subspace of the same dimension as

containing

and clearly of

consequently ( 3 6 ) describes precisely the

;

subalgebra

On the other hand it is a simple exercise, to see that ( 3 6 )

using the Jacobi identity in the Lie algebra describes a Lie subalgebra

for any tensor

subalgebra is an extension of

this

by

contains the element

,

and

Therefore all the subalgebras

with

and with kernel

described b y ( 3 6 ) as

ranges over

; and

same subalgebra precisely when

are

and

describe the

so it suffices to allow

to range merely over coset representatives in

There are thus

four general classes of such subalgebras when depending on the choices of the kernel

two when ;

and within each class the

possible subalgebras are parametrized b y the vector space Actually for the purpose of classifying pseudogroups it suffices merely to consider subalgebras ,

when

n = 1.

when

That can be seen quite conveniently b y

examining the subalgebras and having

, or subalgebras

associated to integrable subgroups one of the forms already determined, and then

applying Lemma 1 ; and that naturally leads to the consideration of four cases, (i)

Suppose first that

has the form ( 3 6 ) with note that

is a subalgebra such that ; thus dim

since

.

whenever for a l l

j

then by

Lemma 1 j

and thus

If

-28-

dim

.

then

On the other hand

so b y (36) for all

hence b y

j,

; and thus

is determined uniquely b y

so

is similarly determined uniquely b y

dim

.

Since

have the same dimension it follows from Lemma 1 that the same pseudogroup.

and

A

and and

define

The same argument applies inductively in

k , the

obvious analogue of (36) showing that belongs to

so that

define the same

pseudogroup for all for

(ii)

Suppose next that

,

is a subalgebra such that

,

as in (28); thus

has the form (36) with and it can

be assumed that

then for all

j ;

j

b y Lemma

for all

and writing this condition out explicitly b y using (27),

Setting

i = j

but since Thus again

and summing over all values of

necessarily and

that

it follows that

and consequently

dim

iim

.

then uniquely b y

i

,

u p to an element of

; and

and hence

On the other hand if

so that

is determined for all

i , so

-29-

Writing this condition out explicitly b y using (26) and (27), and recalling that

so that

and consequently

for all

j ,

it follows that Setting

Since

i = j

n > 1

and summing over all values of

the expression

and consequently

note that

is thus determined uniquely b y

is also determined uniquely b y

dim and

i ,

; and therefore

so it follows again from Lemma 1 that determine the same pseudogroup.

If

determine the same pseudogroup for all next that

for

has the form ,

(iii)

Suppose

as in (28); thus

is the kernel of for every

general that

and

is a subalgebra such that

and it can be assumed that

A

then since

the further analysis reduces to that in case (i) ; hence

since

,

lote incidentally that

£2 b y definition and since b y an easy calculation, it follows in

-30-

(37) If

' all

Lemma 1 ; using (37) this condition can be implies that

, indeed

,

nontrivial element in

j

by

r e w r i t t e n h e n c e so that eith

or

as can be shown b y exhibiting any one

, For this purpose note that for any elements

there must exist elements in

;

the bracket product

then also belongs to

, and

it follows readily from (13), or better from (lU) through (16), that where

(38)

Thus it is only necessary to observe that on

when

is a nontrivial bilinear form

and to see t h a t , merely note that

is the tensor having

as its only nonzero component then

It follows from these observations that

dim

the other hand if

On then

and

for all

be rewritten as

j ;

and the latter assertion can

or using (26) and (37)

equivalently as

Thus

is determined b y

dim that

up to an element of

so that

so again it follows from Lemma 1 A

and

determine the same pseudogroup.

The kernel

now

-31-

being

,

the same argument applies for all indices

as well; it

suffices merely to note that the nontriviality of follows upon considering the bracket product of elements and

of

and

are quite arbitrary. and

Thus once again

define the same pseudogroup for a l l

(iv)

Suppose finally that If

k = 3

is a subalgebra such that then for any tensors

exist elements product

there must in

_ . _

JSL ; the bracket

then also belongs to (51 , and it follows readily from

(13) as before that

where

it is t h e n easy to see that

has the form (38). and hence that

indeed it is only necessary to find elements

such that

nontrivial and is not contained in either

.

take for and for Z

has

If

the tensor having

is

For this purpose

as its only nontrivial components

the tensor having

as its only nontrivial component; then

as its only nontrivial components, and is easily

seen to have the desired properties.

If

n = 1

is trivial, so that there are two possibilities:

the bilinear expression (38) either

or

In the first case there is a nontrivial subalgebra but as in (i) it is not necessary to consider subalgebras with

for values

If

; and in the second case is a subalgebra such that

then for any tensors elements

and

there are and their

-32-

"bracket product for for

is ail element of

the tensor having

even for

having

its only nontrivial component as its only nontrivial component,

n = 1 ; and since it is easily seen that

either

Taking

as its only nontrivial component and

the tensor h a v i n g a s

yields the tensor

.

Z^

is not contained in

it follows that Thus for the remainder of the discussion the only subalgebras

that need be considered are the subalgebras

described b y

(36) and the one additional class of subalgebras that

and

.

algebras more explicitly note that when one-dimensional, so an element complex numbers

such

To describe this last class of n = 1

each tensor space

T^

is

is described b y the three

; and the bracket operations (l^t) through (16)

have the simple form

(39) There must exist an element complex constant If

; and

of the form is even uniquely determined, since

is any element of also belongs to

that

for some

then the expression ; but it follows easily from (39)

hence

and

C^o) This last equation describes a linear subspace of the same dimension as

,„. ,

,

which is of

and which must consequently coincide with

_;

-33-

and it is a straightforward matter to verify that (Uo) defines a subalgebra of

for any complex constant

,

so this additional class of

subalgebras is parametrized b y Rather than determining at this point exactly which of the subgroups described b y the subalgebras (36) and (1+0) are integrable, it is more convenient to describe some simple necessary conditions the parameters Ag

in (36) must satisfy in order that the associated subgroup be integrable;

that these necessary conditions are actually sufficient w i l l then follow easily after a discussion of equivalence of pseudogroups.

Suppose therefore

that the subgroup

associated to the subalgebra

defined b y (36) is integrable.

The 3-jets of all mappings in

compose an integrable subgroup subalgebra

such that associated to

,

then has the property that

and it follows from Lemma 1 that

exist an element the subalgebra

A'

,

the

where

; and since

.

There must describes it follows

that

(1+1)

There are only two cases in which this condition leads to any interesting consequences for the tensor

.

(i)

First suppose that

.

In

this case (4l) can be rewritten more explicitly using (26) as

The left-hand side of the above equality is fully symmetric in the indices

-3-

so the right-hand side must h e also; and that is clearly equivalent to the assertion that

(42)

Thus (!+2) is a necessary condition that the tensor for integrability.

(ii)

Hext suppose that

in this case it can also be assumed that for all that

Y = S •Y ,

j .

must satisfy

Since

and recall that and hence that

consists of a l l tensors

such

condition (l+l) can be rewritten more explicitly using (26)

and (27) as

On the one hand setting

and summing over

i

it follows that

and on the other hand the left-hand side is symmetric in the indices s o

the right-hand side must be also; and upon combining these two

observations it follows that (U3)

Thus (1+3) is a necessary condition that the tensor for integrability in this case.

is symmetric :

must satisfy

-35-

To describe the subgroups of subalgebras (36), for any t

e

n

s

o

corresponding to the r

i

n

t

r

o

d

u

c

e

the complex

analytic mapping

defined b y

w

where

for a

n

y

U

s

i

n

g

(6) and (7) it

is easy to see that

for any two elements

i and recalling (2U) this can be

rewritten equivalently as

It is an immediate consequence of (1*5) that the zero locus of the mapping , the subvariety

(h6)

is actually a subgroup of of the mapping

Furthermore if

with the projections of

spaces of the representation also define subgroups of

n > 1

the compositions

to the invariant sub-

satisfy equations analogous to (^5) and hence That is to say, the mapping

-36-

satisfies

so that

loc

is a subgroup of

; and. similarly for

Ct8) How if all

is a one-parameter subgroup of

t ;

for all t = 0

or more explicitly, recalling

t.

then

for

(V+),

Upon differentiating this identity with respect to

and recalling (26),

t

at

it follows that the element

in the Lie algebra

associated to this one-parameter subgroup

satisfies

hence the Lie algebra of the subgroup

loc

is the subalgebra (36) for

is a one-parameter subgroup of for all

loe

loc

Similarly if

then

t , and it follows correspondingly that

for the element

associated to this one-parameter subgroup ;

hence the Lie algebra of the subgroup (36) for

loc

is the subalgebra

and dually the Lie algebra of the subgroup

loc

is the subalgebra (36) for corresponding to the subalgebra (36) for

full group

The subgroup of is of course the

itself. Finally, to describe the subgroup of

the subalgebra (^0), for any complex constant

corresponding to introduce the complex

-37-

analytic mapping

defined b y (49) for any

Using (6) through (8) it is easy to see

that

for any two elements

in

; and recalling (2k) this can be

rewritten equivalently as (50) The zero locus

loc

is then of course a subgroup of

considering the one-parameter subgroups of

loc

;

and

as above it follows readily

that the Lie algebra of this subgroup is the subalgebra (1*0). In summary t h e n , as defining groups of all possible tangentially transitive restricted Lie pseudogroups it suffices to consider the following subgroups of

:

(51)

l o c w h e r e s a t i s f i e s

(1*2) ;

(52)

l o c w h e r e s a t i s f i e s

(1*3) and

(53)

l o c w h e r e

(5*0

loc

where

and

and

n = 1 .

;

In addition to the pseudogroups having the above defining equations there is the trivial case of the pseudogroup of all complex analytic local homeomorphisms. There remain to be handled the questions whether these equations do indeed define tangentially transitive Lie pseudogroups, then whether the pseudogroups so defined are actually distinct, and finally just what are these pseudogroups; the analysis leading to this list merely guarantees that any possible tangentially transitive Lie pseudogroup can be defined by one of these equations. though.

It is more convenient to handle these questions indirectly,

-39-

§5-

The classification of tangentially transitive pseudogroups: analytic aspects There are the four general classes of possible defining groups

for tangentially transitive Lie pseudogroups, described b y the four classes of equations (5l)> (52), (53) s and (5I+) respectively; and within each class the defining groups are parametrized b y a linear space of tensors

A.

It

is useful to introduce a notion of equivalence among the defining groups in each class separately; but the definition and elementary properties of this relation are formally almost the same in the different eases, so for convenience w i l l only be discussed in d e t a i l for the class given b y equation (51). In that case the defining groups equivalent, written g

and

loc

w i l l b e called

if there exists a complex analytic homeomorphism

from an open neighborhood

g(0) = 0

loc

U

of the origin in

to

g(u)

such that

and

(55) this is actually an equivalence relation, as is readily verified b y using the basic identity (^S). observe that the equation

For this and other purposes it is convenient to can be written in the form

(56) where

is the particular case of this equation corresponding to

A - 0 ; thus (55) can be rewritten

(57)

-1*0-

The situation in the other three classes is quite analogous To see the significance of this notion of equivalence suppose that

so that there exists a complex analytic homoeomorphism

satisfying (57), and consider an element f

g

i ; it can "be assumed that

is also a complex analytic homeomorphism between two open neighborhoods

of the origin and that

f(o) = 0.

The condition that

can be

rewritten using (56) in the form

for all points

z

in an open neighborhood of the origin.

The composition

is also a complex analytic homeomorphism. between two open neighborhoods of the origin such that

:

; and using (1*5) and (55)

it follows that

hence that

Thus whenever

coordinates near the origin in into the pseudogroup

there exists a change of

which transforms the pseudogroup in the sense just indicated; so for the eventual

purposes of this paper it is quite sufficient merely to consider one defining group from an equivalence class.

The advantage of stating the definition of

equivalence in terms of the defining groups rather than of the pseudogroups is that it avoids any questions of integrability in the definition or elementary properties of the equivalence relation. Now in each of the four general classes of possible defining

-ill-

groups for tangentially transitive Lie pseudogroups the defining groups corresponding to different values of the parameter

A

are actually all

equivalent; thus in place of considering four general classes of defining groups it is sufficient merely to consider four explicit defining groups, say those corresponding to the value

A=O

in each case.

That these four

equations do define distinct tangentially transitive Lie pseudogroups is then easily verified, by determining the corresponding pseudogroups quite explicitly; and that will complete the classification of these pseudogroups. The demonstration of the equivalence is most easily accomplished by showing that

A ~ 0 for any admissible parameter value

A ; and recalling (57) that

merely amounts to showing that there exists a complex analytic mapping from the origin to the origin in !E

n

singular and

g

θ(32β(ζ)) = - a ,

groups.

such that the Jacobian of g

g

is non-

satisfies the system of partial differential equations or the corresponding system in the case of the other pseudo­

It is convenient as a preliminary to recall the following rather

classical integrability theorem, and the subsequent useful particular observa­ tion.

Lemma 2.

Let

λ

neighborhood of the origin in analytic mapping

g

be a complex analytic mapping from an open χι

&

into

2

T (n,it).

There exists a complex

from the origin to the origin in In

has any specified value and

where

j2g(z) = ξ(ζ) , if and only if the expression

such that ξ^Ο) 0

-1*2-

(59)

is symmetric ill the indices

Proof.

This is a simple exercise in applying the classical

integrability conditions of Riquier and Janet

[25]

or their modern counter-

parts; but since the proof is so simple in this case an outline w i l l b e included here for the sake of completeness.

Repeated differentiation of (58)

and then the use of (58) to simplify the results shows b y an easy induction that if

g(z)

satisfies (58) then the higher-order terms of the jet satisfy

(60)

where

is the given mapping

and for

(61)

Thus if there exists a function

g(z)

of the desired form then the expressions

defined inductively b y (6l) must be symmetric in the indices for all

Conversely if all these expressions (6l) are

symmetric then having chosen

the formula (60) determines the coefficients

of a formal Taylor expansion estimates [25]

g

satisfying (58); and the usual

show that this series converges in some neighborhood of the

origin, hence represents a function having the desired properties. symmetry of all the expressions formula (6l) reduces to (59); while for

note that for v = 3

As for the the general

b y iterating (6l) it follows

-1+3easily that n

so b y induction this is symmetric in all indices.

Thus it is sufficient

merely to assume that (59) is symmetric, and that concludes the proof.

Lemma 3-

If

g

is a complex analytic mapping from the origin to

the origin in

is the

Jacobian determinant of the mapping

g,

then

(62) where

Proof. matrix

Letting

denote the j-th column vector of the

note that the determinant

is a multilinear function

of the column vectors so that

Expanding this determinant in cofactors of the k-th column, noting that the cofactor of in the original

is the same as the cofactor matrix

it follows that

of the element

-kk-

but as is w e l l known

so the desired result has "been

demonstrated. Turning then to the equivalence assertions, separate arguments are needed for the four separate cases.

Considering first the defining

group (51),in order to show that

it is necessary to show that there

exists a nonsingular analytic mapping such that

.

g

from the origin to the origin in

Setting

explicit form (kb) for the mapping

and using the ,

this equation can be rewritten

in the form

It follows immediately from Lemma 2 that there exists a solution desired whenever the expression since (k2) that the parameter

g

as

is symmetric in the indices

are constants; but that is precisely the condition A

is required to satisfy, hence

as desired.

Considering next the defining group (52), in order to show that it is necessary t o show that there exists a nonsingular analytic mapping g

such that

that

or where

equivalently such

Using the explicit

formulas (27) and (V+) and recalling Lemma 3 it is readily verified that

(63)

where

is the Jacobian determinant of the mapping

g

at the point

z ;

so setting

the equation for the mapping

g

can be

rewritten as (6k)

where

(65)

It is easy to see that if

satisfies (61*)

is of the form (65) for some function is necessarily a constant, and hence

where

then g(z)

satisfies the desired conditions;

so to demonstrate the existence of the desired function

g(z) it is only

necessary to show that there exists some analytic function the equations (6k) are Integrable.

1

for which

H o w using Lemma 2 this integrability condi-

tion readily reduces to the condition that the expression

is symmetric in the indices

but since

A

satisfies (1*3) this

is equivalent to the condition that

is

symmetric in the indices

equivalent to the condition that

and this in turn is clearly

-k6-

(66)

for all indices

.

Thus the problem is reduced to that of determining

whether there exists an analytic function (66).

Setting

satisfying the equations

reduces (66) to the linear system of partial

differential equations

(67)

for which the integrability conditions are classical [25] and can be obtained b y arguing as the proof of Lemma 2. function

satisfying (67)

Indeed if there exists a

then b y repeated differentiation and simpli-

fication it follows inductively that the function

t(z)

satisfies

(68)

for all

where

is the given t e n s o r , and inductively

(69) for

If the expressions

in the indices

are then equations (68) determine the Taylor

coefficients of the desired solution. indices

symmetric

That

is just the condition (^4-3) that

is symmetric in the A

is assumed to satisfy,

and it is a simple calculation to verify that that in turn implies the symmetry of the expression

; and as in the proof of Lemma 2

- k

7

-

the iteration of (69) then yields expressions showing the desired symmetry whenever

Thus the equations (67) are integrable, and as noted that

implies that 11

Considering thirdly the defining group (53)> i- order to show that

Lt is necessary to show that there exists a nonsingular analytic

mapping

g

such that

Recalling (63), this

equation can be rewritten as

(70)

If (70) holds then it is readily verified that

(71)

and conversely if (71) holds then since that (70) holds;

b y assumption it is clear

thus (70) is equivalent to (71), and since the latter

equation obviously has solutions it follows that

Considering finally

1

the defining group (5 *) in the one-dimensional c a s e , in order t o show that it is necessary to show that there exists a holomorphic function that

and

g

such

; but this is an analytic ordinary

differential equation, for which there always does exist such a solution [3], so that in this case too Now it is an easy matter to determine explicitly the pseudogroups having the listed defining equations when the parameter has the special value

A = 0 ; but that too requires the consideration of four separate cases.

—US—

First for the defining group (51) with

A = 0

the associated pseudogroup

consists of those nonsingular complex analytic mappings ; but writing

f

such that

and using the explicit form (iA)

this clearly reduces to the condition that hence that

f

is an affine mapping

(72) for some constants

.

is just that the matrix (52) with

A = 0

The condition that is nonsingular.

f

be a nonsingular mapping

IJext for the defining group

the associated pseudogroup consists of those nonsingular

complex analytic mappings and writing

f

such that and recalling (1A) and (63) this readily reduces

to the condition that

(73)

for the mapping

where f.

is the Jacobian determinant of

Using Lemma 3 it is an easy calculation to see that if

f

is

any nonsingular mapping satisfying a condition of the form (73) for some function

then

thus to find the desired mappings

is necessarily a constant; f

it is sufficient merely to find solutions

of the system of partial differential equations (73) for arbitrarily chosen functions

Of course these functions

the system is integrable; but from Lemma 2 with

must be chosen so that

-1+9-

it follows readily that the integrability condition is just that the expression

is symmetric in the indices

Thus the function for some constants

hence equivalently that

°"(z) must be such that b

and

b., .

The equation (73) equivalently can then be rewritten or yet

therefore that (7*0

f

for

some constants

, so

must b e a projective mapping

As is familar, the condition that matrix

f

be a nonsingular mapping is that the

-50-

is nonsingular; and as noted earlier

must be a

constant, so that the Jacobian determinant of the mapping

f

must be of

the form

(75) for some constant

c.

Thirdly for the defining group (53) with

A = 0.

the associated pseudogroup consists of those nonsingular complex analytic mappings

f

such that

; and writing

and recalling (63), this equation readily reduces to the condition that (5U) with

is constant. A = 0

Finally for the defining group

the associated pseudogroup consists of those nonsingular

complex analytic mappings

f

such that

;

classical Schwarzian differential operator [2o]

so

f

but this is the

must be a one-dimen-

sional projective m a p p i n g , otherwise known as a linear fractional or Mdebius transfomation.

These and the preceding results can then be summarized as

follows.

Theorem 1.

Up to equivalence there are only the following

tangentially transitive restricted Lie pseudogroups of complex analytic transformations in (i)

:

the pseudogroup of nonsingular complex affine mappings (72), having the

defining group

loc 8

or alternatively characterized b y the

partial differential equations (ii)

;

the pseudogroup of nonsingular complex projective mappings (7^-) for having the defining group

loc

characterized b y the partial differential equations

or alternatively

-51-

(iii)

the pseudogroup of nonsingular complex analytic mappings with constant

Jacobian determinants for loc

having the defining group

or alternatively characterized b y the partial differential

equations (iv)

the pseudogroup of nonsingular complex projective mappings (7^) for

n = 1,

having the defining group

loc

or alternatively

characterized b y the ordinary differential equation (v)

the trivial pseudogroup, consisting of a l l nonsingular complex analytic

mappings. For

n = 1

(v) ; while for and (v).

there are just the three pseudogroups (i), (iv), and there are the four pseudogroups (i), (ii), (iii),

The pseudogroups of projective transformations are listed separately,

in the two cases

and

n = 1,

since the defining groups or partial

differential equations differ so much in the two cases; and the pseudogroup of nonsingular complex analytic mappings with constant Jacobian determinant in the case

n = 1

coincides with the pseudogroup of nonsingular complex

affine mappings. It should be pointed out that equivalence does not involve an arbitrary change of coordinates, but rather involves a change of coordinates which transforms one of the pseudogroups listed in Theorem 1 into another pseudogroup which can also be defined b y a subgroup of equivalent pseudogroups still have rather special forms.

these To give merely the

simplest example, any pseudogroup equivalent to (i) in the case have a defining group of the form

loc

n = 1

for some constant

will A,

hence will consist of those analytic mappings equation

f"/f' = A(f'-l ) ;

f

satisfying the differential

and a simple calculation shows that if

A ^ O

these mappings have the form

f(z) = -A"1 log(e~Az + C) +

for some constants

C and

c'

c ' . The set of all these mappings do form a

pseudogroup, as can easily be verified directly; and all pseudogroups equivalent to (i) but not coinciding with (i) in the case some nonzero parameter

A

η = 1 have this form for

characterizing the pseudogroup.

It does not seem

worthwhile here to try to describe explicitly all the pseudogroups equivalent to those listed in Theorem 1 though.

-53-

§6.

Pseudogroup structures and their associated connections As is of course w e l l known, an m-dimensional topological manifold

M

is a Hausdorff space each point of which has an open neighborhood homeo-

morphic to an open subset of such a manifold is a covering of

A coordinate covering M

b y open subsets

which there is a homeomorphism subset

j the sets

homeomorphisms

p

and an open

are called coordinate neighborhoods, and the

are then homeomorphisms between subsets of

at

for each of

between

are called local coordinates.

coordinate transitions.

of

The compositions

, and

For any point

are related b y

,

called the

the two local coordinates j

and for any point

the three coordinate transitions at

p

are related b y

The union of two coordinate coverings is clearly another coordinate covering, the coordinate neighborhoods and local coordinates of which are the unions of those of the two initial coordinate coverings, but the coordinate transitions of which clearly include many more mappings than are in the union of the coordinate transitions of the two initial coverings. The manifold

M

can b e reconstructed from knowledge of the subsets

and the mappings

alone, b y taking the disjoint union of the sets

and identifying points If coordinates

m = 2n

and and

whenever

is identified with

can be viewed as mappings into subsets

then the local and the

coordinate transitions f

CC ρ

coordinate covering

1/£,

as mappings between subsets of

a;11.

The

is called a complex analytic coordinate covering

if the coordinate transitions are complex analytic mappings.

Two complex

analytic coordinate coverings are called equivalent if their union is again a complex analytic coordinate covering; this is easily seen to be an equivalence relation in the standard sense, using the fact that the composi­ tion of two complex analytic mappings is also complex analytic, but is a nontrivial equivalence relation.

An equivalence class of complex analytic

coordinate coverings of a manifold

M

is called a complex structure on

M ;

and a manifold together with a particular complex structure is called a complex manifold. The same construction can be employed to impose other structures on topological manifolds, using in place of the nonsingular complex analytic mappings any family of local homeomorphisms of ]Rm

closed under composition.

For example considering coordinate coverings for which the coordinate transi­ tions are

Cco mappings leads to

C00 structures on manifolds and to

C°°

manifolds, another very familiar and much studied structure and class of manifolds.

Since complex analytic mappings are

complex analytic coordinate covering is also a

C00 it is evident that any C00 coordinate covering and that

equivalent complex analytic coordinate coverings are also equivalent as coordinate coverings; so a complex analytic structure on a manifold M

C00 is

naturally subordinate to a C00 structure on M, or equivalently, a complex analytic manifold is also in a natural manner a all

Cco manifold.

Of course not

C°° manifolds admit complex analytic structures; and if a Cto

manifold

does admit a complex analytic structure it may well carry a number of inequiva-

Ient complex analytic structures.

The study of the relationships between

these structures is a fascinating and active enterprise. The main topic here however is the investigation of the structures associated to the various pseudogroups of complex analytic mappings described in Theorem 1. For the pseudogroup (i), considering coordinate coverings for which the coordinate transitions are nonsingular complex affine mappings leads to complex affine structures on manifolds and to complex affine manifolds·, similarly considering the pseudogroups (ii) and (iv) leads to complex projective structures on manifolds, and considering the pseudogroup (iii) leads to complex flat canonical structures on manifolds.

The pseudogroup (v) of

all nonsingular complex analytic mappings of course merely leads back to complex analytic structures on manifolds.

The mappings in all of these pseudo­

groups are complex analytic mappings, so all of these structures are subordinate to complex analytic structures; thus among other problems are those of determin­ ing which complex analytic manifolds admit any of these finer structures and of classifying all these additional structures on any particular complex analytic manifold.

In addition any complex affine mapping is in particular

both a complex projective mapping and a mapping having a constant Jacobian determinant; so a complex affine structure is subordinate both to a complex projective structure and to a complex flat canonical structure, and similar questions can be asked about the relations between these structures. All these proper subpseudogroups of complex analytic mappings are described by systems of partial differential equations which behave in a particularly simple manner when applied to compositions of mappings; the pseudogroup of nonsingular affine mappings for example consists of those non-

-56-

singular analytic mappings explicitly b y (W+) for

f

A = 0

for which

where

9

is given

and satisfies the basic relation (U5), and

the other pseudogroups are described similarly as noted before.

This

description permits a very simple formal splitting of the problem of investigating the existence and classification of these pseudogroup structures into two parts,at least one of which is quite readily expressible in terms of now standard machinery in complex analysis; and the formal part of this splitting proceeds in exactly the same w a y for all of the first four pseudogroups listed in Theorem 1 , and indeed for m a n y other pseudogroups as w e l l [19 3 j so it suffices to describe the reduction in detail only for the pseudogroup of complex affine mappings and then merely to note the results in the remaining cases. Consider then a complex manifold coordinate covering

M

with complex analytic

having local coordinates and coordinate transitions

In order that

M

admit a complex affine structure there must exist, after a refinement of the covering

if necessary, complex analytic homeomorphisms

such that for the new local coordinates

the

coordinate transitions mappings. tion that

That

are complex affine is a complex affine mapping is equivalent to the condiwhich b y (^5)

c a n

written in the form

-57-

and introducing the complex analytic mapping

defined b y (76) condition can be rewritten the (77) since the representation If

depends only on the one-jet of its argument.

is another set of complex analytic homeomorphisms such

that the local coordinates structure on (77)-

M

also describe a complex affine

then the functions

also satisfy

These two complex affine structures are equivalent precisely when the

compositions

are complex affine mappings; and b y (k^) that is just

the condition that

or equivalently that

To introduce a convenient terminology, a

complex analytic affine connection for the complex manifold

M

w i l l be

defined to b e a collection of complex analytic mappings for some complex analytic coordinate covering these mappings satisfy (77) in any intersection

of

M,

such that

; such a collection

of functions induces a corresponding collection of functions on any refinement

-58-

of the covering

and a l l these w i l l naturally be identified.

Such a

connection w i l l be called integrable if after passing to a suitable refinement of the covering

there exist complex analytic homeomorphisms

satisfying (76); of course this is equivalent to the condition that after passing to a suitable refinement of the covering merely nonsingular complex analytic mappings (76).

there exist satisfying

With this terminology the preceding observations can be summarized as

follows; this result is due to Matsushima [3**], and is also discussed b y Vitter in

[1*5].

Theorem 2.

On any complex manifold

M

there is a natural one-to-

one correspondence between the set of complex affine structures on the set of integrable complex affine connections on

M

and

M.

Having made these simple observations it is apparent that the corresponding result holds for the other pseudogroup structures on replacing

e

M,

b y the appropriate differential operator in the definitions and

assertions.

Thus a complex analytic projective connection for a complex

manifold

of dimension

M

merely

is a collection of complex analytic mappings

for some complex analytic coordinate covering of

M

such that

(78)

whenever

and such a connection is integrable if after passing

to a suitable refinement of the covering analytic mappings

VI

such that

there exist nonsingular complex

-59-

(79) whenever

A complex analytic canonical connection is correspondingly

a collection of complex analytic mappings

for some

complex analytic coordinate covering

such that

(80) whenever

; and such a connection is integrable if after passing

to a suitable refinement of the covering analytic mappings

vt

there exist nonsingular complex

such that

(81) Finally a one-dimensional complex analytic projective connection is a collection of complex analytic mappings

such that

(82)

whenever

,

and all these are necessarily integrable, [20].

is interesting to note incidentally that if affine connection then connection and conversely if

It

is any complex analytic

is a complex analytic projective is a complex analytic canonical connection; and

is a complex analytic projective connection and

complex analytic canonical connection then analytic affine connection.

is a

is a complex

The analogue of Theorem 2 holds for all these

other pseudogroup structures as w e l l , with formally the same argument. Wow using Theorem 2 the problem of investigating the existence

-6o-

and classification of these pseudogroup structures can clearly "be split into two parts: (i) the problems of determining which complex manifolds admit any connections at all and then of classifying these connections; and (ii) the problem of deciding which of these connections are integrable.

The

first part leads to some purely linear problems, readily expressible in terms familiar to complex analysts; these problems are quite interesting in their own right, for the bare existence of a complex analytic connection is often by itself a nontrivial property and can usually be viewed as a weaker form of pseudogroup structure on the manifold.

The second part is really

an integrability problem in the standard sense.

For both parts any detailed

results really require a case-by-case analysis; but at least the reduction of the first set of problems to a more familiar form is a simple formal exercise and can be carried out for all cases in basically the same manner.

Here too

the detailed description will only be given for the pseudogroup of complex affine mappings, and the corresponding results noted in the other cases.

As

a preliminary it is convenient first to review some of the relevant auxiliary machinery, in order to establish notation and terminology. Returning therefore to the complex manifold M analytic coordinate covering CC

:

CC

> V

Oi

IfL = (u } having local coordinates

and coordinate transitions

can be associated the nonsingular

(83)

ηχ η

f „ , to each point Ρ ε U CCp

matrix

τ α β ( ρ ) = ^1 f a p ( z p ( p ) ) I αβ y β

having entries (8h)

with complex

i τ αβ J

CC

Π UQ

ρ

-6l-

with the obvious notation.

This defines a complex analytic mapping and if

then these mappings

clearly satisfy

; therefore these mappings

describe a complex analytic vector bundle M.

This bundle

of rank

n

on the manifold

is called the complex analytic tangent bundle to

and is evidently independent of the choice of coordinate covering.

M, For any

complex analytic group homomorphism

the composite mappings

then describe a

complex analytic vector bundle

of rank

U

on the manifold

M.

The sheaves of germs of holomorphic sections of the bundles and

w i l b e denoted b y

and

The definitions and

standard properties of coherent analytic sheaves and of cohomology groups with sheaf coefficients can be found in most recent textbooks on functions of several complex variables, so nothing further need be noted here in general; but it is perhaps helpful to insert a few notational remarks about the particular cohomology groups

In terns of a given covering

a q-cochain

consists of a collection of

sections

and the coboundary

mapping

is defined b y

whenever

.

The cohomology groups for the covering

-62-

can then be defined for

by

where the space of q-cocycles is defined b y

and the space of q-coboundaries is defined b y

and the cohomology groups

can be defined as the direct

limits of the cohomology groups coverings

of

M.

over the directed set of

How over any coordinate neighborhood

U^

the bundle

is naturally a trivial b u n d l e , so a section

can

naturally be identified with a complex analytic mapping Similarly a section

can be identified

with a complex analytic mapping from the intersection ;

but there are

into

different ways of making this identification,

depending on which of the coordinate neighborhoods is hthen trivialization chosen e r e . naturally Wto i t hdescribe thisbe over systematic identified the the trivialization last convention with coordinate a collection ofa neighborhood the q-cochain bundle of complex analytic fill Henceforth always mappings be thecan used

-63-

and. the coboundary mapping

has the form

(85)

whenever

In particular

(86)

(

(87)

(BE

Now whenever

it follows that hence -using (U5) that

where

are the complex analytic mappings defined b y

(88) that is to say, recalling (87), the mappings

describe a cocycle

T h e n , recalling (86), the defining equation (77) merely asserts that a complex analytic affine connection is a cochain such that

Thus the condition that

- 6 -

there exists a complex analytic affine connection is just that the cohomology class

is zero in

; and if there exists at least one

complex analytic connection then the difference between any two such connections is a cocycle in

In summary

therefore these observations can be rephrased as follows.

Corollary to Theorem 2.

On any complex manifold

M

there exists

a complex analytic affine connection precisely when the cohomology class is trivial in

;

and if there exists a complex

analytic affine connection then the set of all such connections is in noncanonical one-to-one correspondence with the vector space

The corresponding assertions of course also hold for the other pseudogroups, merely replacing operator and representation.

and Thus if

b y the appropriate differential there exists a complex analytic

projective connection precisely when the cohomology class is trivial in

,

while if

n = 1

there exists a complex

analytic projective connection precisely when the cohomology class is trivial in

;

and if there exists one

such connection then the set of all such can be put into one-to-one correspondence with the vector space if

n = 1.

or the vector space Similarly if

there exists a complex

analytic canonical connection precisely when the cohomology class is trivial in

;

and if there exists one

such connection then the set of all such can be put into one-to-one correspondence with the vector space

It may be worthwhile to point out here another interpretation of these connections.

On any complex manifold M the principal bundle

associated to the complex analytic tangent bundle analytic fibre bundle

with fibre

transformations (83) as those of higher order analogues jets of order bundle

k

τ

is the complex

GL(n,(E) and with the same coordinate

τ . it is possible to introduce the

τ^ of the bundle

simply by considering

of the coordinate transitions of the manifold M.

(k) Ta is the complex analytic principal bundle over M

Thus the

with fibre

β^(η,ίϊ) and with coordinates transformations defined analogously to (83) by

ϊβ(ρ) = 3kWVp))

τ

for any point ρ e Uq Π

. It is easy to see that a complex analytic affine

connection really amounts to a reduction of the structure group of the bundle (2)

to the subgroup Ioc θ C G,-,(n,ffi) , while an integrable complex analytic

affine connection amounts to a reduction which can be realized by a complex analytic change of coordinates on

M ; thus, to parallel the terminology used

in discussing the existence and classification of complex structures on differentiable manifolds, a complex analytic affine connection may well be called an almost-affine structure on the complex manifold

M . The corresponding

assertions and terminology can also be introduced for the other pseudogroup structures. Pinally it is perhaps useful to include here a few remarks about the behavior of these structures under automorphisms of the complex manifold M.

If T : M

>M

is a complex analytic homeomorphism and

Ί/ί, =

is a complex analytic coordinate covering, witn the notation as before, then

-66-

T

can be represented b y the coordinate mappings

where

here

is a complex analytic mapping between subsets of the open sub-

domains

and

in

.

To any such automorphism

analytic affine connection translate of the connection tion

on s

by

M

T ,

.

the complex analytic affine connecwhere

It is a straightforward consequence of the

properties of affine connections and of the operator independent of that

,

hence is w e l l defined throughout

It also follows readily that of

that ,

is and further

does define another complex analytic affine connection on

Similarly it can easily be verified that

T

and any complex

there can be associated the

defined b y coordinate functions

whenever

T

M ,

M .

is integrable whenever

s

is.

for any two automorphisms

S

and

so that there is thus defined a representation of the group of

complex analytic automorphisms of

M

complex analytic affine connections on

as a group of operators on the set of M ;

the latter is even a linear

representation for any identification of the set of all such connections with the complex vector space representation.

indeed is the obvious linear

These observations depend only on the formal properties of

affine connections and of the operator

8,

so carry over immediately to the

-6γ-

other connections and the corresponding partial differential operators. If Γ

is a group of complex analytic automorphisms of M

the complex analytic affine connection sT = s

for all elements

T ε Γ·

s

is invariant under

Γ

then

if

I t i s readily verified that any p-invariant

complex analytic affine connection on

M

induces a complex analytic affine

connection on the quotient space Μ/Γ , whenever Γ

acts as a properly

discontinuous group of automorphisms having no fixed points so that

Μ/Γ

is also a complex analytic manifold; and any complex analytic affine connection on Μ/Γ

can conversely be viewed as a Γ-invariant complex analytic affine

connection on

M.

In particular if M

admits a unique complex analytic

affine connection then it must be invariant under any complex analytic auto­ morphism of M j

hence must induce a unique complex analytic affine connection

on any quotient space

Μ/Γ . Since integrability is preserved the same

results hold for integrable complex analytic affine connections, hence for complex affine structures; and these observations too extend immediately to the other structures considered here.

-68-

§7.

Complex analytic affine connections To begin the more detailed discussion of some properties and

applications of the complex analytic connections associated with the various pseudogroup structures, consider the complex analytic affine connections. A s might be expected from the terminology, these are essentially just the complex analytic analogues of the classical affine connections in differential geometry, but there is one point of difference which must be kept in m i n d . If

is a complex analytic affine connection then recalling

(2b) and (1*1*) the defining equation (77) can be written out explicitly in the form

(89)

and that is the complex analytic analogue of the familiar condition that the components

are the Christoffel symbols of a symmetric or torsion-

free affine connection, [27],

The Christoffel symbols of a general affine

connection are required to satisfy (89), but are not required to be symmetric in the lower indices; the differences

are the

components of a tensor called the torsion tensor of the affine connection. Furthermore the complex analytic affine connection

is

integrable precisely w h e n , after passing to a refinement of the covering if necessary, there are nonsingular complex analytic mappings such that such that

where

,

or recalling

(VO

; b y Lemma 2 the necessary and sufficient

-69-

condition that there exist such mappings

is that the expression

(90)

is symmetric in the indices

and that is evidently equivalent

to the vanishing of the expressions

(91) However the expressions

defined b y (90) and (9l) are just the

complex analytic analogues of the components of the curvature tensor of the affine connection, [27]; thus the integrable complex analytic affine connections are the complex analytic analogues of symmetric or torsion-free affine connections having zero curvature. That the classical Christoffel symbols, unlike the coefficients of the complex analytic affine connections, are not normally required to be symmetric in the lower indices, reflects the fact that they are not normally introduced in the investigation of affine structures on differentiable manifolds but rather in the investigation of a different but closely related problem; and that problem too has a complex analytic analogue. M

If the complex manifold

has a complex affine structure then there is a coordinate covering for which the coordinate transitions

are complex affine

mappings; and in terms of this coordinate covering the coordinate transformations

defining the complex analytic tangent bundle are constants.

The coordinate transformations

can be viewed as describing a flat

complex vector bundle, a fibre bundle having as structure group the group

-70-

GL(n,l) M

with the discrete topology; and in these terms, if a complex manifold

has a complex affine structure then the complex analytic tangent bundle

is analytically equivalent to a flat complex vector bundle.

Conversely if

the complex analytic tangent bundle is analytically equivalent to a flat complex vector b u n d l e , and if this equivalence is exhibited b y a suitable choice of coordinates on the manifold, then the manifold has a complex affine structure.

Thus the problem of whether the complex analytic tangent bundle

is analytically equivalent to a flat complex vector bundle is related to but somewhat weaker than the original problem of whether the manifold has a complex affine structure. To investigate this other problem briefly, the bundle

is

analytically equivalent to a flat vector bundle precisely w h e n , after passing to >a refinement of the covering if necessary, there are complex analytic mappings

such that ; and using the exterior differential operator

to the condition that

hence to the condition that

(92) where (93)

are constants in d

that is equivalent

-71-

Here

is an

of degree 1 in

matrix of complex analytic differential forms , and

is an

differential forms of degree 1 in matrix

matrix of complex analytic

. Writing the entries

of the

out in the form

( since there may be too few complex analytic tensor fields.

For this reason the property ( 9 8 ) , or in the differ-

ent table case the property (99), is often taken as the definition of an affine connection, identifying the affine connection with the associated covariant differentiation. Turning next to topological properties, the primary topological

-75-

invariants of complex vector bundles are the well known Chern classes, which can be defined as follows, [ 2 7 ] , [35]• sisting of

Choose any

matrices

con-

differential forms of degree 1 in the coordinate neighborhoods

and satisfying the condition (92) in each intersection

; that

there exist some,such matrices follows from a familiar argument using partitions of unity.

Then introduce the

matrices @ a

consisting of

ential forms of degree 2 in the coordinate neighborhoods of the chosen matrices

as in (95)-

UQ

Cf° differ-

defined in terms

It follows readily upon taking the

exterior derivative of (92) and then using (92) again to simplify the result that

(101) in each intersection an

, next if x

is an indeterminate and

X

is

matrix of indeterminates note that there is an expansion of the form

(102) where of the matrix

is a homogeneous polynomial function of degree X ; and

r

for any matrix

Therefore upon recalling (101) it follows that the expression globally defined

in the entries

differential form of degree

2r

is a

on the manifold M.

Following Weil it can be shown that these differential forms

are

closed, and that up to exact differential forms they are independent of the choice of the connection forms the differential form

, [ 2 7 ] , [35] ; thus by de Rham's theorem determines a cohomology class

which is also independent of the choice of the connection forms

. These

-76-

cohomology classes

are the Chern classes of the manifold

alternatively of the complex tangent bundle

over

M.

M ,

or

This construction

can be used to introduce the Chern classes of any differentiable (but not necessarily complex analytic) complex vector bundle over M ; but for complex analytic vector bundles, such as the complex analytic tangent bundle further refinement is possible. matrices

There exist positive definite

in the coordinate neighborhoods

, a

Hermitian

such that

in each intersection

, by using again the

familiar argument with partitions of unity; and since the matrices complex analytic functions then

, and hence as is easily verified

satisfy ( 9 2 ) in the intersections

the matrices using these functions

are

. Then

it follows directly that ( 9 5 ) takes the simpler

form

so that the differential forms making up the matrix

are not just of

degree 2 but of bidegree (l,l) ; and correspondingly the differential form representing the Chern. class

is of bidegree (r,r).

follows that no matter what differential forms

It therefore

were originally chosen the

Chern class is represented by the component of type (r,r) of the differential form

Now since the matrices

are holomorphic the matrices

consist of differential forms of bidegree ( 1 , 0 ) ; and hence in ( 9 2 ) there is no loss of generality in assuming that the matrices of differential forms of bidegree (1,0).

ea

also consist

Then in constructing the Chern

classes it is only necessary to consider the components of bidegree (l,l) in the matrices

0 ; so in place of (95) it suffices merely to take the

simpler expressions

(103)

θα = δθα ,

and the Chern classes are represented by the differential forms

(§^) ·

σ Γ

Finally it should be noted that the Chern classes can also be introduced as integral cohomology classes, so the differential forms '

cr (q ) rv α

0r

cr (@ ) rv α

have integral periods; but throughout the later discussion the Chern classes will only be viewed as real cohomology classes. There have been several investigations of the topological properties of flat vector bundles; a survey of some results in this direction and a useful bibliography can be found in [26].

Although only in a few cases is

there really a topological characterization of flat vector bundles, nonetheless it is not difficult to show that the Chern classes of flat complex vector bundles are all trivial.

As1 an extension of this, with the observations just

made it is also easy to show that the Chern classes are trivial for any complex analytic vector bundle admitting a complex analytic (nonsymmetric) affine connection; indeed if there is a complex analytic (nonsymmetric) affine connection

θ

then the matrix

hence the Chern classes

(¾)

defined by (103) is identically zero, and are

trivial.

For reference then, the results

described here can be summarized as follows. Theorem 3·

Xf

M

is a complex analytic manifold which admits a

complex analytic (nonsymmetric) affine connection then the Chem classes of M

-78-

are all trivial; and for each complex analytic affine connection the associated covariant derivative (100) determines a sheaf homomorphism

(98) and a complex linear mapping

(99) for any indices

p, q .

-79-

§8.

Complex analytic projective connections. Turning next to the complex analytic projective connections, there

is a well developed but perhaps not so well known classical theory of projective connections [11]; a particularly readable recent survey of that theory can be found in [28].

Here too the complex analytic projective connections

are just the complex analytic analogues of the classical projective connections, indeed more so than in the case of the complex analytic affine connections since for the projective connections symmetry is normally presumed. For the ease

n = 1

complex analytic projective connections were discussed in [2o],

and it was shown there that all such connections are necessarily integrable. For the case

complex analytic projective connections have also been

discussed in [13].

In this case a complex analytic projective connection is

described by complex analytic functions neighborhoods (2k),

such that

in the coordinate for all indices

j ; and recalling

(¥0, (W), and ( 6 3 ) , the defining equation ( 7 8 ) can be written out

explicitly in the form

(1 GL(N,!E) is one of

the standard analytic linear representations, the tensor product of the p-fold tensor product of the identity representation with the q-fold tensor product of the dual of the identity representation.

The representation

ρ

is

not generally irreducible, but is at least equivalent to the direct sum of a number of irreducible representations; the bundle ρ(τ) splits accordingly into a direct sum of complex analytic vector bundles, and the tensor fields split into direct sums of complex analytic sections of these component bundles. The covariant derivative can be decomposed accordingly as a direct sum of linear differential operators between sections of these various bundles. for

Now

η = 2 the irreducible analytic linear representations of the general

linear group are all of the form A^(0,-q) for in the notation of [hj]; here Δ

q > 0 and

ρ

arbitrary,

is the scalar representation given by the

determinant, and (θ,-q) is the dual of the symmetric representation

(q,o) of

-87degree

q.

Letting

denote the complex analytic vector bundle

where

a complex analytic tensor field

thus described by coefficients functions in

is

which are complex analytic

and are symmetric in the i n d i c e s a n d

which

satisfy

(119) in

, where

;

or equivalently of course

(120)

where

and

are complex analytic tensor fields

of the appropriate types and are skew-symmetric in the and symmetric in the

q

indices

pairs of indices Using (120) it

is easy to verify that the covariant derivative (118) of such a tensor field f

takes the form

(121)

This covariant derivative vector bundle

where

is a

section of the complex analytic

is the representation

Up to equivalence of representations, ; and since the representation

can be decomposed

-88-

into the direct sum of irreducible representations as in [36], it follows that Therefore and

can be decomposed into tensors ; and it is apparent that these constituents can

be written in the forms

(122)

(123)

or using (121) in the equivalent forms

(124)

(125)

A useful alternative notation exhibits the parallelism between these differential operators more closely.

(126)

Setting

where

and recalling that the tensors

of the indices are

1

and

,

are symmetric, the expressions (12l+)

-89and (125) can "be rewritten

(127)

(128)

In either form it is evident that the terms involving vanish whenever

3p = 2q , hence that

in the operator

transforms complex analytic

tensor fields into complex analytic tensor fields in those cases; and that the terms i n v o l v i n g i n the operator

vanish whenever

3P = = 1 ,

and the

other numerical invariants can easily be calculated using Theorem 3 and formula (13) of [32] and remembering that

c.2 = p = 0 ; J§

the results are

dim r(M, & 1 ' 0 ) = 0 ,

dim H^Mjffi) = 1 ,

dim ir^M,^ ) = 1 ,

dim H^(M,3l) = 0 .

Substituting these results into the exact sequence (1^2) it follows immediately that the mapping

d

is trivial, indeed that

H^M,

These

observations can be summarized as follows.

Theorem 6.

A compact complex surface M

canonical structure precisely when

admits a complex flat

c^ = 0 ; and if

c^ = 0 the set of all

such structures can be put into one-to-one correspondence with the points of the space

r(M, (&-1'0) of complex analytic 1-forms on M ,

since all complex

analytic canonical connections on a surface are integrable.

As for the question of which compact complex surfaces have

c^ = 0 ,

it can be noted first that all such surfaces are necessarily minimal, in the sense that they contain no exceptional curves of the first kind. an exceptional curve of the first kind on M complex analytic submanifold to the projective line

1

3P

CC M

Recall that

is a connected one-dimensional

such that

C

is analytically equivalent

(is rational) and has self-intersection number

2 C

= - 1 ; these are precisely the irreducible analytic subvarieties of M

that can be blown down to regular points, [15], [30]. one-dimensional analytic submanifold of M

If

C

is any connected

then the adjunction formula [31

page ll8] shows that the canonical bundle of

C

is isomorphic to

K^ ® [C]^, ,

where and

k

is the restriction to

C

of the canonical bundle

[c]

is the restriction to

C

of the line bundle [c] of the divisor

on M .

C

is an exceptional curve of the first kind then the Chern class of fc] 2 c = -1,

k

has trivial Chern class, since

of M

C

is

The bundle

K

hence the Chern class of

class of the canonical bundle of diction.

Thus M

K ® [c]„ C C

C = Jp1

is

= o.

If

is -1 j but the Chern

-2 , and that is a contra-

contains no exceptional curves of the first kind.

The compact complex surfaces having analytically trivial canonical bundles were classified by Kodaira in [32, Theorem 19]. K3

They are: (i) the

surfaces (the surfaces of Kodaira's class II ) 5 (ii) the complex tori

(the surfaces of Kodaira's class I H ) ; and (iii) the Kodaira surfaces, those 2 elliptic surfaces representable as quotients of

$

by properly discontinuous

groups of affine transformations having complex Jacobian determinant 1 and no fixed points (a proper subset of the set of surfaces of Kodaira's class VI ). The compact complex surfaces having bundles are characterized by

c^ = 0 but having nontrivial canonical

c, 1 = 0 , pg = 0 ; and the determination of the

other numerical invariants using Theorem 3 and formula (13) of [32] or the results contained in Part 1 of [5] shows that there are only three possible types in this case also.

They are: (iv) the Enriques surfaces, with

b-, — 0

(algebraic surfaces, a proper subset of the set of surfaces of Kodaira's class I Q ) i (v) the hyperelliptic surfaces, with

b^ = 2

(algebraic surfaces,

another proper subset of the set of surfaces of Kodaira's class I ) ; and (vi) those surfaces of Kodaira's class VIIQ having

c^ = 0.

The numerical invariants

in all these cases are listed in Table 1 for ready reference. noted that in cases (iv) and (v) although the bundle

K

It should be

is not itself trivial

-ίο6-

nonetheless

12 K is trivial; indeed in case (iv) the bundle

already trivial [5]· then

K

12

If the bundle K

2 K is

is reduced to the trivial bundle

itself will have as coordinate transformations some twelfth roots

of unity, hence constants; so this exhibits the reduction of bundle in a rather special way.

K

to a flat

Flat bundles can be described by homomorphisms

of the fundamental group, as discussed in [ 2 o ] among other places for instance. If

12

κ

= 1 then the representation

ρ

describing the flat bundle

12 satisfies ρ = 1 ; the kernel of the representation finite index

v

divisor of 12, a

ρ

v-sheeted covering of

M

of M

K

where

ν

is a

described by that kernel is

on which the induced bundle

trivial. It is apparent that

also

is a subgroup of

in the fundamental group of the surface M , and the covering space

K

/? = ρ ^(κ) is

is the canonical bundle of the surface

M;

so that the surface if must be a surface of one of the classes (i), (ii), or (iii). Indeed since

C^

is the Euler class [5], [32] and satisfies

Cg = ν· C^ , it follows that an Enriques surface has a

K3 surface as a two-

sheeted unbranched covering space, and that a hyperelliptic surface has either a torus or a Kodaira surface as a finite-sheeted unbranched covering space; and since a finite covering space of an algebraic variety is again an algebraic variety, a hyperelliptic surface actually has an algebraic torus as a finitesheeted unbranched covering space.

Thus surfaces of classes (ii), (iii), and

(v) can all be represented as quotients of

I

by properly discontinuous groups

of affine transformations having no fixed points. The surfaces of class

VIIq

have been studied by Kodaira, lnoue, and others [5], [2.b], [32], but are still not completely known. For these surfaces it is not always the case that is analytically trivial for some

m ; but since

Km

H1(M) Θ-1'0) = 0 all complex

analytic line bundles are analytically equivalent to flat line bundles. Finally something should be said about the case of noncompact complex surfaces, or at least about Stein manifolds. H (Ms 0

For a Stein manifold

[IB], so from the exact sequence (lb2) it

follows that hV,

(S^'0) ~

rfW)

;

indeed this isomorphism associates to the cohomology class ff^M, ( 9 ^ ° )

the cohomology class

in this case also c^ = 0 .

M

{d log A^g} €

c ± = c 1 (A) = - C ^ k ) e H^M,!),

so that

has a flat complex canonical structure precisely when

Hot all complex analytic canonical connections are integrablej but

there are a vast number of integrable complex analytic canonical connections.

-108-

TABLE 1

COMPACT COMPLEX SURFACES WITH

^ = 0

1

0

0

0

22

24

(ii) torus

1

2

2

it

6

0

(iii) Kodaira surface

1

2

1

3

(iv)

0

0

0

0

10

12

0

I

1

2

2

0

0

1

0

1

0

0

(i)

K3 surface

Enriques

0

surface (v)

hyperelliptic surface

(vi) VII with o c1 = 0

§11.

Complex affine structures on surfaces. Although complex affine structures are more complicated than flat

canonical structures, in part because of the nonlinearity of the defining partial differential equations, there is nonetheless a great deal known about such structures.

Complex affine structures are of course subordinate to

complex flat canonical structures, so that to determine which complex mani­ folds admit complex flat canonical structures it is only necessary to run through the list of complex manifolds with complex flat canonical structures and see which admit this finer structure.

Thus the only compact complex

surfaces that can possibly admit complex affine structures are those with Chern classes

c^ = 0 ,

with Chern class

C2 = 0 ; and these surfaces are among those surfaces

= 0 listed in Table 1.

Referring to that table, the

complex tori (type ii), Kodaira surfaces (type iii), and hyperelliptic surfaces (type v) all have

c

= 0 ; and as noted in the preceding section all these

surfaces can be represented as quotients of

2

£t

by properly discontinuous

groups of affine transformations having no fixed points, hence do admit complex affine structures.

That leaves the surfaces of type (vi), which are among the

notorious surfaces of type VH0 in Kodaira's classification, to be considered; but before turning to that topic a few further general comments and references are perhaps in order. The set of all complex affine structures on a surface, or equivalently the set of all integrable complex analytic affine connections, is not generally parametrized by a complex vector space, since integrability is a nonlinear condition; and the problem of determining and describing all such structures is therefore quite interesting.

For the surfaces of types (ii), (iii), and (v)

in Table 1, and for some of the surfaces of type (vi), this analysis was carried, out by A. Vitter in [Ά-], ['+5], and nothing further will "be said here about that analysis in detail.

As already noted in the one-dimensional case,

not all such structures actually correspond to representations of the surface as quotients of

2 !E by properly discontinuous groups of affine transformations

having no fixed points. universal covering space

However if M M , so that

is a complex affine surface with M = M/r

where

Γ'= 7Γ (M) is the

covering translation group, then the complex affine structure on M to a r~invariant complex affine structure on JT > and since M

lifts

is simply

connected this structure is necessarily trivial, in the sense that it can be defined by global affine coordinates, [20].

Thus there are a nonsingular

2

complex analytic mapping f : M —> it , the geometric realization or development of the surface M ,

and a homomorphism

ρ : Γ —> A2(Ol) from the group Γ

into the two-dimensional complex affine group such that f(Tp) = p(T)'f(p) for every point subset of

ρ e 0Γ and element

TeT-

The image

D = f(M) is an open

2 it which is mapped onto itself by the affine transformatxons

p ( r ) C A2(Ol) ; but the group

p(r)

of transformations acting on D. 2 plane Ϊ ; and although

need not be a properly discontinuous group The domain

f : M —> D

D

is not necessarily the full

is a nonsingular complex analytic mapping,

hence a local homeomorphism, it is not clear that

f

need always be a covering 2

mapping. The complex affine structures for which D = 31

and f is a homeo­

morphism, hence for which M = D/p(r), can be characterized as the complete affine structures, in the differential-geometric sensej there is an extensive literature on such manifolds, in the real or complex ease, [12, 2k, and further references cited there], but completeness is too restrictive an assumption in

the complex analytic case.

The classification of all compact complex

surfaces of the form

where

T

is a properly discontinuous group

of affine transformations with no fixed points, was carried out by T. Suwa in [1+0] and had earlier "been analyzed by Fillmore and Scheuneman in [12 ]; that too is a topic that will not be discussed any further here. As already noted, the canonical bundle

K

for a surface of type

(ii) or (iii) is analytically trivial, while for a surface of type (v) it is only the case that surface M M

of M

for some integer

for a

of type (vi) then some unbranched m-sheeted covering surface

will have a trivial canonical bundle and trivial second Chern

class, hence M

must be a surface of type (ii) or (iii); but

a Kaehler manifold, since M Thus any surface M

is not, so that U

cannot be

must be of type (iii).

of type (vi) for which

for some integer

must be the quotient of a Kodaira surface automorphisms of

M

by a finite group of

having no fixed points; and arguing as in the proof of

Theorem 39 in [32], it follows that M all these surfaces for which

must have an affine structure. for some

Since

can be taken as fairly

well known it is really sufficient to limit further consideration here generally to those surfaces having m > 0;

for any

all these surfaces are of type (vi) of course.

It is convenient to

introduce here the following simple observations about the complex analytic tangent bundle Lemma

T

and its dual bundle

T

for such surfaces.

On a compact complex surface M for any integer

then

T*

for which

c^ = 0 ,

if dim

can be represented by coordinate transformations of the forms

for some integer can be represented by

On a compact complex surface for which it is only assumed that for some complex analytic line bundle can be represented by

unless either

Proof.

Anv element

complex analytic functions J

is represented by in the coordinate neighborhoods

they satisfy

In

; or viewing

matrices of complex analytic

functions in. The, traces eq^uivalently in UQ, H Up and determinants of these matrices thus satisfy in

. If

then the line "bundles

and

have trivial

Chern classes but are not analytically trivial, hence can have no nontrivial complex analytic sections; and therefore F

0

there must exist matrices

coordinate neighborhoods

UQ

The matrices

• H„ F„ HQ p P P

1

such that

det

If

of complex analytic functions in the det

and

are also coordinate transformations which represent T

the bundle

Ha

and

; and since it follows readily that

(1^7)

where t

. Furthermore h

a

t

,

The functions

a - describe a C£ p

-llU-

complex analytic line "bundle analytic vector bundle line bundle

a

contained as a subbundle in the complex

T , and the functions

d^^

describe the quotient

d ; so (1V7) represents the vector bundle

analytic extension of the line subbundle

T

as a complex

a by the line bundle

d.

It is

well known that the set of all complex analytic equivalence classes of such extensions is parametrized by the cohomology group

with

the cocycle describing the extension

so in this

particular case the possible extensions (1^7) are described by the cohomology group

.

The Chern classes of the manifold

M

and of the

-v line bundle

K

are all zero, so by the Riemann-Roch-Hirzebruch formula

[23] as extended to arbitrary complex manifolds by Atiyah and Singer [1] it follows that

and by the Serre duality theorem

Thus if

-1 then

and the extension (1^7) is trivial;

thus it can be supposed that (lV+).

If

and that gives the first part of

the extension (1U7) may not be trivial} but at least the

second part of (lV+) holds. can only be asserted that

then at the beginning of the proof it

tr

and

det

are constants.

there is always one nontrivial element of by the identity matrix

I

that represented

in each coordinate neighborhood

another linearly independent element

Of course

then

Ua.

If there is

is a nontrivial

element for any complex constant

c ; so if

dim ri

there

is evidently always at least one nontrivial element det

There exist matrices

coordinate neighborhoods

Ua

the matrices 2

Hp1

Ha

of complex analytic functions in the

such that

for some nonzero complex constant

^ for which

c.

det

and

In the first case arguing as before = a

have the form (1^7) with

ap

and

-1

a Q p = K^p , which is the second part of (1^4-3). In the second case arguing have the form (1U7) with

similarly the matrices and

=0

which is the first part of (1^3)• Next if

is a nontrivial element, for some

complex analytic line bundle functions

bap

f^

a , then

f

is represented by complex analytic

in the coordinate neighborhoods

Ua

and these functions

satisfy

in

. It is easy to see that the functions

common zeros in divisor

U .

g^ represent a nontrivial complex analytic

section of some complex line bundle X

analytic section.

have no

Indeed if these functions have a nontrivial common

g^ then the functions

Chern class of

and

A. over M ;

is trivial, and hence

X

Thus the common zeros of

but since bg = 0 the

cannot have any nontrivial complex 1 2 f

and

hence represent the second Chern class of the bundle

f^

are isolated, and but

since

and

T

have trivial Chern classes it follows that

f^"

and

2 f

can have no common zeros at all.

There thus exist matrices

complex analytic functions in the coordinate neighborhoods det

Ha

of

such that

and

The coordinate transformations the form

are then easily seen to be of . The possible extensions (1^7)

parametrized by the cohomology group

are

and by the Riemann-

Roch-Hirzebruch theorem and the Serre duality theorem dim

Thus

dim

; and except for

these two cases it can be assumed that

That suffice to complete

the proof of the lemma.

A complex analytic vector bundle of rank 2 over M

is called

reducible if it contains a complex analytic line bundle as a subbundle, or equivalently3 if it can be represented by coordinate transformations of the form

(148)

and otherwise the bundle is called irreducible.

The bundle is called decompos-

able if it can be written as a direct sum of line bundles, or equivalently,

if it can "be represented by coordinate transformations of the form =

with bundle

T

0 ; and otherwise the bundle is called indecomposable.

is reducible then

for the line subbundle

a ; and it follows from Lemma ^ that if whenever bundle

If the

c^ = 0

and

c^ = 0

then conversely

for some complex analytic line bundle

T

is reducible and contains

a

as a subbundle.

a

the

The same result

*

is of course true for the dual bundle manifold

T

;

indeed on a two-dimensional

M

(11*9)

as is obvious since for

If M

2 x 2 matrices

is a two-dimensional compact complex manifold for which and

for any integer

and if M

has a

complex analytic affine connection, then the curvature tensor is a complex analytic tensor field on M thus

r

and is skew-symmetric in the indices

can be viewed as a section

It now follows from Lemma

that if

is nontrivial then

represented by coordinate transformations of the form

T

can be

Thus either

a ® K = 1,

covering surface trivial. bundle of

M

The bundle

or after passing to an unbranched two-sheeted

over M

the bundle on M

τ induces over M

M ; hence either M

induced by

a ®K

is

the complex analytic tangent

or the two-sheeted covering surface

will have a complex analytic tangent bundle

M

τ that can be represented by

coordinate transformations of the form

The dual bundle M

or M

τ

must then have the same form; and consequently either

has a nontrivial complex analytic section of the bundle

nontrivial complex analytic one-form.

However since

M

τ* , a

is a surface of

type (vi) in the list given in Table 1 then as noted in that table Γ(Μ,

= 0.

c^ = 0 and

On the other hand the covering surface

c^ = 0,

also has

and no power of the canonical bundle on M

trivial either; consequently M Γ(Μ, C?"1'0) = 0.

M

can be

is also a surface of type (vi) so that

That is a contradiction, and consequently

trivial; so every complex analytic affine connection on

M

r must be

has zero curvature

tensor, and is therefore integrable. Turning next to the problem of the existence of complex analytic affine connections on these manifolds, it is quite easy to calculate the cohomology group

H^M, ©-(ρ(τ)) when the tangent bundle τ

Hote first that the Chem classes of M trivial, since the Chern classes of τ

and of the bundle

is reducible. p(f) are all

are trivial, so it follows from the

Riemann-Roeh-Hirzebruch theorem [23] that

-119-

From the Serre duality theorem [23] note that and since by (ll+9) necessarily T

=(150) p( ) j it follows that T

If the tangent bundle

is reducible, so can be represented by coordinate

transformations of the form (1^8), then as noted in the proof of Lemma lithe obstruction to reducing the entries

b^R

to

0

lies in the group

; so it can be assumed that

unless either

Leaving aside for the moment the cases in which b^p

, if

then it is clear that an element

is described by complex analytic functions neighborhoods and

,

f ^ .

in the coordinate

and that these functions are symmetric in the indices J

jg, and satisfy

Since the line bundles with

a

and

d have trivial Chern classes on any surface

it follows that

r(M,

or bundle

unless either

furthermore, since T

is symmetric in

a

and

d when b = 0 ,

really

a = 1 and the

r(M,

0-(P(T)))

= 0

unless either b = 0 then

T*

is also reducible and contains 1 as a subbundle, so that but that is impossible for a surface of

type

as observed in Table 1.

Therefore if

is reducible and

then

Next if

,

for any section

so that

, then no matter what

f € r(M,

it follows first that

2 ^ ^ = 0 ; then using this it further follows that

hence that

2 hence that

fal2

it is finally concluded that h

e

n

c

e

a

, n

d

=

® ' an O, all admit complex affine M for which

- O , Cg = O ,

m > O, are of type (vi) in Table 1; for such

surfaces all complex analytic affine connections are integrable; and there exist complex analytic affine connections whenever the complex analytic tangent bundle unless

τ

is reducible, indeed there exists a unique such connection

τ = a®ο;

cannot admit a complex analytic projec­

tive connection. The surfaces of type (ii) as listed in Theorem 8 were already examined during the analysis of complex affine structures in the preceding section. All these surfaces admit complex analytic flat canonical connect tions, by Theorem 6; and since the direct sum of a complex analytic projective connection and a complex analytic flat canonical connection is a complex analytic affine connection, the existence of complex analytic projective connections is equivalent to the existence of complex analytic affine connec­ tions. As for surfaces of type (iii) as listed in Theorem 8, it is convenient to begin by examining a special subclass of this class of surfaces. First suppose that M

is a minimal elliptic surface having no singular fibres.

Then Kodaira's analytic invariant

jj.

is everywhere holomorphic and therefore

constant, so that all the fibres are analytically equivalent; and M

is

consequently a complex analytic fibre bundle over a compact Eiemann surface Δ of genus g, the fibre being a nonsingular elliptic curve F

and the

structural group being the group of complex analytic automorphisms of F , [31].

When

F is represented as the quotient of the complex line !E by

a lattice subgroup

the automorphisms of

F

linear transformations

are all represented by

where

such that

for a general torus

F

X

is a complex number

the only possibilities are

but for those special tori with complex multiplication it is possible that

X

is a complex number with

Suppose further then

that the structural group of the bundle M

over A

can be reduced to the

subgroup of translations, so that only those automorphisms of the form appear.

That means that M

, where

admits a coordinate covering

is the product of a coordinate neighborhood

the Riemann surface A

with local coordinate

hood

F

U j on the torus

representation

F = £U/

with local coordinate

U^

on

and a coordinate neighborinduced from the

; and the coordinate transitions are of the form

(157)

where covering

are the coordinate transition functions of the induced coordinate UL' ~ ^U^} of A

analytic mappings.

and

are some complex

Actually the values taken on by the functions T^g

really be viewed as only determined modulo

can be

considered as describing a cohomology class is the sheaf of germs of holomorphic mappings from A the functions T)ag p

varies over

are

can

where into

F;

but when

viewed as complex-valued functions then as the point

O^ (1 UJ fl

, which can be assumed connected, clearly is a fixed element of

^ , and these

elements describe a cohomology class This is of course merely a special case of the constructions introduced by Kodaira in [31] to handle general elliptic surfaces. the first Betti number of the surface M

As in that treatment

is given by

(158)

Using (158) and the facts that of the surface M

c^ = c^ = 0 , the other numerical invariants

can be readily determined to be the following:

It is clear from this table that M and

is nonalgebraic when

then the mappings

a cohomology class

^TQ,^

can

r^g = 0

and hence that

This surface has the obvious projective structure.

VII

so that M ; but in that case

been considered. functions

f^

If

be "viewed as describing

; but for the projective line

so that it can been be assumed that

then

If

If

and

is one of the surfaces of Kodaira's class

so this can be rejected here as having already and

_

then the coordinate transition

can be assumed to be affine mappings, and the mappings

which represent an element in

r^g

can be assumed to be constant; thus

the given coordinate covering of M on M , indeed

M

already exhibits an affine structure

is a complex torus.

If

g = 1

and

then M

is a nonalgebraic minimal elliptic surface with

thus

is a Kodaira surface, hence admits an affine structure. ing the trivial case

that M

Thus after eliminat-

and those surfaces for which

can further be assumed that

g > 1.

M

It then follows that

necessarily has a nontrivial canonical bundle and

c^ = 0 it

p^ > 1 , hence ^

/ 0 .

Now for this special subclass of surfaces the investigation of complex analytic projective connections is an utterly straightforward matter. Indeed for the given coordinate covering it follows from (157) that

(159)

and

so recalling (10*0 a complex analytic projective connection on M by complex analytic function's on M

such that

+

is described

in the coordinate neighborhoods =0

for all

j

and that in

Ua

Equation (l6l) shows that

s

a2

2

i nde P enden "t

z a

2

an
1, the fibre being a nonsingular

and the structural group being merely the group of

translations of F .

This surface has

0, c^ =

= 0, and the other

numerical invariants are as in Table 2, depending on the structural invariant C(TJ) S

(Δ,

connections on

.

If ο(η) =

M.

If

0

there are no complex analytic projective

ο(η) ^ 0 the nonalgebraic surface

M

has complex

analytic projective connections parametrized by a complex vector space of dimension

^g-3 > and all these connections are integrable, hence describe

complex projective structures on

M.

This result is probably primarily of interest for whatever light it may shed on the question of the extent to which topological restrictions alone guarantee the existence of complex projective structures.

For the case

of complex flat canonical structures the topological condition c^ = 0 is both necessary and sufficient for the existence of those structures, by Theorem 6j while for the case of complex affine structures the topological

conditions c^ = 0 and

C^ = 0 are necessary and almost sufficient for the

existence of those structures by Theorem 7, the only instances in which sufficiency may be in doubt being for surfaces of a quite special type and of uncertain existence.

For the case of complex projective structures the

topological condition Cg =

1

2 C^

is necessary but not sufficient for the

exlstence of such structures.

Indeed the surfaces of type (i) in the list in

Theorem 8 do satisfy this topological restriction, but fail to admit complex projective structures since they contain exceptional curves of the first kind; so it might still be expected that the topological restrictions together with the nonexistence of exceptional curves of the first kind would suffice to guarantee the existence of complex projective structures.

However that

is not the case as evidenced by Theorem 9» there are further conditions required, apparently also topological conditions in this case though. Turning then briefly to a general surface M

of type (iii) as

listed in Theorem 8, there is a finite branched analytic covering branched only along fibres, so that M no singular fibres; so M finite group Γ

M —> M ,

is a minimal elliptic surface having

can be represented as the quotient of M

of complex analytic automorphisms of

M, [31]·

by a

Then M

is a complex analytic fibre bundle over a compact Eiemann surface Δ genus

g , the fibre being a nonsingular elliptic curve

group being the group

G

F

of

and the structural

of complex automorphisms of F. The group of

translations of F is a subgroup of finite index in G ; so by passing to a iw in­ finite unbranched covering Δ —> Δ the structural group of the induced fibre bundle can be reduced to the subgroup of translations of F. there is a finite unbranched covering

M —> M

so that M

Thus

is one of the

restricted subclass of surfaces of type (iii) just considered; and M can be iW represented as the quotient of M by a finite group Γ of complex analytic automorphisms of M .

The complex analytic projective connections on

M

are related to the Γ - invariant complex analytic projective connections on M; and that reduces the problem to an analysis of the behavior of the complex

analytic projections on M "under automorphisms of

M , the details of

which perhaps need not be pursued further. Finally the surfaces of type (iv) as listed in Theorem 8 have been investigated by S.-T. Yau, [>9].

He has shown that those surfaces that

also have an ample canonical bundle can be represented as quotient spaces 2 of the unit ball in it by properly discontinuous groups of projective trans­ formations, hence admit complex projective structures; and he has asserted that his method of proof will quite likely extend to cover all surfaces of type (iv). Examples of surfaces of this type were considered earlier by A. Borel,

[6].

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