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
C© 1978 by Princeton University 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
-1-
PREFACE
These are notes based on a course of lectures given at Princeton University during the Fall term of 1976, incorporating some material from lecture courses given during the year 1963-64 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 The particular pseudogroup structures considered, and Riemann surfaces. 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 A more detailed overview of as a unifying and limiting principal theme. the topics covered and the point of view taken is given in the introductory There remain many fascinating open questions and likely avenues chapter. 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. R.
Princeton, New Jersey
C. Gunning
-11-
CONTENTS Page §1.
Introduction
Part I: §2. §3. §4. §5.
§6.
§7. §8. §9.
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:
7
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:
1
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
..............................................
137
Bibliography
-1-
§1.
Introduction
The general uniformization theorem for Riemann 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 dimensions 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 Riemann 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.
Recent works (surveyed in [2]) have demonstrated the existence
and importance of a considerable array of different representations of compact Riemann 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 Riemann surface, a rather finer structure than the complex analytic structure [20].
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-
-2-
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 [41], 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 Riemann 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 Riemann 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
-3-
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 deformation 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 pseudogroups 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
- s-
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], [14) ; 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 Dart, 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. Cartan in a series of fundamental papers, [7]
;
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
-5-
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), 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-
§2.
The group of k-jets and its Lie algebra.
Consider the set of all germs of complex analytic mappings from .n
the origin to the origin in the space k-jet of such a germ
terms of order < k
denoted by
f,
of
n
complex variables.
The
is defined to consist of the
jkf,
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 will be slightly modified in that the constant terms in the Taylor expansion, the terms of order k-jet.
Upon identifying a k-jet with its Taylor coefficients the set
Mk = Mk(n,C)
of all such k-jets can be viewed as a finite-dimensional complex
vector space; indeed Mk
can be viewed as the direct sun
Mk=T1®... ®Tk,
(1)
where
will not be considered as part of the
= 0,
Tp = T(n,M) p
is the complex vector space of dimension
= nrn+pp1)
consisting of the Taylor coefficients of order choosing any germs of complex analytic mappings
= p . fi
E Mk
If
such that
then)
Pi = ikfi
define
El E2 = jk(fl ° f2)
(2)
,
noting that the k-jet of the composite mapping k-jets of the individual mappings operation (2) the set
Mk
fl of2
depends only on the
It is readily verified that under the
fi.
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
Gk C Mk
form the group of invertible elements in
of germs of local homeomorphisms
Mk ; this group
Gk = Gk(n,M)
will
be called the general k-fold group or the group of k-jets, the special case being of course the general linear group.
G1 = G1(n,T)
e Mk
consists of all the jets nonsingular space
Mk,
n x n
matrix; thus
Gk
The group
such that the terms of order
= 1
form a
is a dense open subset of the vector
Gk
and with the natural manifold structure inherited from that
vector space it is evident that
Gk
is a complex Lie group.
It is a quite simple matter to write the group operation in
Gk
explicitly in terms of the natural global coordinates provided by the encom-
passing vector space Mk , 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 Mk ,
since there are various possibilities.
It seems most convenient for the present purposes to view Tp = Tp(n,M) the subspace of the (p+1)-fold tensor product
an ® ... ® an
as
consisting of
those tensors which are fully symmetric in the last
p
indices; the first
index will be written as a superscript and the last
p
indices as subscripts,
so an element p
e T p
is a tensor
(3)
Ep
which is symmetric in the
p
set of
k
lower indices.
Ek}
E =
f
An element
C e ML
is then the
tensors
(4+)
If
l "jp
P
,
where
Ep E Tp
is the germ of a complex analytic mapping from the origin to the origin
in e and is given by the the k-jet
E = jk£
n
coordinate functions
fi(zl, ..., zn)
then
will be taken to be the element (4) with components (3)
-8-
given by apfi(z)
i
_
(5)
8z. ...8z. p
1
z=0
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 by repeated
applications of the
chain rule for differentiation.
jkf,
= jk(f G g) = Q
r1
In particular if
it follows readily that i =EkkTij i k j
(6)
i
(7)
jlj2 i
(8)
=
i
k1
i
T1j
E
k2
k1
k k1k2k3
T1j
l
k3 T1j
2
3
kl
1
£
i
kl
k
+ Ek Ek1k2(T1j1j2T1j
+
i
k k k T1jlj2
k2
rjj1
k £k1k2
=E
j lj 2j 3
and so on.
and
T) = jkg,
3
k
kl
j
k2
+ T1j 1 3 71j 2 + T1j 2j 3T>j 1 )
k
k k T1j lj 2j 3
Formula (6) is just the usual matrix product; and while the
ensuing formulas are somewhat more complicated, their general pattern is quite i
transparent.
of the form
Indeed
C
.
jl...jp
Pk
is a sum of where
terms, the q-th of which is
p
1 p q 1- q nomial function of the components of the tensors E1
k
k
(r1)
k (TI)
P k
denotes some poly-
q ri
.
That polynomial is in
-9-
k1
k
where
turn a sum of terms of the form J ... r1Jq 1
the indices
such that
jl, ..., jp
of the indices
(j1, ..., i
)
are various subsets
Jr
q
(J1, ..., Jq)
is a permutation of
; all possible sizes of subsets
appear, since
Jr
p
all such differentiations appear upon iterating the chain rule, and the sum must be formally symmetric in the indices
£p (9)
-
-
-
v1 > v2 > ... > vq > 1 so on,
consisting of
Jr
kq
J T1J1 ... 'lJq
v1 + ... + Vq = P ;
and
Thus in general
kl
S
denotes a sum over all sets of integers
£v
-
£
E
q=1 k -k1...kq V
j1...jp
Here
jl, ..., jp .
v1, ..., Vq
such that
Ji = (jl, ...) iv
)
and
1 yr
of the indices
jl, ..., jp ;
and
jl, ..., j
a sum over some set of permutations of the indices
.
SJ
denotes
Actually
p
consists of the minimal sum needed to ensure the formal symmetry of Si
in the lower indices, taking into account the symmetries of the tensors and
Tl
be proved.
;
but that is a finer point than is really needed here, so will not Indeed the general formula is not really needed, and it is an easy
matter to verify any particular case of the formula. p = 4,
For example in the case
the next case after (8), the formula is
k1
(10)
J1j2j3j4
k2
k3
k4
k k1k2k3k4 TIjl T1 02 T1j3 '1j4
k1k2k3 + £k Ei
S
k1 2 T k2 1 rljlj . 3 '1jk3 4
kl i k2 + k Ek1k2(S2 r1j1j2j3 r1j4 + s3
i k +£k£k1jlj2j3j4
k1 k2 Tj1j2 1J3j4
-10-
where and
S3
6
is a sun over
S1
is a sum over
3
symmetric in the indices j3
and
j4
permutations,
permutations; for j1
and
j2,
is a sum over
S2 S1
4
permutations,
the expression is already
and is also symmetric in the indices
is symmetric), so the summation is only extended
(since
over a set of permutations in the symmetric group on
letters which
4
represent cosets of the subgroup describing this symmetry, and similarly in the other cases.
The structure of the Lie group
Gk
can be described in general
terms rather easily, without making much use of the preceding detailed form of the product operation;
describing subgroups of
but more details will be needed later in Gk .
Note that for any integers
possible to consider the J,-jet
it is
e Gk ; this defines a
of a k-jet
j
1 < t,< k
mapping jz :
Gk(n,T.) - GG(n,M)
,
which is evidently a surjective group homomorphism.
In terms of the represen-
tation (4) of course
For the special case
t = k-l
the kernel of this group homomorphism can be
identified with the vector space is clearly the point set
Tk ;
indeed the kernel of this homomorphism
S ® 0 ® ... ® 0 S Tk
in the decomposition (1),
1
where
S1 a T1
is the identity matrix (the identity element in
81 S 0 0 ... G; 0),
Gk
being
and it follows easily from (9) that in this subgroup the
group operation amounts to addition in the vector space
Tk .
There thus arises
-11-
an exact sequence or groups
jk-1
0>Tk-> Gk->Gk-l-> 0 for any index k > 2,
showing that
Tk by Gk-l ; and
group
G1
Gk
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
Gk
it is a straightforward matter to derive a correspondingly explicit form for of that Lie group. k = ' k(n,M) 0 it is clear that as a vector space
the bracket operation in the Lie algebra Since
41k Gk,
Gk
is a dense open subset of Mk
can be identified with Mk ; if
fi(t)
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
X = dE(t)/dt
t=0
Furthermore the associated right-invariant differential operator on the manifold
Gk
is
DXf() = dt f(e(t)-
where
f
It
is any differentiable function in an open neighborhood of the point
E E Gk, [22] ; and writing
for the global coordinates (3) for short,
{E J}
and recalling that the product
E(t)-
is linear in the first factor, it
follows that
DXf(E) = EJ(X- E)J
'(£)/)EJ
.
If Y is another vector in the Lie algebra 4k then the bracket is the element of 4k such that
[X,Y]
-12-.
f(O ;
D
D
x
hence
(Y' )J - EK(Y' )K
([X,Y]' VJ = EK(X' 0K
(12)
aK
aEK If
J = (
then the left-hand side of (11) is
)
j
1
(X' )J
p
k [X,Y]1
k + ...
jl ...
kl...k
,
p
p
where the unwritten terms involve i
[X,Y]k for q < p ; so to compute k 1 q it suffices merely to calculate the coefficient of the multi-
k
[X,Y]k 1
klp
nomial
k
...jp on the right-hand side of (12).
For this purpose, considerp ing initially only the first part of the right-hand side of (12), the only 1
terms in (X E)K which need be considered are those which involve
multinomials in the tensor components S and the only terms in (Y )J which need be considered are those which involve the products of r
multimomials in
¢s
;
so writing
K = (m
m
1
)
with
EK with
1 < q < p,
the only
q
terms in the first part of the right-hand side of (12) which need be considered are
nl
y
p
Eq=l tmn
Cnq
n ...nq
sl
a
71...fmq
1
Sr
Es Ys1...sr SJ
-
where indices
r = p-q+l
and
j1, ..., jp .
nl...nq
SJ
sl...sr
-01'*'Jq jq+l
n
n C
Eq=1 mns SJ
2
l
..
q
a
s
q a£K Jl
p
s
s ..
@.
Jq £Jq+l
denotes a sum over some set of permutations of the
When
q > 1
,
'
the only nontrivial terms are those for
p
-13-
j1.s
which
K =
1
(
..jq
while when
),
the only nontrivial terms are those
q = 1
(sv
for which
K =
v
) for
;
i
n
n
s
j1 l ...
jq q
jq+1 2
s Eq=2 Ens SJ Xn1 ...n
-
Ys 1
1
+ EP
V=1
n where
V
E ns
so this expression simplifies to
V = 1, ..., p
Xsv Y1 n1 s1...s
s
r
Eip
s
nl
s1
jv
...
V
j1
P
..
P
indicates that the v-th term in the product is omitted.
This can
be rewritten as
p
S
J
q=2
i
Q,1...kq Y,kq+1...kP
k1
kP
j1
jp
k1 kp Ejp
p
+ Ev_1 K Xkv
and the contribution from the second part of the right-hand side of (12) is of the same form, but with
X
and
interchanged and a negative sign.
Y
Con-
sequently
(13)
where
[X,Y]k
SK
1- kp
Eq=1 E
SIC ("k
1... kq Y1k
q+1...
kp
Yk1...kq Xlk
denotes a sum over some set of permutations of the indices
q+1...
kp
k1,,.., k . P
In particular, for some small values of explicit form
(14)
LX,YJj'
= Zk( j Yk i - Y3 Xk)
p
the bracket operation has the
(15)
(16)
Ek(Xk
[X,Y]
[X'Y]
jj
1 2 3 =
yj2 +
YkiI + Xjlj2 Yk)
i j 2j 3
K(Sl X j 1 Yk
+ S2 Xj
Ykj 1 2
+ Xj 3
jj
1 2 3
ik
where the unwritten terms in (15) and (16) are obtained by inter-
and so on,
changing
3 l
X
and
Y
in the first terms; the symmetrizations
are both summations over three terms.
S1
and
S2
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
Tn
either directly from the defini-
,
tion or by 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 machinery, than some others.
The structure of the Lie algebra lk in general terms of course parallels the structure of the Lie group
Gk
.
The Lie group homomorphisms
jt induce surjective Lie algebra homomorphisms
it : fk(n,e) -> ft' (n, Q:)
of the same form whenever
1 < t < k.
For the special case t = k-1
the
kernel of this Lie algebra homomorphism can be identified with the vector space
Tk
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 the point set
0 ® ... @1 0 (D Tk
in the decomposition (1); and
it follows easily from (13) that the bracket operation in this subalgebra is trivial.
There thus arises an exact sequence of Lie algebras
Jk-1
(17)
0->Tk> ?k> 4k-1->0
for any index k > 2, Lie algebra linear group.
showing that 4 k
Tk by I k-1 ; and 1 1
is an extension of the abelian
is the Lie algebra of the general
-16-
The pseudogroups defined by partial differential equations.
§3.
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. k
of partial differential equations of order
morphisms from
Tn
to
Mn
subvariety A C Gk(n,M).
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 a: n
into
In
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
Gk(n,Q)
A,
Gk(n,a) ; and a closed subgroup of the Lie
is necessarily a Lie subgroup.
Lie pseudogroup of order
of mappings in
k
are closed under composition
In
A complex analytic restricted is defined to be the set of
all complex analytic mappings
f
jkf(z) E A
in the domain of definition of
for all points
is a Lie subgroup of
z
Gk(n,T)
from subdomains of
iCn
into
n f,
such that
where A
called a defining group for the pseudogroup;
the pseudogroup defined by a subgroup
A C Gk(n,T)
will be denoted by
?(A).
All the mappings in a pseudogroup f (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
F(A) 3'(A) also
-17-
T(A)
belongs to
whenever well defined.
For the classification of these pseudogroups it is not necessary to consider all subgroups
A C Gk(n,C.),
for distinct values of
subgroups of
Gk(n,M)
pseudogroup.
For any subgroup
the k-jets of all elements of
may well define the same
there is a naturally associated
3'(A).
A subgroup A C Gk(n,el)
A = A* ; equivalently a subgroup A C Gk(n,T)
integrable if
from some open neighborhood jkf(z) e A
of the origin in
U
z e U
for all
pseudogroups of a fixed order
and
k
Tn
jkf(0) _ ¢.
will be called is integrable
to
Gk(n,T)
f
such
Thus when examining Lie
values of
may still define the same pseudogroup.
but integrable subgroups of
;
f(U)
n
it suffices merely to consider integrable
subgroups of k
consists of all
there exists a complex analytic homeomorphism
e A
if for any element
that
A C Gk(n,M)
k,
defining the same pseudogroup: A*
A* C A
minimal subgroup
since distinct subgroups, even
Gk(n,T)
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
>1 j
:
,
k(n,R) -> Dk-l( n,T)
which associates to an element Xp =
jp}
the element
X = {X1, ..., Xk} C J k(n,O:) X.X e
k-1(n,M)
with
with
1 < j < n
-18-
(18)
X)j ...j
1
= X,
p
1
...j
;
p
and then to any linear subspace pL C Gk-1(n,a)
associate the subspace
ARC /Tlk(n,T) defined by
AX = jkll(') f
(19)
so that AF = {X e I k(n,T)
Lemma 1.
A C Gk(n,.)
Let
161 C A11k(n,Q)
If
A
where
be a Lie subgroup with associated Lie algebra
k > 1.
A
is integrable then
In an open neighborhood
0 ;
and
jk-1 g C k"k-l(n,U)
jk-lA
the mapping
0
(20)
jk-l'OZ = {X E
before.
is the tangent space to the
in the sense that
E
XJ
a0
JI
denote the natural global coordinates (3) in
If A is integrable then for any fixed point
Mn
in
at the identity, so can be defined by the differential of
a complex analytic homeomorphism origin in
V of the identity in Gk-l(n,T) the
is the set of common zeros of the component functions
ik-lA
submanifold
{Q
and
can be defined by an analytic mapping 0: V -> MN,
the sense that V ( of the mapping
A _ Ajk-1p
define the same Lie pseudogroup.
submanifold jk-lA
where
If
is integrable and P = Ajk-l,UZ then A C Gk(n,T)
ik-lA C Gk-l(n,(C)
Proof.
k-1X VZ and X.X e,UL for j = 1,...,n}
to
f(U) C Mn
f
`t
=1
= 0} ,
Gk-1(n,M) e A
as
there exists
from some open neighborhood U of the
such that
jkf(z) E A
for all
z E U
and
-19-
and
jkf(0) = t .
¢t
If
for all
jk-lf(z) E V n jk-1A near
near
z
and consequently
0,
e(jk-lf(z)) = 0
and upon differentiating this identity with respect to
z = 0 ;
and setting
is near enough to the identity then
z.
it follows that
z = 0
0 = J (xj t)T -
(21)
E=fit
This last identity holds in particular at all points t of any one-parameter subgroup
{Et}
of
for
A,
sufficiently small; and upon differentiating
t
this identity with respect to
and setting
t
so kj% = 0
and that
algebra
t=0 corresponding to the subgroup
0 = L (x.x) J
(22)
is the element of the Lie
= X E ,(2Z
aEt/atl
it follows that
a
.
J
J( E=fit
Upon comparing (20) and (22) it follows that hence that
r( C Ajk-l A ,
recalling that o = 1
t = 0,
whenever
XjX e jk-1
X E
)rL
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
f e f(jk-A)
F (B) = F (Jk-lA)
form an integrable subgroup
and since
;
necessarily A C B.
f (A) C I(jk-lA) = T(B)
Now the elements
E B
evidently satisfy (21), since the mappings satisfy
(jk-1f(z))
tangent vector
X
0
to the subgroup
B
d/C Ask-l.D
.
Zr to the subgroup
and
A
is integrable
near enough to the identity of which these are the k-jets
by definition of the subgroup
B
;
hence as above any
at the identity satisfies (22).
comparing (20) and (22) and recalling that the tangent space
f
such that
B C Gk(n,T)
B
jk-1B = jk-lA
Upon
it follows that
at the identity satisfies
If it is assumed that Ajk-lQZ = ,(
then
C J and
-20-
consequently
B C A ; but then in view of the previously obtained contain-
ment it follows that desired.
B = A,
and hence
F (A) =
F (jk-lA) =
That suffices to complete the proof of the lemma.
F(B)
as
-21-
§4.
The classification of tangentially transitive pseudogroups: algebraic aspects The detailed classification of pseudogroups will only be attempted
here for the special case of the tangentially transitive Lie pseudogroups, those for which all the defining groups
A C Gk(n,a)
have the property that
j1A = G1(n,M) ; 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
j1A = G1(n,M),
for all
k > 1,
A C Gk(n,M)
and then determining which of these
subgroups describe the same pseudogroups; but it is actually a considerably simpler matter than might be expected. If
A C Gk(n,M)
is an integrable subgroup for some
k > 1
then
is an integrable subgroup; and the exact sequence (11) induces
Jk-lA C Gk-l(n") an exact sequence
0->Kk->A->jk-lA->0 ,
(23)
where
of the tensor space
Tk
whenever
(9) that
The kernel
.
k} E A
C = e7)f-1 E Kk ;
and writing
C _ {81, 0, ..., 0, Ck}
(24)
and
Thus when
is a normal subgroup of
Kk
Tj _ {&1, 0, ..., 0, rjk} E Kk
the product
£j1 ... Ejk )1k
is viewed as a linear subspace
and any matrix
so
where
jl... k Kk
A,
Q-1 = {1, ..., Qk} , it follows readily from
T',
T)k E Kk
can be viewed as a linear subspace
Kk = A 0 (S1 G; 0 G) ... G 0 (; Tk)
E1 E j1A = G1(n,Z)
Kk C Tk
the tensor
then for any tensor Ck E Tk
given by (24)
-22-
must also be contained in Ck
as a function
Now the expression (24), when viewed as giving
Kk.
of the matrix l and the tensor
Ck
describes a representation
of the group
Pk
transformations on the vector space
two matrices El,l .
as a group of linear
for (24) is clearly linear in
Tk
and it is easily seen that
G1(n,k)
,
T1k
for any
(Pk(e1) - T1k)
T1k =
Indeed the representation
Pk
Tyk,
is one of the classical
symmetry representations of the general linear group, the representation (kn-1)
Pk = (1) ® (-k) = _ k(1) 0 that
Kk
be a normal subgroup of
in the notation of [47].
A thus amounts to the condition that Pk
be invariant under the representation
when
is viewed as a subspace
Kk
The condition
of the general linear group on
Kk Tk,
The same conclusion can of
Kk C Tk .
course be obtained by considering the Lie algebras of the groups involved. kL'
If
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
0 -> X k -> R -> jk-1'¢ > 0 ,
(25)
where k =
U? n (0 ff ... (P 0 G Tk)
space of the tensor space k
The kernel
Tk
product
Z = [Y,X] E
0, Zk}
where
Thus when
ZJ
1'' k
Xk
can also be viewed as a sub-
and then coincides with the subspace
is an ideal in the Lie algebra , ,
X = {X1, ..., Xk} ER and
(26)
,
;
e
so whenever the bracket
k
1Xk ; and it follows readily from (13) that
E
k
Y = {0, ..., 0, Yk}
Z
Xi YJ
k
1'' j k - Ev-1 E
XJv YJ
is viewed as a linear subspace
Kk C Tk.
..
J
Z = {0,
k
Wk = Ilk _ Tk
then for any
J
-23-
Yk E
tensor
and any matrix l E jl,Qj
Ook
=
o
given by (26) must also be contained in
Zk E Tk
describes a Lie algebra representation
k
The expression (26)
Z = d Pk (X) Y ,
merely the differential of the representation
the tensor
,0;1(n,C.)
Pk
,
which is indeed
as follows immediately
upon differentiating the expression (24); for when considering a one-parameter subgroup
necessarily
fi(t) C Gk(n,M)
Gk(n,U)
formulas.)
The invariant subspaces of
Tk
under the group representation
coincide with the invariant subspaces of
representation
When
n > 1
n = 1
the space
the representation
the direct sum
under
Pk,
the kernel Kk = Tk.
and
under the Lie algebra
Kk C Tk
Pk
for any
k > 2
or
Kk = Tk.
is decomposable into
Tk , Tk
are invariant and irreducible
Pk = PklTk ; and the only possibilities for
and
are either
Kk = 0
or
or
Kk = Tk
Kk = Tk
or
This decomposition can be described conveniently and explicitly by
a projection operator
2
commuting with the representation
by a linear mapping II: Tk -> Tk such that 2 = 0 Pk(£1)' 0
Kk = 0
Thus there is a direct sum decomposition
where the subspaces
Kk C Tk
are either
k; so the
of two irreducible representations, as is
Pk = Pk T_ Pk
Pk = pkITk'
is one-dimensional for any
Tk
demonstrated for instance in [36]. Tk = Tk &' Tk ,
Tk
d Pk .
only possibilities for the kernel When
(The reversal
with right-invariant vector fields to simplify the
algebra of
Pk
= (-t).
reflects the identification of the Lie
Z = [Y,X]
of order in the bracket
fi(t)-l = F(t)
for any
El E Gl(n,M)
;
0
and
Pk,
that is,
0' Pk(Y
and any nontrivial such mapping can be
used, for
Tk = 11 'Tk e (I-Q) Tk
-24-
is then a nontrivial decomposition of under the representation
into subspaces which are invariant
Tk
so must coincide with the above decomposition.
pk
It is a straightforward calculation to verify that the linear mapping
Q
defined by
where
t jv
= (n+k-l)-1 El=1 E
(ax)l
i1...ik
(27)
X =
{Xi
il "3k
81
has the desired properties; so let
} E Tk,
Tk = S2
Tk = kernel (I-f2),
(28)
Tk = (I-Q) Tk = kernel R
As a brief digression, but for use at a later point, an interesting alternative description of this decomposition of the representation should be noted here. subspace
Sk
In addition to the tensor space
n(,
of the k-fold tensor product
fully symmetric tensors; thus an element
X E Sk
X = {X.
(29)
... (
consisting of the
is a tensor
}
k
indices
ill ..., jk .
there can then be introduced the linear representation linear group which associates to any element
(30)
n
jk
which is symmetric in the
X E Sk
Tk
pk
introduce the
the vector rk(A) X
On this vector space Qk
A E GL(n,Q)
and any vector
having components
(rk(A) . X)jl...ik
E
X,el...tk A
l
of the general
A.
-25-
1
where as usual
1
this is again one of the classical symmetry
(A
representations of the general linear group, the representation in the notation of [47].
k > 1
For any index
Ok = (-k)
there is a natural linear
mapping
P
(31)
Tk -> Sk-l ,
:
the contraction mapping, which associates to any tensor the tensor
PX = {(PX)j
j -
}
Xi .
Jl-Jk
defined by
E Sk-1
(PX)jl...jk-l
Z
It follows readily from the definition (32) of the linear mapping the descriptions (24) and (30) of the representations
P Pk(A) ' X =
(33) for any element linear mapping
A E GL(n,C) P
and
that
rk
The image of the
X E Tk(n,C).
and any tensor
is therefore a linear subspace of
ak-1
Pk
and
P
Qk-l(A) ' PX
invariant subspace of the representation nontrivial while
} E Tk
k 1
1
(32)
X =
ok-l
;
which is an
Sk-l
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
is the entire space
P P
Sk-1 .
is a linear subspace of
Further-
which
Tk
is an invariant subspace of the representation
Pk ; and since this kernel
is clearly a nontrivial proper subspace of
and is contained in
Tk
Tk
as is evident upon comparing (27) and (32), it follows that the kernel of the linear mapping
P
coincides with the subspace
Tk
.
Thus (31) can be
-26-
extended to the exact sequence of linear mappings P
0->Tk->Tk->Sk-1 ->0
(34)
which commute with the appropriate linear representations; and this also
tuents.
Furthermore this argument shows that
into irreducible consti-
Pk
exhibits the decomposition of the representation
Sk-l
is isomorphic to
Tk
in such a manner that
(35)
Pk
ak-1 '
an observation which will eventually be useful but which will 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
A C Gk(n,T)
by listing the corresponding subalgebras a( C ,(1
Consider first a subalgebra ,oj C J 2(n,C) with with kernel A e A2
K2 C T2
of the form
,
in the extension (25).
A = {Sl, A2}
where
S1
jl R = 41(n,T)
k(n,M).
and
There must be an element
is the identity matrix; here
is determined uniquely up to the addition of an arbitrary element of
W2 C T2
.
For any element
X e,7
the bracket product
Y = [A,X]
e
and it follows readily from (13), indeed from the special cases (14) and (15), that
Y = {Y1, Y2}
where
Yl = 0
the Lie algebra representation
Y e X2 ,
(36)
dP2
and
Y2 = X2 + [A2, X1] = X2 + dP2(X1). A2,
being given explicitly by (26).
or equivalently
X2 + ddP2(X1) A2 e
K2
C T2
.
Thus
-27-
X = {X1, X2}
This last equation is a linear equation in the variables and describes a linear subspace of
R ;
the same dimension as
subalgebra ,( C
On the other hand it is a simple exercise,
using the Jacobi identity in the Lie algebra describes a Lie subalgebra
C 42 (n,M)
subalgebra is an extension of
K2 C ,C
A C'§ 2(n,M) with jl ,1 =
.
I1(n,M)
ranges over
for any tensor A2 e T2 ;
by j1,G`j = Z 1(n,Q)
this
and
,
and with kernel 2 C T2 are and
T2
A2-A2 e
same subalgebra precisely when
to see that (36)
,(i.2(n,Q),
Therefore all the subalgebras
contains the element A = {81, A2}.
A2
and clearly of
consequently (36) describes precisely the
42(n,M).
described by (36) as
,A
containing
T2
2
to range merely over coset representatives in
and
A2
so it suffices to allow A2
T2/ ) 2
2
There are thus
.
four general classes of such subalgebras when n > 1, depending on the choices of the kernel
describe the
A2
two when
n = 1,
and within each class the
possible subalgebras are parametrized by the vector space
T2/ K2
Actually for the purpose of classifying pseudogroups it suffices merely to consider subalgebras
hr
C AT 3(n,R)
when
A C Gk(n,Q)
and having
or subalgebras
n > 1,
That can be seen quite conveniently by
n = 1.
examining the subalgebras
when
,4Z C
,(2 C j2 ,4T
4k(n,C)
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 (36) with note that
3 = 0,
X = {0, , X3} J
c /c k(n,R)
is a subalgebra such that
2 = 0 ; thus dim j2, since
2 = 0
whenever for all
j
= dim
X = {0,0,X3} e
by
Lemma 1
;
1 1,5
3
.
then
and thus
If
j2 tZ
k = 3
-28-
dim R = dim j2 R = dim 41 . then
so by (36)
j2X C j2,fL,
X ,X e j2 1R,
J
hence by
for all
X1
so
j,
belongs to
for
n > 1,
K 2 = T2
,
j
;
k > 3
Suppose next that
(ii)
.
as in (28); thus A2 e T2
x.x = {O,x.x } e T2
and
j2A
k = 3
for all
j
Aj2 T-
define
k, the
X1'" 1""'X k-1) -1)
C ,{; k(n,M),
has the form (36) with
j2 A(,
and it can
X = {0,0,X3} e3 then
and
by Lemma 1, so that
X.X3 = n >,.X-
for all
and writing this condition out explicitly by using (27),
X..
JJlj2
Setting
= (n+l)
E (S
X Vjl)
' Jl
J2
n > 1
i
it follows that
= 2(n+l)-1 Et
xvVlj2
xvV 102
but since
j2
and summing over all values of
i = j
E
j
necessarily
= 0
and consequently X.
,
Thus again
3 = 0,
X = {X1,X2,X3} E Aj2 uniquely by
X1
and
dim AZ then
= dim j2,CC
j2X e j?A ,
up to an element of
TZ
;
and
On the other hand if
.
so that
and hence
X2
X X e j2 /C 0
XjX3 + dp2(XjX2)' A2 E T2
= 0.
1j2
1J2
that
X2
define the same
)
m
dim j2 C? = dim F 1 + dim T2 ,
If
.
j2A C G2(n,,
and
is a subalgebra such that
be assumed that
A
Xk + (some expression in
A C Gk(n,a:)
so that
pseudogroup for all
and
The same argument applies inductively in
obvious analogue of (36) showing that k,
/U
Since
2
and
X1,
is similarly determined uniquely by
X3
have the same dimension it follows from Lemma 1 that the same pseudogroup.
X = {X1,X2,X3}E
is determined uniquely by
X2
dim Aj2 R = dim
and thus
;
On the other hand if
is determined for all
j, so
-29-
X.X3 + dp2(X.X2) - A2 = s
[ 0X3 + dp2(XJX2) ' A2]
Writing this condition out explicitly by using (26) and (27), and recalling that
so that
A2 e T2
L
and consequently
A2 = 0
for all
0
j
it follows that (n+1)-1
X'01)
)c'.j2 +
t(8 + Et(Xjj1AiV2 + X j2Ai-
i = j
Setting
and summing over all values of
V lj2 = 2(n+l)
,
l
Xijt j lJ2)
-
i
.
note that
,
-1
t V 132
m (Xmj1A.2 +
+
Since
n > 1
is thus determined uniquely by X2 ' 102 is also determined uniquely by X2 ; and therefore
X3
t
dim Aj2 P, = dim j2 T = dim L and
j2A
so it follows again from Lemma 1 that
,
determine the same pseudogroup.
If
k > 3
the further analysis reduces to that in case (i) j2A C G2(n,T)
k(n,l),
has the form (36) with + dim T2 since
Tk
for
R2 = T"
n > 1, ,
hence
is the kernel of
i2
A2 E Tz
.
dim j2,
V
.
2)
general that
k
for every
by definition and since
X E T,
(iii)
and
Suppose
= dim
+
Note incidentally that E (S2X)''V2... ''
E
X3 = 0
A C Gk(n,O:)
k > 3.
A
is a subalgebra such that j2p
as in (28); thus
and it can be assumed that
,
;
then since
determine the same pseudogroup for all
next that C C
Xm
E A
the expression
and consequently
MJ2AV1 -
_
jk by an easy calculation, it follows in
-30-
Tk = {XETk:
(37)
k = 3
If
and
for all
0
Z
X = {0,0,X3} E "3
then
j2,...,jkI
{O,ajX3} E T
JX =
Lemma 1 ; using (37) this condition can be rewritten
E
.
for all
by
j
2
A.V.
hence
= 0,
t implies that n
X e T3
indeed3 = T3 ,
;
nontrivial element in
or
YZ3 = 0
as can be shown by exhibiting any one For this purpose note that for any elements
(3
p2 there must exist elements
of
in
so that either
,
n
3 = T3
X2, Y2
3 C T3
Thus
.
the bracket product
,
X = {O,X2,X3} ,
y = {O,Y2,Y3}
then also belongs to M
Z = [X,Y)
,
and
it follows readily from (13), or better from (14) through (16), that Z = {O,o,Z3} E
where
3
(38)
Ek
SJ(xj.`lj2Ykj 3 - Yjlj2 xkj3)
Thus it is only necessary to observe that on
T2 X T2
when
n > 21
is the tensor having
and to see that, merely note that if
;
as its only nonzero component then
X11
It follows from these observations that the other hand if and
be rewritten as
E j2IFL
then
for all
j
X X3 + dP2(XjX2)' A2 e T2 ,
X2 E T2
Z112 = X11 22
dim j2 R + dim T3
dim
X = {X1,X2,X3} E Aj2 n
X.X = {x.x2, x.x3}
is a nontrivial bilinear form
Z3
;
On
j2X = {X1,X2} E j2 P and the latter assertion can
or using (26) and (37)
equivalently as
Thus
X3
dim Aj2 that
A
t
nV
is determined by
X2
= dim j2 and
j,A
Jl
ij l Aim
.&,m
up to an element of
+ dim T3 = dim ,ff
;
T3
,
so that
so again it follows from Lemma 1
determine the same pseudogroup.
The kernel
K 3
now
-31-
k > 3
the same argument applies for all indices
T3 ,
being
k
suffices merely to note that the nontriviality of
for
follows upon considering the bracket product of elements
Yk_1 E
k-1 =
(iv)
.
exist elements
If
(13) as before that
n > 1
,
k > 3
Z = {O,O,Z3} e 3 where
and hence that X2, Y2 or
T3
= X21
there must
has the form (38).
Z3
3 = T3,
X12
T2
, and it follows readily from
then also belongs to AR
the tensor having
.
in JR ; the bracket
y = {O,Y2,Y3}
nontrivial and is not contained in either X2
and
is a subalgebra such that
indeed it is only necessary to find elements
take for
= T2
Thus once again A C Gk(n,G)
k(n,M)
it is then easy to see that
k > 3
X = {O,X2,...,Xk}
k = 3 then for any tensors X2, Y2 of
X = {O,X2,X3}
Z = [X,Y]
product
2
define the same pseudogroup for all
Suppose finally that a( C
J2 ,Ql = ,(J}.2
X2 e
are quite arbitrary.
k-1
Tit
j2A C Gk(n,M)
and
where
of
Y = {o,...,O,Yk-l,Yk}
and
as well; it
T3
3
such that .
If
Z3
is
For this purpose
as its only nontrivial components
Y2222
and for Z
has
as its only nontrivial component; then
the tensor having
Y2
Z221 = Z212 = 2122
as its only nontrivial components, and is easily
seen to have the desired properties.
If
n = 1
the bilinear expression (38)
is trivial, so that there are two possibilities:
X3 = 0
either
In the first case there is a nontrivial subalgebra
W 3 = T3 .
but as in (i) it is not necessary to consider subalgebras ,Z C
with j3 ,07 = A 3 for values k > 3 If k > 3
and R C
- § k-l(n,(t) elements
k(n,(t)
and
3 C
3(1,a),
k(1,C)
and in the second case
is a subalgebra such that Jk-lP
then for any tensors
X = {O,X2,...,Xk}
;
or
X2 e T2
and
Yk-l e Tk-l,1, there are
Y = {0,...,O,Yk-1,Yk}
in
'
and their
-32-
bracket product
the tensor having
for
X2
for
Yk-1
either
Tk
n = 1 or
Yi
having
Zk
as its only nontrivial component
1
Zi
as its only nontrivial component,
1
and since it is easily seen that
;
it follows that
Tk
Taking
k .
as its only nontrivial component and
X11
the tensor having
yields the tensor even for
is an element of
Z = [X,Y] = {0,...,0,Zk}
Zk
and
}"{k = Tk
is not contained in
= % .
Thus for the remainder of the discussion the only subalgebras that need be considered are the subalgebras
C jj,2(n,T)
i
A? C
(36) and the one additional class of subalgebras that
j2
2(l,M)
3 = 0
and
algebras more explicitly note that when one-dimensional, so an element complex numbers
{X1,X2,X3}
.
3(l,a)
such
To describe this last class of n = 1
3(1,x)
X e
described by
each tensor space
Tk
is
is described by the three
and the bracket operations (14) through (16)
;
have the simple form
[X,Y] = {O,X1Y2-X2y1, 2X1Y3-2X3Y1} .
(39)
There must exist an element complex constant If
X = {X1,X2,X3}
A3
A3 ; and
of the form
hence
(40)
Y e
;
3 = 0.
3
but it follows easily from (39)
so Y= 0
and
x3 = A3X1 .
This last equation describes a linear subspace of the same dimension as
for some
then the expression
also belongs to
Y = {0,0,2X3-2A3X1},
A = {1,0,A3}
is even uniquely determined, since
is any element of
Y = [A,[A,X]] - [A,X] that
A e A
,ll
,
9 3(l,C.)
which is of
and which must consequently coincide with Xi? ;
-33-
and it is a straightforward matter to verify that (40) defines a subalgebra for any complex constant
of 41 3(1,M)
subalgebras is parametrized by A3 e
A3
so this additional class of
,
M .
Rather than determining at this point exactly which of the subgroups described by the subalgebras (36) and (40) are integrable, it is more convenient to describe some simple necessary conditions the parameters in (36) must satisfy in order that the associated subgroup be integrable; A2
that these necessary conditions are actually sufficient will then follow easily after a discussion of equivalence of pseudogroups.
Suppose therefore
that the subgroup A C G2(n,1)
associated to the subalgebra
defined by (36) is integrable.
The 3-jets of all mappings in
compose an integrable subgroup
A' C G3(n,T,)
subalgebra
£ ' C Q] 3(n,O)
P = j2 AZ'
,
exist an element
such that
associated to
A'
and it follows from Lemma 1 that A' _ {81,A2,A3} e
/Q,
where
,
AI
the subalgebra A by (36) ; and since
T (A)
A = j2A'
;
the
then has the property that ,t7
' C A 1
.
A = {81,A2}
e A for all
X.
C ,V2(n,T)
j3
There must describes it follows
that
(41)
X
j 3A3
+ dp2(Xj3A2) - A2 E
R2
for all
j3
.
There are only two cases in which this condition leads to any interesting consequences for the tensor
A2
(i)
.
First suppose that
2 = 0
.
In
this case (41) can be rewritten more explicitly using (26) as
i Ajlj2j3 =
k
i
k
(Ajlj3Akj3 + Aj2j3
k
j lj,
ki 3A
The left-hand side of the above equality is fully symmetric in the indices
-34-
jl' j2' j3 ,
so the right-hand side must be also; and that is clearly
equivalent to the assertion that
Ak 1j2
£k Akj
(42)
is symmetric in
j1 j2 j3
.
3
Thus (42) is a necessary condition that the tensor for integrability.
(ii)
Next suppose that
in this case it can also be assumed that Akj = 0
that
Y = Q
for all Y ,
j
.
Since
T
A2 E T2
X 2 = T2
A2 E T2
,
must satisfy
and recall that
and hence that
consists of all tensors
Y E T2
such
condition (41) can be rewritten more explicitly using (26)
and (27) as
Aj123 jj
=
(n+1)-1 Ek
(sjj 23j + Sj 1
xJ13 j
2
k i Ajlj3 k i Ajlj2 k + £k (Akjl Aj2j3 + Aj2 - Akj3
On the one hand setting
i = j3
and summing over
(n+1)-1 £k Akjlj 2 = 2(n-1) -1 £k
t
i
it follows that
AV 1 k2
and on the other hand the left-hand side is symmetric in the indices j3
ji, j2,
so the right-hand side must be also; and upon combining these two
observations it follows that
(43)
k
Akj
Ak
+ (n-l)-1
01j2
81
Ak.
Fk' j3 Tji
Thus (43) is a necessary condition that the tensor for integrability in this case.
is symmetric in
A2 E T2
jl,j2,j3.
must satisfy
-35-
To describe the subgroups of G2(n,@) subalg ebras (36), for any tensor
corresponding to the
{A,lj2} 1 E T2(n,M)
A2
introduce the complex
analytic mapping
9A: G2(n,1) -> T2(n,Q)
defined by
_
_
where
k k £k Aklk2 j1 £j2 +
eA(E)jlj2 = Ek Ek jlj2 -
(44)
for any
E-1 =
2
Using (6) and (7) it
_ (El'E2) a G2(n,M).
is easy to see that
lj2 = k
9A(
for any two elements
E, r
k k 1 Ij2 ')k 9A(e)klk2 Ij1
+
G2(n,T) ; and recalling (24) this can be
in
rewritten equivalently as
(45)
eA(£' TO = P2(TI-1)
'
eA(E) + eA(r1)
It is an immediate consequence of (45) that the zero locus of the mapping 6A , the subvariety
loc eA = {E C G2(n,l)
(46)
is actually a subgroup of of the mapping
9A
G2(n,a).
also define subgroups of
P2
9.
:
T2(n,M)
n > 1
the compositions
to the invariant sub-
satisfy equations analogous to (45) and hence
G2(n,Q).
e,' = 2
eA(E) = G}
Furthermore if
with the projections of
spaces of the representation
(47)
:
That is to say, the mapping
G'(n,d) -> T,,(n,T)
-36-
satisfies (45')
eA(E' TI) = p2(Tj-1) ' eAO + eA(T))
so that
1oc OA
is a subgroup of
(48)
eA = (I
Now if all
t
E(t)
G2(n,M)
;
and similarly for
A)' OA : G2(n,e.) -> T2(n,G)
is a one-parameter subgroup of
loc OA
then
or more explicitly, recalling (44),
;
k q 1k2 A
( t)k
for all
t = 0
t.
0
Upon differentiating this identity with respect to
in the Lie algebra
,(; 2(n,M)
t
at
X = dE(t)/dtIt=0
associated to this one-parameter subgroup
X2 + dp2(Xl)' A = 0 ; hence the Lie algebra of the subgroup
satisfies
loc OA C G2(n,T)
is the subalgebra (36) for
is a one-parameter subgroup of
loc OA
then
2 = 0
Similarly if
E(t)
eA(E(t)) = 4. OA(E(t)) = 0
t, and it follows correspondingly that SZ (X2 + dp2(X1). A2) = 0
for the element
X E
42(n,C.)
associated to this one-parameter subgroup
hence the Lie algebra of the subgroup (36) for
2 = T"
loc eA C G2(n,I) G2(n,T)
k
(t)ai E(t)j2 +
and recalling (26), it follows that the element
for all
for
9A(E(t)) = 0
,
loc OA C G2(n,T)
is the subalgebra
and dually the Lie algebra of the subgroup
2 = T;
is the subalgebra (36) for
.
The subgroup of
corresponding to the subalgebra (36) fort = T2
full group
G2(n,T)
is of ccurse the
itself.
Finally, to describe the subgroup of the subalgebra (40), for any complex constant
A3
G3(1,1)
corresponding to
introduce the complex
-37-
analytic mapping
G3(l,a) -> T
defined by
GA(E) = E3 Ell -
(49)
for any
3
=
4331 E3 G3(l,T) l
.
1
2
A(El - 1)
-12 -
2
Using (6) through (8) it is easy to see
2 '
that * (TI)
Tj) = T12 0*(E) + GA for any two elements
e,
TT
in
G3(l,C) ; and recalling (24) this can be
rewritten equivalently as
(50)
eA
The zero locus
loc 6A
T') = p3(T1-1)
'
eA(E) + eA (Tl)
is then of course a subgroup of
considering the one-parameter subgroups of
loc eA
G3(1,T)
;
and
as above it follows readily
that the Lie algebra of this subgroup is the subalgebra (4o). In summary then, as defining groups of all possible tangentially transitive restricted Lie pseudogroups it suffices to consider the following subgroups of
Gk(n,R):
(51)
loc GA
where
A2 E T2(n,T)
satisfies (42)
(52)
loc eA
where
A, E T2(n,M)
satisfies (43) and
(53)
loc eA
where
A2 E T7(n,M)
and
(54)
loc eA
where
A e I
and
n = 1
n > 1
;
n > 1
-38-
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-
The The classification of tangentially transitive pseudogroups: analytic aspects
There are the four general classes of possible defining groups for tangentially transitive Lie pseudogroups, described by the four classes of equations (51), (52), (53), and (54) respectively; and within each class the defining groups are parametrized by 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 cases, so for convenience will only be discussed in detail for the class given by equation (51). loc 0A
In that case the defining groups
from an open neighborhood U
g(0) = 0
will be called
if there exists a complex analytic homeomorphism
equivalent, written A ~ B, g
loc eB
and
of the origin in
Mn
to
g(U)
such that
and
0B(32g(z)) = B-A for all
(55)
z e U ;
this is actually an equivalence relation, as is readily verified by using the basic identity (45).
observe that the equation
(56)
where
For this and other purposes it is convenient to eA
can be written in the form
0A(E) = e(E) + A - p2((-1)- A
o(¢) = e0(e)
is the particular case of this equation corresponding to
A = 0 ; thus (55) can be rewritten
(57)
e(J2g(z)) = p2(Jlg(z)-1)
-
B- A
-4o-
The situation in the other three classes is quite analogous To see the significance of this notion of equivalence suppose that
B -A,
so that there exists a complex analytic homoeomorphism
satisfying (57), and consider an element f
f e
'P(A)
;
g
it can be assumed that
is also a complex analytic homeomorphism between two open neighborhoods
of the origin and that
The condition that
f(O) = 0.
f e T(A)
can be
rewritten using (56) in the form
9(j2f(z)) = p2(jlf(z)-1)
for all points
f* = g e f . g-1
z
'
A- A
in an open neighborhood of the origin.
The composition
is also a complex analytic homeomorphism between two open
neighborhoods of the origin such that
f*(0) = 0 ; and using (45) and (55)
it follows that
e(j2f*(g(z))) = 0(j2g(f(z))j2f(z). j2g(z)-1) =
hence that
f* e
T(B).
p2(j2f*(g(z))-1)' B-B Thus whenever
coordinates near the origin in into the pseudogroup
T (B),
n
A - B
,
there exists a change of
which transforms the pseudogroup
° (A)
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
-41-
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 = 0
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
for any admissible parameter value
A -. 0
A ; and recalling (57) that
merely amounts to showing that there exists a complex analytic mapping from the origin to the origin in singular and
g
9(j2g(z)) = -A , groups.
,n
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 observation.
Lemma 2.
Let
A
neighborhood of the origin in analytic mapping
g
be a complex analytic mapping from an open In
into
There exists a complex
T2(n,T).
from the origin to the origin in
has any specified value and
(58)
Where
j2 (z) j'g(z) = @(z)
,
k E1k (z)
= E
k J1J2 (z)
,
if and only if the expression
,n
such that
1(0)
-42-
s (59)
j 1j 2j3
(z)
j j
a
is symmetric in the indices
(z) + Ek X
(z) x
12
J3
3
(z)
1j 2
ill J2l J3
This is a simple exercise in applying the classical
Proof.
integrability conditions of Riquier and Janet
[25)
or their modern counter-
parts; but since the proof is so simple in this case an outline will be Repeated differentiation of (58)
included here for the sake of completeness.
and then the use of (58) to simplify the results shows by an easy induction that if
satisfies (58) then the higher-order terms of the jet
g(z)
(z) = jg(z)
satisfy
(6o)
where
jl...jv(z) = E
X01j, (z)
Ek(z)
is the given mapping
X
and for
v > 3
.
1 (z) jl...Jv
(61)
= azj V
X1
(z) + E
j1...Jv_1
Thus if there exists a function l,
(z)
1
g(z)
1
k
iv(
(z). z) Xk J1...Jv_l
of the desired form then the expressions
defined inductively by (61) must be symmetric in the indices
jv
j1,...,jv
for all
v > 3.
symmetric then having chosen (0)
Conversely if all these expressions (61) are e1(o) the formula (60) determines the coefficient'
of a formal Taylor expansion
estimates [25)
satisfying (58); and the usual
g
show that this series converges in some neighborhood of the
origin, hence represents a function having the desired properties. symmetry of all the expressions
kl
Jl...jv
formula (61) reduces to (59); while for
(z)
note that for v = 3
v > 4
As for the the general
by iterating (61) it £oi_ows
-43-
easily that 2
1
...j
v
(z) = az
j ...j v -2 (z) az. 1 iv Jv-1 a
+
v-1 (z) az jv 1
+
> j ...j1-2 (z) 1
+ . v(z) 3z
`'
j
1...
iv-1
y 2
(z) ))
(z) 1 (Z) 1.-.Jv-2Jv-1Jv
k
so by 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.
the origin in
In,
is a complex analytic mapping from the origin to
g
If
and if n(z) = det{E'(z)}
j2g(z) = (z),
if
Jacobian determinant of the mapping
then
g,
A(Z) Ek
(62)
S
L
J
where
{EJ (z)}
_
Proof.
matrix
{{1(z)},
{EJ(z)}-1
is the
(Z) EkJ(Z)
.
Letting
(z)
denote the j-th column vector of the
note that the determinant
A(z)
is a multilinear function
j
of the column vectors so that
as z)
= Ek det { 1(z),...' k-1(z),
8Ek(z)/8zj1Ek+l(z)...3En(z)}
Expanding this determinant in cofactors of the k-th column, noting that the
cofactor of 8(z)/8z.
is the same as the cofactor k(z)
J
(z)
in the original matrix
as Z) _ £k,
J
it follows that
8Ek(z)
z
j
k(z)
of the element
-44-
but as is well known
=k (z) = 1(z) ¢' (z),
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
A ^ 0
exists a nonsingular analytic mapping such that
Mn
e(j2g(z)) = -A .
from the origin to the origin in
g
(z) = j2g(z)
Setting 0 = 00 ,
explicit form (44) for the mapping
and using the
this equation can be rewritten
in the form
11
(z) = i
k Ek(z)
-
2
A.
1J2
ymmj
It follows immediately from Lemma 2 that there exists a solution desired whenever the expression since
j2' j3'
Ask
(42) that the parameter
E
1
k Akj3
g
as
is symmetric in the indices
A`
1 lj2
are constants; but that is precisely the condition A
is required to satisfy, hence
A ^-0
as desired.
Considering next the defining group (52), in order to show that A - 0 g
it is necessary to show that there exists a nonsingular analytic mappin_
such that
that
O" (j2g(z)) _ (I - S2) 0(j2g(z)) = -A
where
0(¢(z)) = R0(¢(z)) - A
equivalently such
or
,
Using the explicit
8(z) = j2g(z).
formulas (27) and (44) and recalling Lemma 3 it is readily verified that
= 2 0(C(z))i Jlj2
(63)
where
Zl(z)
1
Csi
nn+l
IL jl
alog'
azj2
z)
+
&i
J2
is the Jacobian determinant of the mapping
alog0 z)1 1 azjl
g
'
at the point
z
;
-45-
0(z) = (n+l)-1 log L(z)
so setting
the equation for the mapping
can be
g
k
l2 (z) _ j
1jlj2 (z)
k Ek(z) X j (z) 1 2
sl
= S1 "' +
ao z) -
jl az
j2 6z j.
j22
It is easy to see that if
Al jlj2
l
satisfies (64)
e(z) = j2g(z)
where
X
(z)
j1 2
is of the form (65) for some function is necessarily a constant, and hence
r(z)
then
(n+l)-1 log t(z) - o(z)
satisfies the desired conditions;
g(z)
so to demonstrate the existence of the desired function
g(z) it is only
necessary to show that there exists some analytic function
o(z)
for which
Now using Lemma 2 this integrability condi-
the equations (64) are integrable.
tion readily reduces to the condition that the expression
E p1 k
G73
Ak
aQ z - ago z) + Si r 33L azj azj azj _(z) azj
l2
1
is symmetric in the indices
1
2
j1, j2, j3
-
jlj2
azkz )
2
A
but since
;
aQ
Ak k
k
satisfies (43) this
is equivalent to the condition that
81
j3
[l
n1,t Ak
A.
V1 kj2
pk
+
jlj2
6-(z) - a-(,) 3-(z) +
a2v
azj
azj azj
azk
azj 1
is
symmetric in the indices
j1, j2, j3
equivalent to the condition that
,
2
1
2
and this in turn is clearly
(66)
a2a(z) azj1 azj2
for all indices
as z
as z
azj1
azj2
ji, j2
+E
6-(')
Ak
k
1
+
1
n1
J1j2 azk
0
Ak
ijl Al J2
Thus the problem is reduced to that of determining
.
whether there exists an analytic function
a(z)
satisfying the equations
e_a
(66).
reduces (66) to the linear system of partial
Setting '¶ =
differential equations 2
a T(z)
(67)
+£
k az jl
as z_ 1
k
k
nn-1 k A'P
azk
k Ajlj2 a2
.j 2
z () z = 0
1
for which the integrability conditions are classical [25] and can be obtained by arguing as the proof of Lemma 2.
function
¶(z)
satisfying (67)
Indeed if there exists a
then by repeated differentiation and simpli-
fication it follows inductively that the function avc(z) (68)
for all
a
(-1 )v+l kE A1...jv j
azj1 ...az3v
where
v > 2 ,
aj
j 12
Ak
A = 5
Akj
Vl
k,
satisfies
k + (-l)" Ek A j l...jv_kakjvT (z)
azk )
= (n-1)-1
¶(z)
,
2
Aj'
12
is the given tensor, and inductively
(69)
Ajl...jv
for v > 3.
+ k Ajl...3V_2
Ek Aji...jv_1 Akjv
If the expressions
.v
and
ji, ..., iv
.v_1
jv
are symmetric v then equations (68) determine the Taylor
Al
1
in the indices
Ek A
3v-1
coefficients of the desired solution.
That A
12.3
a.
1
is symmetric in the
j
indices
ji, j2, j3
is just the condition (43) that
A
is assumed to satisfy,
and it is a simple calculation to verify that that in turn implies the symmetry of the expression
Ek
akj3
;
and as in the proof of Lemma 2
-47-
the iteration of (69) then yields expressions showing the desired symmetry v > 4.
whenever
Thus the equations (67) are integrable, and as noted that
implies that A ^- 0 . Considering thirdly the defining group (53), in order to show
that A ^-0 mapping
g
it is necessary to show that there exists a nonsingular analytic e'(j2g(z)) = 4 e (j2g(z)) = -A.
such that
Recalling (63), this
equation can be rewritten as
alog
si
(70)
1 6z
J2
z)
Ai
jlJ2
az
Jl
j2
If (70) holds then it is readily verified that
clog z) 3zj
(71)
k
=
-Fk Akj
and conversely if (71) holds then since that (70) holds;
A = CA
by assumption it is clear
thus (70) is equivalent to (71), and since the latter
equation obviously has solutions it follows that
A ^ 0
.
Considering finally
the defining group (54) in the one-dimensional case, in order to show that A - 0 it is necessary to show that there exists a holomorphic function that
g'(0) # 0
and
g
such
e*(j3g(z)) = -A; but this is an analytic ordinary
differential equation, for which there always does exist such a solution [3], 80 that in this case too
A - 0
.
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.
-48-
First for the defining group (51) with
A = 0
the associated pseudogroup
consists of those nonsingular complex analytic mappings
f
such that
6(j2f(z)) = 0 ; but writing (z) = j2f(z)
and using the explicit form (44)
this clearly reduces to the condition that
E1
(z) =
2f.(z)/az. az. = 0, 1 J1 J2
f
be a nonsingular mapping
JlJ2
hence that
f
is an affine mapping
fi(z) = £,j aij z. + ai
(72)
for some constants
aij, ai
is just that the matrix
(52) with A = 0
The condition that
.
{aij} is nonsingular.
Igext for the defining group
the associated pseudogroup consists of those nonsingular
complex analytic mappings
f
and writing (z) = j2f(z)
such that
6"(j2f(z)) = (1- 2) O(j2f(z)) = 0
and recalling (44) and (63) this readily reduces
to the condition that )
( z
(73)
_ E1
1J2
i(
(z) aQ z
j1
+
Z.
z
)
aQz)
az
j2
J2
for
Q(z) = (n+l)-1 logA(z),
the mapping
f.
where
Jl
is the Jacobian determinant of
A(z)
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
0(z)
then
is necessarily a constant;
0(z) - (n+l)-1 log A(z)
thus to find the desired mappings
f
it is sufficient merely to find solutions
of the system of partial differential equations (73) for arbitrarily chosen functions
o(z).
Of course these functions
must be chosen so that
o(z)
the system is integrable; but from Lemma 2 with
ao z)
Jlj2 (Z) = slJ1 azj 1
2
+ siJ? azj
a01(z
)
1
-49-
it follows
readily that the integrability condition is just that the
expression
si
ago Z)
J
z. azj2
Jl
Jl
j (z) = -e-Q(Z)(bj
l2
and
l
bJ
that
f
U(Z)
aZ Jl
Z) J2/
=o
The equation (73) can then be rewritten
.
or yet equivalently
(z) + bj ¢i (z)),
2
2
a2
e-
ao
aZ
= b + E. b.z. J J J
l
(e-o(z)
3z. 8z. Jl J2
therefore
3r(z)
must be such that
o(z)
b
hence equivalently that
,
az . az . \ a2o(z) J1 J2
e-0(Z)
for some constants
j,
-e-o(z)
e o(z)
J2
Thus the function
Z
az. J
61Y
jl, j2, j3
is symmetric in the indices
a2
3Q(Z)
J1
3
adz :
-
=
fl(z))
some constants
for
f .(z) = a + E, a., z. 1 1 J 1J J .
0 .
ai, aij
,
so
must be a projective mapping
f(z) =
(74)
1
As is familar, the condition that (n+l) x (n+l)
a.
b+
bJ J
f
.
j
be a nonsingular mapping is that the
matrix
{a.1 } b
.zj z,
a.+
{a 1J .. } {b.} J
-50-
is nonsingular; and as noted earlier
must be a
o'(z) - (n+l)-1 log A(z)
constant, so that the Jacobian determinant of the mapping
f
must be of
the form
A(z) = c e(n+1)o(z) = c(b+F-
(75)
for some constant
c.
bjzJ)-(n+l)
Thirdly for the defining group (53) with
A = 0.
the associated pseudogroup consists of those nonsingular complex analytic mappings
such that
f
(z) = j2f(z) dition that
e'(j2f(z)) = SI9(j2f(z)) = 0 ; and writing
and recalling (63), this equation readily reduces to the conis constant.
s(z) =
(54) with A = 0
Finally for the defining group
the associated pseudogroup consists of those nonsingular
complex analytic mappings
f
such that
e*(j2f(z)) = 0 ;
classical Schwarzian differential operator [201
so
f
but this is the
must be a one-dimen-
sional projective mapping, otherwise known as a linear fractional or Moebius These and the preceding results can then be summarized as
transformation. follows.
Theorem 1.
Up to equivalence there are only the following
tangentially transitive restricted Lie pseudogroups of complex analytic transformations in (i)
In
.
the pseudogroup of nonsingular complex affine mappings (72), having the
defining group
partial differential equations (ii)
n > 1,
or alternatively characterized by the
loc e C G2(n,T)
e(j2f) = 0
;
the pseudogroup of nonsingular complex projective mappings (74) for having the defining group
loc e" C G2(n,itl
characterized by the partial differential equations
or alternatively 9"(j2f) = (I- R)9(j2f) _
'%
-51-
the pseudogroup of nonsingular complex analytic mappings with constant
(iii)
Jacobian determinants for loc 8' C G2(n,M)
n > 1,
having the defining group
or alternatively characterized by the partial differential
e'(j2f) = ae (j2f) = 0 ;
equations
the pseudogroup of nonsingular complex projective mappings (74) for
(iv)
n c 1,
having the defining group
loc 6* C G3(1,2)
characterized by the ordinary differential equation (v)
or alternatively 0*(j3f) = 0 ;
the trivial pseudogroup, consisting of all nonsingular complex analytic
mappings.
For (v) ; while for and (v).
n = 1 n > 1
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
n > 1
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 by a subgroup of equivalent pseudogroups still have rather special forms.
Gk(n,T);
To give merely the
simplest example, any pseudogroup equivalent to (i) in the case have a defining group of the form
loc 6A C G2(1,T)
these
n = 1
for some constant
will A,
-52-
hence will consist of those analytic mappings equation
f"/f' = A(f'- 1)
;
f
satisfying the differential
and a simple calculation shows that if
A # 0
these mappings have the fore
f(z) =
for some constants
C and
C'
-
.
A-1 log(e-Az + 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
n = 1
nave this form for
characterizing the pseudogroup.
worthwhile here to try to describe explicitly all the to those listed in Theorem 1 though.
it does not seem equivalent
-°,3-
Pseudogroup structures and their associated connections
S6.
As is of course well known, an m-dimensional topological manifold
M
is a Hausdorff space each point of which has an open neighborhood hom.eo-
morphic to an open subset of
A coordinate covering
Mm.
such a manifold is a covering of M by open subsets which there is a homeomorphism subset
V. _ Il
; the sets za
homeomorphisms
Ua
za :
= za c
at
p
are related by
p e UU '1 U5 '1
and an open
The compositions
za(UafU
z(Uan
For any point
V
and
p e Ua fl Ug
z a(n) = fa R(z(p)) ;
Va ,
called the
the two local coordinates
and for any point
the three coordinate transitions at
T.1
Ua
are called coordinate neighborhoods, and the
are then homeomorphisms between subsets of coordinate transitions.
for each of
Ua C M,
Ua -> Va between
are called local coordinates.
faL
^V. ={U2} of
p
are related by
The union of two coordinate coverings is clearly
(z,Y(p))) =
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 and the mappings
can be reconstructed from knowledge of the subsets fa p
and identifying points
alone, by taking the disjoint union of the sets za e "a
and
zo e V
whenever
za
fa 13
If m = 2n coordinates
za
Va C Ikm
and
II?m
is identified with
,n
can be viewed as mappings in.-t... subsets
(za)
then the local VC, C M'
and the
Va
-5L+-
coordinate transitions
as mappings between subsets of
fa
fin.
The
P
coordinate covering Z.
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 composition 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 e closed under composition. For example considering coordinate coverings for which the coordinate transitions are
C00
mappings leads to
C00
structures on manifolds and to
C°C
manifolds, another very familiar and much studied structure and class of manifolds.
Since complex analytic mappings are
C00
it is evident that any
C' coordinate covering and that
complex analytic coordinate covering is also a
equivalent complex analytic coordinate coverings are also equivalent as
coordinate coverings; so a complex analytic structure on a manifold M naturally subordinate to a
C°C
structure on
M,
analytic manifold is also in a natural manner a all
C0C
is
or equivalently, a complex C00
manifold.
Cm manifolds admit complex analytic structures; and if a
of course not C0D
manifold
does admit a complex analytic structure it may well carry a number of inequiva-
_55-
lent 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 an 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 determining 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 Paeudogroup of nonsingular affine mappings for example consists of those non-
singular analytic mappings explicitly by (!.t) for
f
A = 0
for which
P(j2f) = G,
where
e
is given
and satisfies the basic relation (45), 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 way for all of the first four pseudogroups lisoe,i
in Theorem 1, and indeed for many ether pseudogroups as well [19], 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 M coordinate covering
I/!
7. a: U,,, > Va C (Cn
and coordinate transitions
= {Ua}
with complex analytic
having local coordinates
faP
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
wa = fa ° za: Ua -> Wa
coordinate transitions ffP = wa -W sl = fa mappings.
That
tion that
e(j2fQ,0) =
=
fU
fa: Va > Wa the
faE ° f51 are complex affine
is a complex affine mapping is equivalent to the oonci0,
(J2fa' 62''a5
which by (45) can be written in the form
1
P2(i2fs){P2(J2fa g)-1 e(j2fa) + 9',j_fa )-
3
-57-
and introducing the complex analytic mapping
Ua > T2(n,e)
sa :
defined by
Sa(p) = e(j2fa(za(p)))
(76)
the condition can be rewritten
e(j2fa
(77)
6
(z(p))) = s13(p) - p2(jlff3
since the representation
if fa: Va -> wa
p2
depends only on the one-jet of its argument.
is another set of complex analytic homeomorphisms such
that the local coordinates a =
a
structure on M then the functions (77).
a (za(p)))' sa(p)
za
also describe a complex affine also satisfy
sa(p) = B(j2fa(za(p)))
These two complex affine structures are equivalent precisely when the
compositions
fa ° fat
are complex affine mappings; and by (45) that is just
the condition that
C = 0(j2f0
j2fal)
= p2(j,fa) {e(j7fa) - -,)(j2fa)}
or equivalently that c
sa = sa .
,
To introduce a convenient terminology, a
lex analytic affirm connection for the complex manifold M will be
defined to be a collection of complex analytic mappings for some complex analytic coordinate covering
U-
tese mappings satisfy (77) in any intersection
a
sa:
= {-J I
UP ;
of
-> T2(n,a), such that
such a coliection
functions induces a corresponding collection of functions on any refinement
-58-
of the covering U ,
and all these will naturally be identified.
Such a
connection will be called integrable if after passing to a suitable refinethere exist complex analytic homeomorphisms
ment of the covering 7/
ff: Va -> W. satisfying (76); of course this is equivalent to the condition there exist
that after passing to a suitable refinement of the covering merely nonsingular complex analytic mappings (76).
fa: Va -> do
satisfying
With this terminology the preceding observations can be summarized as
follows; this result is due to Matsushima [34], and is also discussed by Vitter in
[45].
On any complex manifold M
Theorem 2.
there is a natural one-to-
one correspondence between the set of complex affine structures on M the set of integrable complex affine connections on
and
M.
Having made these simple observations it is apparent that the corresponding result holds for the other pseudogroup structures on replacing
6
M,
by the appropriate differential operator in the definitions and
assertions.
Thus a complex analytic projective connection for a complex
manifold M
of dimension
sa
Ua -> T2(n,Q)
Z i = {Ua} (78)
whenever
mercy
n > 1
is a collection of complex analytic mappings
for some complex analytic coordinate covering
of M such that e"(j2fa
a
(zP(p))) = sa(p) - p2(jlfp a(za(p))) sa(p)
p e Ua n Ua ;
and such a connection is integrable if after passing
to a suitable refinement of the covering Li there exist nonsingular complex analytic mappings
f a : Va -> n
such that
-59-
sa(P) = e!(j2fa(za(P)))
(79)
whenever
A complex analytic canonical connection is correspondingly
p e Ua .
a collection of complex analytic mappings
Ua -> T2(n,M)
sa :
complex analytic coordinate covering 7 = (Ua)
(80)
whenever
O'(j2fa
15
for some
such that
(zs(P))) = s' (P) - p2(jifo a (za(P))) sa(P)
p e Ua fl US ; and such a connection is integrable if after passing
In
to a suitable refinement of the covering such that
fa
analytic mappings
there exist nonsingular complex
sa(P) = 6,02fa(za(p)))
(81)
Finally a one-dimensional complex analytic projective connection is a collection of complex analytic mappings
(82)
whenever
Ox(j3fa
sa :
Ua -> T3(1,¢)
a(za(P))) sa(P)
(zi(p))) = S*(p) -
D e Ua fl u,
and all these are necessarily integrable, [20].
,
is interesting to note incidentally that if affine connection then connection and conversely if
s
s"
'
a
= Sts
such that
sa = (I - R) sa
a
(sa)
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 aaalYtic affine connection.
sa = sa B sa
s'
a
is a
is a complex
The analogue of Theorem 2 holds for all these
Other pseudogroup structures as well, with formally the same argument.
Now using Theorem 2 the problem of investigating the existence
_up-
and classification of obese 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 oseudogroup structure or 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 forma]. 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. a preliminary it is convenient first to review some of the relevant auxiliar machinery, in order to establish notation and terminology. Returning therefore to the complex manifold analytic coordinate covering
= tU }
fa O ,
can be associated the nonsingular r_ x r. matrix _
'ifa-3
having entries
afa i
vita complex
having local coordinates
Za : Ua -> 11a and coordinate transitions
I`a6 (p)
M
`_ .a =
to each point p E Ua
-61-
This defines a complex analytic napping
with the obvious notation.
Tai
and if
Ua n U -> GL(n,a),
Ta o(p)- T131Y(p) =
clearly satisfy
(p)
then these mappings
; therefore these mappings (Ta
-Y
describe a complex analytic vector bundle This bundle
M.
n UY
p e Ua n U
T
of rank n
T
on the manifold
is called the complex analytic tangent bundle to
M,
For any
and is evidently independent of the choice of coordinate covering.
cmplex analytic group homomorphism
p
GL(n,.) -> GL(N,Q)
:
the composite mappings p o Ta : Ua n US -> GL(N,M) p(T)
complex analytic vector bundle
of rank
N
p(T)
on the manifold
M.
sections of the bundles
The sheaves of germs of and
then describe a
wil be denoted by 3 (T)
and
V ;p(T)).
T
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
14
= {Ua}
sections
v e Cq(L',(p(T)))
a q-cochain
a
a q
o
r:
Hq(M, 5L(p(T))). In terms of a given covering
r(Ua n ... n Ua , q 0
SPping 8 : Cq(21;, _-(p(,))) -> Cq+1( 'i, (8a)a ...a
q+l
o
enever
(p(T))),
and the onboundary
(P(T))) is defined by
(p) (p) = Ec+o(-1)j Qa ...aJ-la o 3+l" ' aq+l
p e Ua n... n o
consists of a collection of
.
U",
c+l
.
The cohomology groups for the covering
Z'`
by
can then be defined for e > 0
(5!(p(T)))
(p(T))) =
O'(p(T)))
where the space of a-cocycles is defined by
Zq(f, (p(T))) _ {Q E
01
Q = 0?
(p(T)))
and the space of q-coboundaries is defined by
if q>0
SCq(7, c (p(T)))
B°(V!,
0
and the cohomology groups
Hq(M,
limits of the cohomology groups coverings p(T)
V?,.
of M.
can be defined as the direct
9_(p(,)))
over the directed set of
10(Z2, 6(p(1)))
Now over any coordinate neighborhood Ua the bundle
is naturally a trivial bundle, so a section
v e i(Ux,
-(p(r)))
can
IN
naturally be identified with a complex analyJ c mapping
1Ja
a
can be identified (p(T)1) E P(Ua n...n ua , q 0 q with a complex analytic mapping from the intersection Ua n...n ua into
Similarly a section
o
q
o
MN ;
but there are
q+l
depending on which of the
different ways of making this identification, q+l
coordinate neighborhoods
is ..., Ua 0 c_ Henceforth the
Ua ,
chosen to describe the trivialization of the bundle
p(r).
trivialization over the last coordinate neighborhood
U.
will always be u:se=
q
here.
With this systematic convention a q-cochain
a e Cq(j , (-P(T)))
can
then naturally be identified with a collection of complex analytic mapaings
Sa ...a c 0
:
Ua 0
n.
n Ua -> q
iN
-63-
has the form
g01 the coboundary mapping
(es)a
(85)
o
...a q+1
j (p) = Eq (-1)j Sa ...a
o
+(-1)q+1 P(ra
q+l aq pEU n ao
whenever
(86)
(87)
.. n uq+1
.
a
J-1
(p))
sa ...a (P) 0
q
In particular
(8s) oal(P) = sa1(P)-P(Talao(p))' Sao(p) for p E Uaon Ual , (8s)acala2(P) = sala2(p) - Saoa2(p)+P(Ta2al(p))' sgal(p)
for p6 Uao nUal nUa2 Now whenever
p C
as
n u
al
n u a2
.
it follows that
j2faoa2(za2(o)) = j2faoa1 (za1(p))'1 fala2(za2(7D)),
2
where
(p)
j+1...aq+l
sa p
: a n US -> T2(n,a)
hence using
that
are the complex analytic mappings defined by
saE (p) = e(j?fad (z13 (p))) that is to say, recalling (87), the mappings or
ft (sa B} E
z1(G[, 0 (p(-r)))
.
{sa3 I
describe a cocycle
Then, recalling (86), the defining equation
(77) merely asserts that a complex analytic affine connection is a cochair_ to - {sa} E C°(vi, C-(P(T)))
such that
RC
0
= Q.
Thus the condition that
-64-
there exists a complex analytic affine connection is just that the cohomology class
Q
is zero in H '(M, G(P(s))) ; and if there exists at least one
complex analytic connection then the difference between any two such connections is a cocycle in
Z°(27, 6-(p(1))) = Ho(M, @-(p(T))).
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
v = {g(j2f00 ))
Ht(M, (9-'p(T)))
;
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
Ho(M, c- p(T))).
The corresponding assertions of course also hold for the other pseudogroups, merely replacing operator and representation.
e
and
p
by the appropriate differential
n > 1
Thus if
there exists a complex analyti^
pivj ective connection precisely wher. the cohomology class is trivial in
Hl(M, u?(p"(T))),
while if
o" "
there exists a complex
n = 1
analytic projective connection precisely when the cohomology class c* = {g*(j3fa
)}
is trivial in
,
H1(M, 0'(p*(T)))
and if there exists cne
;
such connection then the set of all such can be put into one-to-one correspondence with the vector space
H°(M, i(p*(T)))
if
n = 1.
Hs,(M,
S-(p++(T)))
Similarly if
if
n > i
n > 1
or the vector space
there exists a complex
analytic canonical connection precisely when the cohomology class
{e'(j2fa
is trivial in
He(M, i1(p'(T)))
;
and if there exists -Ine
such connection then the set of all such can be put into one-to-one corresp:-n= with the vector space
Ho(M, 6- (p'(T)))
.
-65-
It may be worthwhile to point out here another interpretation of these connections.
On any complex manifold M
associated to the complex analytic tangent bundle analytic fibre bundle
T(1)
with fibre
transformations (83) as those of higher order analogues jets of order T(k)
bundle Gk(n,@)
k
T(k)
r
.
GL(n,T)
the principal bundle is the complex
T
and with the same coordinate
It is possible to introduce the
of the bundle
T(l),
simply by considering
of the coordinate transitions of the manifold
M.
Thus the
is the complex analytic principal bundle over M with fibre
and with coordinates transformations defined analogously to (83) by
Tak (p) = Jkf"P(za(p)) for any point
p a Ua 1 U
.
It is easy to see that a complex analytic affine
connection really amounts to a reduction of the structure group of the bundle T(2
to the subgroup
loc 6 C
while an integrable complex analytic
affine connection amounts cc 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 or differentiable manifolds, a complex analytic affine connection may well be called as almost-affine structure on the complex manifold M
.
The corresponding
assertions and terminology can also be introduced for the ether pseudogrcuro structures.
Finally it is perhaps useful to include here a few remarks about the behavior of these structures under automorphisms of the complex manifold 1[ .
If T : M -> M
is a
complex analytic coordinate covering, with
is a complex analytic hemecmorphism and
= {Ua}
no-a.-ion as before, then
-66-
T
can be represented by the coordinate mappings
s a : za( ari T -) -> z13 (UU,
7%)
,
where
z
TF here
TP a
V0
in
In
analytic affine connection translate of the connection sT
o T ° za
is a complex analytic mapping between subsets of the open sub-
domains a and
tion
-1
.
To any such automorphism
s = (s s
by
T
and any complex
on M there can be associated the the complex analytic affine connec-
T ,
sa
defined by coordinate functions
where
sa(p) = It is a straightforward consequence of the
p e Ua I) T-lU .
properties of affine connections and of the operator independent of that
Ua ,
hence is well defined throughout
Ua ,
sT(p
is
and further
is a) does define another complex analytic affine connection on M.
Similarly it can easily be verified that It also follows readily that T
9 that
of M ,
is integrable whenever
s
is.
for any two automorphisms
S
and
sT
(sS)T = sST
so that there is thus defined a representation of the group of
complex analytic automorphisms of M
as a group of operators on the set of
complex analytic affine connections on M ;
the latter is even a linear
representation for any identification of the set of all such connections with the complex vector space representation.
F(M, 0-(p(-r))),
indeed is the obvicus linear
These observations depend only on the formal properties of
affine connections and of the operator
9,
so carry over imediately to the
-67-
other connections and the corresponding partial differential operators. If
r
is a group of complex analytic automorphisms of M then
the complex analytic affine connection sT = s
for all elements
T e r.
s
r
if
It is readily verified that any r-invariant
complex analytic affine connection on M connection on the quotient space
is invariant under
M/F ,
induces a complex analytic affine whenever
-
acts as a properly
discontinuous group of automorphisms having no fixed points so that
M/1_'
is also a complex analytic manifold; and any complex analytic affine connection
on M/r
can conversely be viewed as a F-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 ,
hence must induce a unique complex analytic affine connection
on any quotient space
M,/J'
.
Since integrability is preserved the same
results hcld for intesrable 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. As 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 mind. If
is
a
} = {s
ajlj2
is a complex analytic affine connection then recalling
}
(24) and (44) the defining equation (77) can be written out explicitly in the form
(89)
azakl
2
azfl
z
azak k0i
ak
_
= sf3jlj2 1
azak2
k
i
saklk2
czP. 1
2
2
and that is the complex analytic analogue of the familiar condition that the components
is 1
12
}
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
t 1
qjlj2
= sl
ajlj2
- sl
are the
4201
components of a tensor called the torsion tensor of the affine connection. Furthermore the complex analytic affine connection
{s
a
is,
aj1J2
}
is
integrable precisely when, after passing to a refinement of the covering if necessary, there are nonsingular complex analytic mappings such that such that
sa(p) = 6(Ea(za(p))) Ei
ajlj2
= £
E i
k ak s aj12
where
Ea(za) = j2fa(za),
fa
:
Va ->
,n
or recalling (44)
; by Lemma 2 the necessary and sufficient
-69-
fa
condition that there exist such mappings
sajlj2j3
(90)
czaj3 sajlj2 +
is symmetric in the indices
is that the expression
r sak j3 kS. Ek
and that is evidently equivalent
jl, j2, j3,
to the vanishing of the expressions 1
1
1
rajlj2j3 = saj3jlj2 - saj2jlj3
(91)
However the expressions
{raJ13203 }
defined by (90) and (91) 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.
If the complex manifold
M has a complex affine structure then there is a coordinate covering
M _ {%}
for which the coordinate transitions
fa
0
are complex affine
mappings; and in terms of this coordinate covering the coordinate transformations sa S = jlfa 6
defining the complex analytic tangent bundle are constants.
The coordinate transformations
{Ta s}
can be viewed as describing a flat
complex vector bundle, a fibre bundle having as structure group the group
-70-
GL(n,T)
with the discrete topology; and in these terms, if a complex manifold
M 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 bundle, and if this equivalence is exhibited by 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
T
is
analytically equivalent to a flat vector bundle precisely when, after passing to ,a refinement of the covering if necessary, there are complex analytic mappings
ha :
Ua -> GL(n,C.)
ha to h-1
such that
13
are constants in
13
Ua n U. ; and using the exterior differential operator
d
that is equivalent
to the condition that
0 = d(ha Taoh-l) = dha Ta h13 l + ha dTa
0
hPl - ha Ta 0
h13
hence to the condition that -1
(92)
where (93)
T_
6 dTa S = es
_
-1
Ta B ea Ta O
dh0 h13l
-71-
to
Here
'ra
of degree 1 in
is an
n x n and
Ua fl u, ,
matrix of complex analytic differential forms
ea
differential forms of degree 1 in
n x n
is an
matrix of complex analytic
Writing the entries
Ua .
e aj of the
matrix ea out in the form
ea . (za) = Ek sa k j ('a) dza k
(94)
sakj(za)
for some complex analytic functions in
Ua
of the local coordinate
(92) easily reduces to the form (89); so the coefficients
za
sakj
are
complex analytic analogues of the Christoffel symbols of a general or nonIf there are complex analytic mappings
symmetric affine connection.
ha such
that (93) holds then 0 = d d ha = d(haea) = ha dea + d ha A ea = ha(d) a+ ea ^ ea) so that
dea + ea ^ ea = 0 ; and conversely it is easy to see by arguing as in
the proof of Lemma 2 that whenever complex analytic mappings be omitted here.
ha
dea + ea - ea = 0
then there exist
such that (93) holds, although the details will
Introducing the
differential forms of degree 2 in
n x n Ua
matrix Oa of complex analytic
defined by
ea = dea + ea ^ ea ,
(95)
and writing the entries O. of this matrix out in the form
Oaj(za) =
(96)
F,,, raj kt(za) dzak - dzat 2
for some complex analytic functions za
in
ra
jk
(z) a
of the local coordinate
Ua , it is a simple matter to verify that the functions
can be expressed in terms of the functions
sa k(za)
rai j kt, (za)
as in (90) and (91).
J
Thus the integrability of the equations (93) reduces to the vanishing of the
-72-
coefficents
raj k ,
so that the complex analytic tangent bundle to M
is analytically equivalent to a flat complex vector bundle precisely when there exists the complex analytic analogue of an affine connection having zero curvature; and each such connection describes a reduction of the complex tangent bundle to a flat complex vector bundle. Several properties of complex affine manifolds are really consequences merely of the analytic equivalence of the complex analytic tangent bundle to a flat complex vector bundle; so the corresponding weaker properties of complex manifolds admitting complex analytic affine connections do not The first such property to
require the symmetry of the Christoffel symbols.
be considered is the existence of a complex analytic covariant derivative.
Recall that a complex analytic tensor field over M, p
and covariant of order
q,
contravariant of order
is a section
f e I'(M, L90(T(P) F *(q)) where bundle
is the complex analytic tangent bundle,
T
T
'r*
(defined by the coordinate transformations
defined by the coordinate transformations
T(p) = T
ri
... C T
,
(ra
(p
)
,
is the dual of the
{tT a-11) 13
when
T
and
factors)
such a section is thus described by its components
il...i p
111
faj which are complex analytic functions in the coordinate neighborhoods and in any intersection Ua fl U.
these components are related by
Ua
is
-7 3-
i ...iP
fajl..
q
_
1
k,
k ...k
i
1
tq
TOajq
Ta6k1... Taskp
azai 1
az a...
k,
ZOkp
If the coordinate transformations
TCO
f
az
azPtl
kl...k
6tl...tq
azaj 1
...
zajq
are constants then upon differentiating
both sides of (97) with respect to the local coordinates and recalling that a/azaj =
T t
a/3zs,
a
it follows that the derivatives
il...i
il...ip
)ail... jqj
fa jl... jq
are the components of a complex analytic tensor field over of order
p
and covariant of order
q+l ;
M, contravariant
thus ordinary differentiation
induces a sheaf homomorphism
(98)
0 :
&(T(P)
T*(q))
_>
(qfT(P)
*(q+l))
&
locally and a linear mapping
(99)
globally.
o : P(M, 9(T(P) ®
T*(q))) I'(M,
pfT(P)
(S
T*(q+l)
More generally a linear differential operator (98) or (99) can be
introduced on any complex manifold which admits a (not necessarily symmetric) complex analytic affine connection derivative [27], defined by
{s
{si
ajlj2
}
;
that is the covariant
-74-
11...i p
f
(100)
= j
f
a p
1
p
iv
£
il...k...ip
k sajk fail...]
v=1
q
il...ip E
Cl
v=1
sk
f
k aj1...k...jq
ajjy
It is a straightforward matter to verify that this differential operator does satisfy (98) and (99), using the defining equation (89) of an affine connection. It should be noted that if
{sajk}
are any functions, not necessarily complex
analytic functions, which satisfy (89) then the expression (100) formally satisfies the transformation equations (97), although the components (100) are of course not necessarily complex analytic functions; the principal property of a complex analytic affine connection in a sense is that the associated covariant derivative preserves complex analytic tensors, hence defines a sheaf homomorphism (98) and linear mapping (99).
Indeed it is not difficult to verify
that any first order linear differential operator 0 which satisfies (98) can be put in the form (100), where
{sajk}
are the Christoffel symbols of a
(not necessarily symmetric) complex analytic affine connection; that is of course not true for the global property (99), since there may be too few complex analytic tensor fields.
For this reason the property (98), or in the differ-
entiable 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, [27], [35]. sisting of
Choose any
n x n matrices 8a con-
differential forms of degree 1 in the coordinate neighborhoods
C°°
Ua and satisfying the condition (92) in each intersection
Uafl u, ; that
there exist some such matrices follows from a familiar argument using partitions of unity.
Then introduce the
n X n matrices Ga consisting of
ential forms of degree 2 in the coordinate neighborhoods
of the chosen matrices 0a
as in (95).
C°°
differ-
Ua defined in terms
It follows readily upon taking the
exterior derivative of (92) and then using (92) again to simplify the result
that ea = Tao e T9a
(101)
in each intersection
Ua fl U. .
Next if
x
is an indeterminate and
X
is
an n x n matrix of indeterminates note that there is an expansion of the form
(102)
where
det(I -
or(X)
of the matrix
i X) = 1 + xo1(X) + ... + xn on(X)
is a homogeneous polynomial function of degree X ; and
ar(C X C-1) = or(X)
for any matrix
C E GL(n,M). ar(Oa)
Therefore upon recalling (101) it follows that the expression globally defined
C°°
differential form of degree
2r
in the entries
r
on the manifold
Following Weil it can be shown that these differential forms
is a M.
or(%) are
closed, and that up to exact differential forms they are independent of the choice of the connection forms
the differential form
or(Ca)
0a ,
[27], [35]
;
thus by de Rham's theorem
determines a cohomology class
Cr
which is also independent of the choice of the connection forms
E H2r(M,(t)I
0
a
.
These
-76-
cohomology classes
Cr
or
are the Chern classes of the manifold M,
alternatively of the complex tangent bundle
over
r
M.
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
fa
in the coordinate neighborhoods TO a
fa = t T
There exist positive definite
in each intersection
C00
T
,
a
Hermitian
such that
Ua
, by using again the
Ua n U
6 a f0
familiar argument with partitions of unity; and since the matrices
Ta
are 13
complex analytic functions then the matrices
sa =
fat
using these functions
afa 8a
0 , and hence as is easily verified
at T
satisfy (92) in the intersections
Ua fl U..
Then
it follows directly that (95) takes the simpler
form
"C' = -fal
afa d
fal
afa =
a0a
so that the differential forms making up the matrix degree 2 but of bidegree (1,1) or(ca)
;
ca
are not just of
and correspondingly the differential form
representing the Chern class
Cr
is of bidegree (r,r).
follows that no matter what differential forms
sa
It therefore
were originally chosen the
Chern class is represented by the component of type (r,r) of the differential form
0, r(Ca).
Ta 6 d Ta
Now since the matrices
T.
are holomorphic the matrices 0
consist of differential forms of bidegree (1,0)
there is no loss of generality in assuming that the matrices of differential forms of bidegree (1,0).
;
and hence in (92)
ea
also consist
Then in constructing the Chern
-77-
classes it is only necessary to consider the components of bidegree (1,1) in the matrices
pa ; so in place of (95) it suffices merely to take the
simpler expressions
ea = a ea
(103)
and the Chern classes are represented by the differential forms
9r(pa)
Finally it should be noted that the Chern classes can also be introduced as integral cohomology classes, so the differential forms
ar(Oa)
or
Or (pa)
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.
As 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
ea then the matrix ca defined by (103) is identically zero, and
hence the Chern classes
or(ca)
are trivial.
For reference then, the results
described here can be summarized as follows. Theorem 3.
If M
is a complex analytic manifold which admits a
complex analytic (nonsymmetric) affine connection then the Chern classes of M
-78-
are all trivial; and for each complex analytic affine connection the associated covariant derivative (100) determines a sheaf homomorphism
O 7(P) ® ¶*(q)) ->
(98)
(
.(P)
0
(q+l))
and a complex linear mapping
(99)
o : r(M,
for any indices
0-(,T(P)
p, q
.
T*(q)))
-> P(M,
(9fT(P)
C1+1))) (S T*(
-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 Here too the complex analytic projective connections
can be found in [28].
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 case
complex analytic projective connections were discussed in [20],
n = 1
and it was shown there that all such connections are necessarily integrable. n > 1
For the case
discussed in [13].
complex analytic projective connections have also been In this case a complex analytic projective connection is
{s" } _ {si
described by complex analytic functions neighborhoods
Ua
such that
Ek s
ak j
= 0
in the coordinate
ajl02 }
a
for all indices
j
;
and recalling
(24), (44), (48), and (63), the defining equation (78) can be written out explicitly in the form
az i £
k
a
2
zak
_ 5i
azak
iEi
(104)
a
01
a6 j2 oz
k
_ 8i
o j2 a6j1
akl
sDjlj2 - k azak saklk2 "FP 01 where
tion
= {sa}
and a 0j
az
ak2
azOj2
= (n+l)-la log oa
/az,,j.
The connec-
6
is integrable precisely when, after passing to a refinement of the
covering if necessary, there are nonsingular complex analytic mappings
-8o-
f
Va -> a:n
such that
where
sa(p) = 0"(Ea(za(p)))
8 (za) = j2fa(za)
and recalling (44), (48), and (63), this condition can be written out more explicitly in the form
lj2 = Ek Ea
(105)
where Aa = det
k Sa lj2
1'9a = (n+l)-
to verify that if a = j2fa case that
as
+ Eaj2 vajl
+
-ajI vaj2
necessarily a constant, and hence
analytic functions
{sa}
{07}.
It is easy
satisfies (105) in the slightly more general
is any prescribed function then
thus the connection
j1Qa =
log Aa , and
Ea = j2fa
oa - (n+l)-1 log 6a
is
satifies the desired conditions;
is integrable precisely when there exist some complex such that the equations (105) are integrable.
(T a
for any given functions
Qa
However
is follows readily from Lemma 2 with
kaj1j 2 = saj1j2 + sj 1 saj 2 + sj 2
Qaj 1
that the integrability condition for equations (105) is just that the expressions
(106)
1j2j3 = za s C4j
3
are symmetric in the indices
(107)
lj2
+ Ek sak j3
j1, j2, j3
_
xaijlj2 _ Ek ak
k
sajlj2
,
salj2
xajli
where
+ oajlj2
+ Qajl vaj2
Now if the expressions (106) are symmetric then k k Ek sajlj2k = Ek sajlkj2
from which it follows easily that
+
-81-
xajlj2 = (n-l)-1( -k
(108)
{sa}
Thus the connection
sak j1
satj2
k rk a
-
aJlj2)
is integrable precisely when the following two
conditions hold: (i) the expressions (106) are symmetric in the indices j1, j2, j3
,
where the expressions
xa.
are defined by (108); and (ii)
.
1 2
there exist functions a
satisfying the equations (107).
Here (i) is in
turn equivalent to the vanishing of the Weyl curvature tensor
1 1 ajlj2j3 = sajlj3j2 -
(109)
1
c4jlj2j3
it is a straightforward calculation to verify that (109) is in general a tensor Next equation (107) can be linearized by of the appropriate type, [11]. -v setting 1a = e a , hence reducing that equation to the form
Taj j
(110)
=£kT
12
ak
s
k
.
12
+ Ta xaj j
l2
here too the integrability conditions can be j2Ta = {Taj, Taj j 1 2 Indeed if there exists a obtained by arguing as in the proof of Lemma 2. where
function
Ta
satisfying (110) then by repeated differentiation and simplifica-
tion it follows inductively that the function
(m)
v > 2, where
aj1j2
s 1
are the given connection coefficients,
a1jare defined by (108), and inductively for 2
saj1 .j v+1
=
k
s
satisfies
k k Ta k sail...jv + Ta xa,71...Jv
Tail...iv
for any index
T.
i
v > 2
i
k
a k .
v+1
saj 1...
jy
+ az
i
saj ...j
X7v+1
1
+ Sj
j v+1 xaj 1. ..v
-82-
a
(113)
x ail- .'v+l =
azajv+lx ajl.jv
k
k akjv+l
+
SCOI...jv
and as usual the integrability conditions for the equation (110) are just that all the expressions (112) and (113) are symmetric in the indices Now by using (112) and (113) twice it follows easily that all
jl'".'jv+1 '
these expressions are symmetric if it is merely assumed that
.
and
saj 1j 2 3
x ajlj2j3
are symmetric; and the condition that
s 1
ajlj2j3
be symmetric is
The symmetry of the
just that the Weyl curvature tensor (109) vanish.
is a more interesting condition, involving when written
expression j
l 2 3
out explicitly the second derivatives of the connection coefficients since _ (114)
(n-1)
a
xjlj2j3
,
k
a
satjl sakj2
azaj3
k
m ajlj2
t
k
m
sakm 5c
k
azak sajlj2
3q sajij2
azaP,
Sakj3
It is easy to verify though that
k
sWlj2j 3
n-2
n-1
a
2
azaj azak
s
k
Cd1i2
3
_ k
+ Eq azak ('OV3 sWlj2 + n-l
(saV1 sakj2 )
8zaj 3
and if
n > 2
then this can be used to eliminate the second derivatives of
the connection coefficients from (114), after which a simple calculation shows that the symmetry of the expressions
',
already implies the symmetry 1j2j 3
-83-
of the expressions
Finally if 1j 2 j 3 matter to verify that the expressions saj recalling that
xaj
sk
it is a straightforward
are necessarily symmetric, 1 2 j3 and writing out these expressions separately;
= 0
k
E k
n = 2
j
so in that case the symmetry of the expression
is sufficient for
xaj j2j
integrability.
{s
In summary therefore, the connection
precisely when: (i) for
n = 2
the expressions
,
3
is integrable defined by (114)
x
001 j 2 j 3
are symmetric, or (ii) for
the expressions
n > 2,
defined by
saj j l23 .
(106) and (108) are symmetric. To express the integrability in another but equivalent way, for
n = 2
introduce the expressions
(115) ai10 203
and for
n > 2
a`jl 203
introduce the expressions .i
(115')
aoli312
= s
ajlj2j3
i
- s
ajlj2j3
i
ajlj3j2
It is a straightforward matter to verify that the expression (lil) are the components of a complex analytic tensor field of covariant order 3 and skewsymmetric in the last two indices, although this does not seem to have been much noted in the literature; indeed it is quite easy to see that
1j2
=
xak1k2
2
1
Ek asksaj1j 2
+ a01 a6j 2
and using this the assertion follows without undue difficul'.y.
- Qasj1j2
On the other
hand the expressions (11`') are the components of the complex analytic analogue of the Weyl curvature tensor [23]. Presisely when: (i) for (ii) for
n > 2
n = 2
the tensor
Thus the connection
the tensor
V
{s'a}
is integrable
defined by (11°) is zero, or
W defined by (11F1) is zero.
-84-
There is an analogue for projective connections of the covariant derivative for affine connections, although really only a rather partial analogue.
On a complex affine manifold the complex analytic tangent bundle
is flat, hence the partial derivatives of any complex analytic tensor
field
again form a complex analytic tensor field; more generally this construction actually only requires the presence of a complex analytic affine connection, in the sense that using such a connection it is possible to modify the partial derivatives by constant terms to provide first order linear differential operators transforming complex analytic tensor fields into complex analytic tensor fields.
On a one-dimensional complex projective manifold Bol av/az
observed [4] that the differential operators
transform some particular
complex analytic tensor fields into other complex analytic tensor fields, and Eichler used this observation [10] to introduce what are now known as the Eichler cohomology groups on such manifolds; this was actually carried out in the context of automorphic functions, but the extension to any complex projective structures is trivial and the role of the projective connection was indicated in [21].
On an n-dimensional complex projective manifold for
n > 1
the general problem of determining all such operators will be left until later, but the particular operators most closely related to the affine covariant derivatives will be considered here.
of dimension
n > 1
there is on
projective connection
M
If M
is a complex projective manifold
the canonical trivial complex analytic
sa a 0 ; hence if
complex analytic canonical connection then
sa
is any
sa = sa
CW is a
necessarily complex analytic complex affine connection. (28), the connection
sa
is described by coefficients
but not necessarily C00
but not
Recalling (27) and
i s'ajlj2
(116)
(bl x
1 nn+l
=
jl
aj2
+ bl x j2 ail
where
k xaj = £k sak
.
and recalling (63) and (80), the condition that is just that the coefficients
(117)
3z
in each intersection any
C00
j
k
log a XBj -
xak Eat j
where na ,3
=
det4a
functions in the coordinate neighborhoods
(117) in the intersections
is a canonical connection
satisfy
xaj
Ua fl u, ,
sa
J}.
S
Ua
Thus if
are
and they satisfy
then (116) is actually a
Ua fl U,
xaj
necessarily complex analytic complex affine connection on M .
C'
but not
In terms of
this connection the covariant derivative (100) takes the simpler form
1l ...1
i ...i
(118)
(Of)ail* ..jPj _ q
jP +
fai
Q
aj
il
p- 4 n+
xaJ fal
1. .. q
...1
JP 1... q
il...k...ip
1
P + 7+_1
1
_
n+l
£v=1
k xa k
bj
fajl...Jq
il...iP £ q v=l xaj, faj1...j...j 4
for any tensor field
f.
Now this linear differential operator does not
generally transform complex analytic tensor fields into complex analytic tensor fields, since the coefficients
xaj
are not necessarily complex analytic
functions; but because of the relatively simple form (118) this operator does transform special classes of complex analytic tensor fields into other classes of complex analytic tensor fields.
This is of course not really surprising,
-86-
since after all the ordinary exterior derivatives are restrictions of the general covariant derivative to special classes of tensors, the skewsymmetric tensors, for which all terms in the covariant derivative involving the coefficients of the affine connection vanish; the point is that there are more such combinations on complex projective manifolds than on arbitrary complex manifolds.
To examine this situation, which in general can get somewhat complicated, let it suffice merely to consider the case of a complex projective
manifold M f
,
of complex dimension
contravariant of order
p
n = 2
.
A complex analytic tensor field
and covariant of order
analytic section of the complex analytic vector bundle the complex analytic tangent bundle and
q , is a complex
p(T)
,
p: GL(n,M) -> GL(N,C)
where
is
T
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
is
p
not generally irreducible, but is at least equivalent to the direct sum of a number of irreducible representations; the bundle
p(T)
splits accordingly
into a direct stmt of complex analytic vector bundles, and the tensor fields
split into direct suns 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
n = 2
Now
the irreducible analytic linear representations of the general
linear group are all of the form A (0,-q)
in the notation of [47]; here A
for
q > 0
and
p
arbitrary,
is the scalar representation given by the
determinant, and (0,-q) is the dual of the symmetric representation
(q,0) of
-87-
q.
degree
where
Letting
TQ
denote the complex analytic vector bundle
a complex analytic tensor field
p = Ap(o,-q),
,thus described by coefficients Ua
f e r(M, &-{TQ)
is
which are complex analytic faj 1
functions in
p(T)
j
q
and are symmetric in the indices
jl,...,jq
and which
satisfy
k
(119)
k
1
fail... jq = was Lk fPk1.kq T6ajl ... where Aa o = det{Ta o
in Ua p Uo ,
}
or equivalently of course
;
12...12 fail, ..jq (120)
q
T13ajq
if p>0 ,
fail...jq
j1...jq
if p<0,
11...12 where
p
fa.
.
fai
and
1... 1-2pj 1...
are complex analytic tensor fields j
q
of the appropriate types and are skew-symmetric in the and symmetric in the
ill...il2pl
q
indices i1 ,jq
pairs of indices
IpI
Using (120) it
.
is easy to verify that the covariant derivative (118) of such a tensor field f
takes the form
121)
Vfajl.jgj zaj
(
fail.
This covariant derivative vector bundle
p'('r),
PT
where
3
is a p'
jq + (P-
) xaj fajl...jq
Egl v x ajv faj1... j... jq C00
section of the complex analytic
is the representation
Up to equivalence of representations,
p' = p (S (0,-l).
p ® (0,-1) = dP(0,-q) ® (0,-l) _
a AP-q-1 (q) 0 (1) ; and since the representation
(q) ® (1)
can be decomposed
-88-
into the direct sun of irreducible representations as in [36], it follows that
(q) ® (1) _ (q+l) ED (q,l)
p' = Ap-q-1(q+l) (B AP-q-1(q,l) = §p(0,-q-1) e
AP-1(0,1-q).
Therefore Vf can be decomposed into tensors 0'f E r(M,C°°(tpq+l), r(M,C00(rp-1
and '7" f E
q-1
)
;
and it is apparent that these constituents can
be written in the forms
(122)
0'fail.jq+l
=
(123)
7" f ajl
_
jq-1
Eq+l17f v=1
ajl...jv-1
jv+l...jq+l jv
. - Pf aJl...jq-112
'7f
or using (121) in the equivalent forms
7'f
(124)
ajl...jq+l
8
= 2q+l v=1
+ (P-
(125)
D" fail, ..jq-1
f
ajl...jv-l
f
2q) 2q+1 x v=1
3
ajv
ail'
jv+l*' jq+1
8
3
=
jv+l...jq+l
azal
f ail...jq-i2
a fail...jq-11
q-1
+ (P- 3 )(xaifaj1...jq-12 - xa 2
fail* ..jq-'1)
.
A useful alternative notation exhibits the parallelism between these differentia operators more closely.
(126)
f
av=
fa1.. . 12. .
Setting
.2
where
and recalling that the tensors
f
v
of the indices are
aj1...j4
1
and
0 < V < a
,
are symmetric, the expressions (12L)
-89-
and (125) can be rewritten
(127)
N7'fay= (1 + (P- 3 ) xal)
-
+(
(128)
p"fav =
a2
a2
\a
(q+l-v) fav, 0 < v < q+l
+ (p- 3 ) xa2)
+ ( p- q31) xal,
Wal
yfav-1
fav
+ (p-q31) xa2)
In either form it is evident that the terms involving V'
vanish whenever
3p = 2q
,
hence that
'7'
0
fav+1 xaj
in the operator
transforms complex analytic
tensor fields into complex analytic tensor fields in those cases; and that the terms involving
xaj
in the operator
17"
vanish whenever
3p = q-1
These observations can be restated as follows.
Theorem 4.
If M
is a complex projective manifold of dimension 2
then the first order linear differential operators
'7, 0"
described by
(124) and (129) or by (127) and (128) determine sheaf homomorphisms
'7' 17 "
:
O-(tpq) -> &-(Tpq+l)
for 3p = 2q > 0
B'(Tpq) > Cr(TPgll)
for
and complex linear mappings
3P = q-1 > 0
-90-
17'
0
:
r'(M, 0(Tpq)) -> r(M, L9{Tpq+l))
for 3P = 2q > 0
r(M, 0(TPq)) > r(M, B{rPgll))
for
It should be noted that when
,
3P = q-1 > 0
p = q = 0
the operator 7'
can be
identified with the exterior derivative taking functions to 1-forms, and that
when
p = q-1 = 0
the operator
can be identified with the exterior
'7"
derivative taking 1-forms to 2-forms.
The iterates of the operators
17', V"
can be used to construct still further interesting tensor fields, although usually not complex analytic tensor fields and sometimes involving fewer
derivatives than might be expected; for instance the commutator 0' 0" -
p"p'
does not involve any differentiation at all.
that if the line bundle A(T)
can be written as a power of another complex then it is possible to introduce
analytic line bundle, say A(T) = or , complex analytic tensor fields number such that and the operators
pr
It should also be mentioned
f e r(M,t9-("p q)
where
p
is any rational
is integral, defining them by (119) with pa P = 0-a p r;
0, '7', 17"
can also be introduced for such tensor fields,
yielding further differential operators on complex analytic tensor fields. most interesting case of course is that in which transitions are of the form (74) with
n = 2
p = p'/3 ; if the coordinate
then by (75)
AaB = cab (baB+ £j baBizPj) for some constants
ca B ,
so that
Al/3
The
3
has a simple natural form if well
defined.
If it is merely assumed that M complex analytic projective connection
s'a
is a complex manifold with a and if
s a
is any
C' but not
-91-
necessarily complex analytic canonical connection then C'
a
sa = sa + s'a'
is
but not necessarily complex analytic complex affine connection on
M.
The associated covariant derivative (100) can then be decomposed as the sum of the expression (118) plus a linear combination of the components of the tensor
with coefficients
f
sit
;
so whenever (118) preserves complex
analytic tensor fields so does the covariant derivative
17
.
In particular
therefore Theorem 4 extends immediately to complex manifolds of dimension 2 which have complex projective connections
with the differential
sa ,
operators
f
(129)
q+l V=l
ail...Jq+l
aza
ajv fail*
- £ q+l 4,v=1
sajµji, k
JV+l...Jq+l
V-1
fajl...(jµ,jt, omitted)...j q+1
µ #v
(130) '7 'if ajl...3q-1
a a1 faj1...Jq-12
q-1 Ek + £v_1
a2 faj1...Jq-1
(fajl.k...j
s
q-11
k
a2jv
-
fajl...k...j412
sit k
-
Turning next to topological properties, the Chern classes of complex projective manifolds, or more generally of complex manifolds which admit complex analytic projective connections, have a very simple form.
If
complex analytic projective connection on a complex manifold M fly
C-
sa
is a
and
sa
is
but not necessarily complex analytic canonical connection then again
so,= s'a+
s'a
connection on M
is a .
C°°
but not necessarily complex analytic complex affine
Writing
s'
a
in the form (116), the matrix differential
-92-
form (94) is given by
eaj(za)
(131)
n+1
(Sj (Pa(za)+xaj(za)dzai) + Ek sakj (za) dzak
where
Wa(za) = Ek xa k(za)dza k
(132)
and since
is complex analytic the matrix differential form (103) is
s'
given by
(133)
eaj(za) = n+l (Sj
(Pa(za) + a xaj(za) A dzai).
It should be noted that as a consequence of (117) the differential forms c(Pa(za)
describe a global differential form of bidegree (1,1) on the manifold
M ; the interpretation of this differential form will appear shortly in the
The Chern classes of M
discussion of canonical connections.
can then be
represented by the differential forms
(134)
cr = vr(®a)
where the functions
v
for
1 < r < n
are defined by (102).
More explicitly, as is
r
evident from (102), for any
n x n
matrix
X = {x l} J
(135)
where
rr (X) _ (1 27r )r
sgn
(kl,...)kr)
kl
E 1
E n
jl...jr 1
rlt
1...
xjk 1
r
kr
jr)
is zero unless
(jl,...,jr)
is a permutation of
and is then the sign of the permutation.
into this explicit formula it follows readily that
Upon substituting (133)
-93-
cr
=(-
-
2rr
n+l)r
i
E
Y,
1
y=0
F(ktl,...,k
)
where
_ r-v
1k...k
1
F(k1,...,ky) =
EJv 1st j1...iV axjl^ dzkl ...^
&j
^ dzk v
v
v: a(xk1dzk1 ^...^ a(xkvdzkv)
=
note that
(a )
is symmetric in the indices
F(kl,...,kv)
k1,...,kv
trivially zero whenever two of these indices are equal.
and is
With the obvious
simplification then r
(2
n+1)
cr =
r
)(vr
(r
'
)
v£0
1
where
F(kl,...,ky)
E
k
r
denotes the usual binomial coefficient and
(
,
kn-l
E
denotes a
v) sumation over distinct indices only; and hence
r
i
=
cr
2
n+l
}
`2
n+1
!
n E F(kl,...)ky) k1,...kv=1
Er (n_r) v v=0
r
r
,P)r
(n-r) (a
v0
(2
n+l ) r
(nrl) (a , )r
In Particular
(136)
i
-0 (P
and with this observation the general formula for the Chern classes of M Can be rewritten as follows.
-94-
Theorem 5.
If M
is a complex manifold which admits a complex
analytic projective connection then the differential forms
the Chern classes of M
(137)
can be taken to satisfy
c,. = (-,I)
(11..y) c,
for
r = 1,...,n
Cr
representing
-95-
§9.
Complex analytic canonical connections The complex analytic canonical connections are in many ways the
As was already
simplest of the three classes of connections considered here.
essentially noted, a complex analytic canonical connection
aJ1J2 Ua where
can be written in the form (116) in each coordinate neighborhood the functions that
sa
Xaj =
£ k sa k k
are complex analytic in
Ua ; and the condition
is a canonical connection is equivalent to (117).
the complex analytic 1-form
CPa
Introducing
U. defined
in the coordinate neighborhood
by
'Pa(za) _ £k xak (za) dza
(132)
the condition (117) can be rewritten
d log pa B = cps - 9a
(138)
where
=
in Ua fl U.
thus a complex analytic canonical connection
P
can be described equivalently as a collection of complex analytic 1-forms in the coordinate neighborhoods Ua n u, .
Ua
`Pa
satisfying (138) in the intersections
Recalling (63) it follows readily that this connection is integrable
precisely when, after passing to a refinement of the coordinate covering if necessary, there exists a nonsingular complex analytic mapping
fa: Va ->
such that (139)
Ta = d log a
where
}
n,, = det{afa/az
.
c4j
Thus if this connection is integrable then the differential forms necessarily closed, that is, satisfy
d ¢a = 0 .
4a
are
on the other hand if the
n
-96-
differential forms
"a are closed then after passing to a refinement of the
covering if necessary there will exist functions
% = d ha
ha
in
U.
such that
and these functions must therefore be complex analytic functions.
;
Then after further refinement if necessary there will exist a complex analytic
ff
function
in
Va
such that
complex analytic mapping 1 fa '
afar za 1 = exp ha ; and introducing the
fa: Va -> an
2
fa
= z2 a '
...,
a =
defined by the coordinate functions
it is apparent that this mapping is non-
Zn a
singular and that a = det{afa/azaj} = exp ha , and the connection is integrable.
hence that
(a = d log a
Thus the complex analytic canonical
connection described by the differential forms
(P
a
is integrable precisely
when these differential forms are closed. As noted in the corollary to Theorem 2, there exists a complex
analytic canonical connection on a complex manifold M cohomology class
o = {e'(i2fa jO
in (35) the representation of degree 1, so that T*
p'(T)
p,
is trivial in
precisely when the
H1(M, &(p'(-r)).
Now as
is equivalent to the covariant representation
is analytically equivalent to the dual bundle
to the complex analytic tangent bundle to M ; hence the sheaf
can be identified with the sheaf 1-forms on
M.
Gl'0 of
CSL(P'(T))
germs of complex analytic
After making this identification the corollary to Theorem 2
can be restated as the assertion that there exists a complex analytic canonical
connection on M trivial in
precisely when the cohomology class
o = {d log A.
}
is
H1(M, &"0) ; and that is of course merely a restatement of (138)
The corollary further asserts that if there exists a complex analytic canonical connection then the set of all such connections is in one-to-one correspondence with the vector space -H0(M,
(90)
of global complex analytic 1-forms on M,
-97-
an assertion which is also an immediate consequence of (138).
Since the
integrability condition for complex analytic canonical connections is a linear condition the corollary can evidently be extended in this case. letting
6_'1,0
denote the subsheaf of
6Ll'O
consisting of closed
differential forms, or alternatively defining
0 -> 6 1,0
X1,0
>
Indeed
by the exact sequence
d> X2,0
it is evident that there exists an integrable complex analytic canonical
connection on M
precisely when the cohomology class
D = {d log a p}
is
H1(M, 6L" O) ; and that if there exists an integrable complex
trivial in
analytic canonical connection then the set of all such connections is in one-to-one correspondence with the vector space closed complex analytic 1-forms on
H0(M, &-l'0)
of global
M.
describe
The functions Aa $ =
a complex analytic
line bundle over any complex manifold; but it is customary and in some ways more convenient to consider in place of this line bundle the dual line bundle
K , the canonical bundle of M,
defined by the coordinate transformations
Kay = aP . Complex analytic sections of the canonical bundle can be identified with complex analytic n-forms on an n-dimensional complex manifold
M,
so that
manifold M
O-(K) =
6n,0
and r' (M, ®(K)) = P(M, 0-n'0).
If a complex
has a complex flat canonical structure then the bundle n and
hence of course the canonical bundle or equivalently are flat line bundles.
equivalently the canonical bundle K,
K
have constant coordinate transformations,
On the other hand the bundle A, is analytically equivalent to a flat
line bundle precisely when, after passing to a refinement of the coordinate
or
-98-
ha: Ua -> Q*
covering if necessary, there are complex analytic mappings such that
are constants, or equivalently such that
ha Aa ,3h
0 = d log(ha 6a h-l)
= d log nsa + d log ha - d log h That implies that {cpa = d log ha}
canonical connection on canonical connection
define an integrable complex analytic
and conversely any integrable complex analytic
M ;
can be written in the form
(Pa
<Pa = d log ha
ha
suitable refinement of the coordinate covering, and the functions
A or
K
after a reduce
Thus each integrable complex analytic canonical
to a flat bundle.
connection determines an analytic reduction of the canonical bundle to a flat bundle, and evidently different connections determine reductions to different flat bundles; so there is a one-to-one correspondence between complex analytic
canonical structures on M
and complex analytic reductions of the canonical
bundle to a flat line bundle.
Note that if for some integer
then
p
That incidentally explains the terminology.
is a complex analytic section
f f
is described by complex analytic functions
in the coordinate neighborhoods
Ua,
Ua fl US
.
If
K
afa
aJ so that
t3fa/8zaj}
fa
and these functions satisfy
P
fa -= Kasf13 in
f e ]'(M, &-(KP))
P
=
a6 fB
is a flat line bundle then
aft = Z K ap6 azBk aJJ _ £1cK
P
aB
af9
describe a section V f E P(M, -(KP 0
k
k 76 a j t*))
.
More general!'
-99-
if it is not assumed that
is necessarily flat but only that there is a
K
Ta(za) = E xaj(za) dzaj
Complex analytic canonical connection
(140)
(
of )
- P xaj fa
_
Cli
then
Q7
are easily seen to be the components of a complex analytic section
The sheaf homomorphism
of a r(M, B-(KK ® T*)).
V0
:
4(KP) -> B-(K® ® S*)
V0
:
r(M,B(KP)) -> r(M, S-(Kp(9 t*))
or linear mapping
defined by (140) is the analogue of the covariant derivative for this pseudogroup structure.
Turning then to topological properties, the Chern classes can be introduced for an arbitrary complex analytic vector bundle over a complex
manifold M, not just for the complex analytic tangent bundle, with the same formalism as in the case of the complex analytic tangent bundle. the Chern class
cl(C)
of a complex analytic line bundle
represented by the differential form differential forms of bidegree in Ua f u,,
and
Ca B
in
such that
Ua
choose a
C'
1
is a
C°°
a
= s' + s"
a
C-
d log a - T Ta
are the coordinate transformations defining the line
COmmection
s
is
'
bundle . Now for any complex manifold M
ea
over M
where Ta are any
2,n aTa
(1,0)
C
In particular
where
,
s'
a
is a
complex projective connection.
the form (116), where the
C°°
Cm
complex affine
canonical connection and
Here again
differential forms
sa
can be written
Ta = Ek xa k dZa k
-100-
satisfy (138)
;
and the coefficients of
k k snakj Setting
9aj = Ek sajk dza k,
=0
sa
satisfy
for all
j
.
the first Chern class of the manifold M
is
represented by the differential form
Cl = 0-1(a eaj
)
a
=
k
ea k
= Ejk(a sCYk j ^ dzaj + a sak ^ dz
j
= E. a xaj ^ dz
=
(141)
and
K
)
aa
and therefore the first Chern class of M the line bundles
o4j
is related to the Chern classes of
by
cl = cl(A) = -C1(K)
.
If M admits a complex analytic canonical connection then this Chern class
must vanish.
-101-
§10.
Complex flat canonical structures on surfaces A discussion of complex analytic pseudogroup structures on one-
dimensional compact complex manifolds can be found in [20]
;
although there
remain several open problems in that case, they mostly have to do with more detailed properties of complex projective structures, and will not be considered here.
Turning next to two-dimensional compact complex manifolds,
it is convenient to change from the order in which the various pseudogroups have been discussed and to begin with the simplest case, that of the complex flat canonical structures.
If the two-dimensional compact complex manifold M complex flat canonical structure then as already noted c1 e H2(M,T)
is the first Chern class of M .
Cl
can be identified with
-cl(K),
denotes the Chern class of the line bundle
K,
c1 = 0
where
cl(K) = 0
so that necessarily
but does at least imply that some product trivial.
Km
The bundles
K
.
c1(K) E H2(M,C:)
c1(K) e H2(M,7L);
does not ensure that the line bundle
cl = 0
such that
As is well known [23], the topological type of the line bundle determined by its integral Chern class
where
,
As for the converse assertion,
consider a two-dimensional compact complex manifold M The class
admits a
K
cl(K) = 0.
is completely
so knowing that
is topologically trivial,
Km = K(9 ... ®' K
is topologically
may or may not be analytically trivial; and to
investigate that question it is convenient to introduce the plurigenera ID = dim P(M,()-(Km))
,
in particular the geometric genus
the dimension of the space of holomorphic 2-forms on M.
pg = Pl If
P
m
which is > 0
then
there exists at least one nontrivial complex analytic cross-section of the line bundle
Km ; but the divisor of that cross-section represents the Chern
-102-
class
and must be trivial [32],
cl(Km)
analytically trivial and when
Km
P
m
= 1.
Thus
hence the bundle 2
m
is
p
= 1
precisely
pg < 1,
and
pg = 1
< 1,
is analytically trivial ; in particular
Km
and
m
precisely when
K
is analytically trivial.
then of course
K
is analytically equivalent to a flat line bundle, and
therefore M
If
K
is analytically trivial
admits a complex flat canonical structure.
In this case
incidentally the set of complex analytic canonical connections is nonempty and hence can be put in one-to-one correspondence with the space
r(M, X1'0)
of holomorphic 1-forms on M, and the set of integrable complex analytic connections is also nonempty and can be put in one-to-one correspondence with the subspace
M
;
r(M, 0-x,0) c r(M,C9.l 0)
of closed holomorphic 1-forms on
but every holomorphic 1-form on a compact complex surface is necessarily
closed [32 Theorem 1], necessarily integrable.
equivalently M
so every complex analytic canonical connection is If
K
is not analytically trivial then
pg = 0
If {spa}
is any
has no nontrivial holomorphic 2-forms.
collection of holomorphic 1-forms in the coordinate neighborhoods
{U"1
,
and
these forms satisfy (138), hence represent a complex analytic canonical then it is apparent that
connection on M , 2-form on M ,
and consequently
d(Pa = 0.
{d Pa}
describe a holomorphic
Thus in this case also every
complex analytic canonical connection is integrable; but there remains the question whether there exist any such connections. To investigate this question introduce the exact sequence of
sheaves over M
of the form
0
a p->0 >
J0
or
-103-
.}.and the associated exact sequence of cohomology groups
0->r(M,(9-1'0)->H1(M,T)->Hl(M,D-) a>
(142)
a*> Hl(M, (91'0)
> H (M, C) -> H2(M,O_)
c
as for the beginning of this sequence, compact and
]'(M, 6-1'0) = P(M, (9-1'0) c
since M is
r(M,M) = 1-(M, 9L)
as already noted.
The necessary and
sufficient condition that there exists an integrable complex analytic canonical connection is that the cohomology class r = td log ni a HI'(M,(9 '0)
is trivial.
is just the Chern class
It is easy to see that the image
cl, hence that
S(r) = 0
in
8(r) E H2(M,a)
since by assumption
cl = 0 ; therefore in order to show that there exists an integrable complex
analytic canonical connection it is enough merely to show that the mapping d*
in the exact sequence (142) is trivial.
If M
is a Kaehler manifold
there is an isomorphism
Hl (M, C) ° F(M, 6
since
Hl(M,T.)
r(M, 6-1'0)
H1(M, 4)
0) e H1(M,
is isomorphic to the space of harmonic complex 1-forms while
is the space of harmonic 1-forms of bidegree (1,0)
and
is isomorphic to the space of harmonic 1-forms of bidegree (0,1),
[46]; so in this case the mapping
d
is trivial as desired.
Kaehler it is still true that the mapping
d*
If M
is not
is trivial, but more use of
detailed properties of compact complex surfaces is required for the proof. By Kodaira's classification theorem for surfaces [34, Theorem 22] the only
°Dpact complex surfaces with
pg = 0
are those of classes I and VII.
The
surfaces of class I are algebraic, hence Kaehler, so need not be considered here.
-1o4-
For the surfaces of class VII the first Betti number is
b1 = 1,
and the
other numerical invariants can easily be calculated using Theorem 3 and formula (13) of [32] and remembering that
c1 = pg = 0 ;
the results are
dim r(M, 61l'0) = 0 ,
dim H (M,T) = 1 ,
dim H (M,&L) = 1 ,
dim H (M,a:) = 0
.
Substituting these results into the exact sequence (142) it follows immediately that the mapping
d
is trivial, indeed that
H1(M, 6-1'0) = 0. c
These
observations can be summarized as follows.
Theorem 6.
A compact complex surface M
canonical structure precisely when
admits a complex flat
c1 = 0 ; and if
c1 = 0
the set of all
such structures can be put into one-to-one correspondence with the points of the space
r(M, 4-l'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
cl = 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 C2
]P1
C C M
Recall that
is a connected one-dimensional
such that
C
is analytically equivalent
(is rational) and has self-intersection number
= - 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 118] shows that the canonical bundle of C
is isomorphic to
KC ® [C)C
-105-
ere
I
And
KC
is the restriction to
C
of the canonical bundle
[C] C
is the restriction to
C
of the line bundle [C]
on M. IC
is
The bundle
KC
has trivial Chern class, since
K
of M
of the divisor c1(K) = 0.
If
is an exceptional curve of the first kind then the Chern class of [C]C
C = - 1,
KC ® [C]C
hence the Chern class of
is -1 ; but the Chern
I
>.ass of the canonical bundle of diction.
Thus
M
C = lPl
is
-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].
X3 surfaces (the surfaces of Kodaira's class IIo) (the surfaces of Kodaira's class 111
0
)
;
They are: (i) the
(ii) the complex tori
; and (iii) the Kodaira surfaces, those
elliptic surfaces representable as quotients of
M2
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 VIo). The compact complex surfaces having
bundles are characterized by
but having nontrivial canonical
c1 = 0
c1 = 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 it this case also.
They are: (iv) the Enriques surfaces, with b1 = 0
(algebraic surfaces, a proper subset of the set of surfaces of Kodaira's class Io)
;
(v) the hyperelliptic surfaces, with
b1 = 2
(algebraic surfaces,
another proper subset of the set of surfaces of Kodaira's class Io) those surfaces of Kodaira's class VII0 having
and (vi)
The numerical invariants
c1 = 0.
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
-106-
nonetheless
K12
is trivial; indeed in case (iv) the bundle
already trivial [5].
then
K
If the bundle
K2
is
is reduced to the trivial bundle
K12
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.
to a flat
K
Flat bundles can be described by homomorphisms
of the fundamental group, as discussed in [20] among other places for instance. If
K12 = 1
satisfies
then the representation
v
divisor of 12,
K
also
is a subgroup of
p
in the fundamental group of the surface M,
where
v
is a
and the covering space M of M described by that kernel is
V-sheeted covering of M
trivial.
describing the flat bundle
P12 = 1 ; the kernel of the representation
finite index
a
p
It is apparent that
on which the induced bundle a = P-1(K) is
is
is the canonical bundle of the surface
so that the surface M must be a surface of one of the classes (i), (ii), or (iii).
c2 = v. c2
Indeed since ,
c2
is the Euler class [_], [32] and satisfies
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
12
by properly discontinuous grc-_17z
of affine transformations having no fixed points.
The surfaces of class
have been studied by Kodaira, Inoue, and others [°], not completely known.
[24+],
[32], but are still
For these surfaces it is nett always the case that
is analytically trivial for some m ; but since
VII_
Y. (M, &-l'0) = 0
K
m
all complex
-107-
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. gp(M, B-) = 0
whenever
p > 0
For a Stein manifold
[18], so from the exact sequence (142) it
follows that
H1(M, m- '0)
n (M,e.)
indeed this isomorphism associates to the cohomology class R 1(m,
the cohomology class
in this case also M C1 = 0.
{d log
cl = cl(A) = -c1(K) e I2(M,a),
Aa
O}
e
so that
has a flat complex canonical structure precisely when
Not all complex analytic canonical connections are integrable; but
there are a vast number of integrable complex analytic canonical connections.
-108-
TABLE 1
COMPACT COMPLEX SURFACES WITH
TYPE
pg = P1 =
dim
? 2
(M, 0 )
cl = 0
Betti numbers
q =
1
dim H (M, a)
dim [' (M, B 1' 0 )
b1
Chern number
b2
c2
(i)
K3 surface
1
0
0
0
22
24
(ii)
torus
1
2
2
4
6
0
1
2
1
3
4
0
0
0
0
0
10
12
0
1
1
2
2
0
0
1
0
1
0
0
(iii) Kodaira surface (iv)
Enriques surface
(v)
hyperelliptic
surface (vi)
VIIo with
c1=0
-109-
411.
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 manifolds 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
cl = 0
with Chern class
,
cl = 0
c2 = 0 ; and these surfaces are among those surfaces
listed in Table 1.
Referring to that table, the
complex tori (type ii), Kodaira surfaces (type iii), and hyperelliptic surfaces (type v) all have
c2 = 0 ; and as noted in the preceding section all these
surfaces can be represented as quotients of
C2
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 VII
0
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)
-110-
in Table 1, and for some of the surfaces of type (vi), this analysis was carried out by A. Vitter in [44], [45], 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
12
by properly discontinuous groups of affine transformatior.;
having no fixed points.
However if M
universal covering space M ,
is a complex affine surface with
so that M = M/P
where
is the
r = sr1(M)
covering translation group, then the complex affine structure on M
lifts
to a 1-invariant complex affine structure on M ; and since M is simply connected this structure is necessarily trivial, in the sense that it can be defined by global affine coordinates, [20]. complex analytic mapping
of the surface M,
f
M -> a2,
and a homomorphism
Thus there are a nonsingular
the geometric realization or developmen`. p
:
r' -> A2(T)
from the group
into the two-dimensional complex affine group such that for every point subset of
T2
p E M
and element
p(r)
of transformations acting on
D.
plane
e.
The image
f(p)
is an open
D = f(in)
; and although
need not be a properly discontinuous group The domain
f :M -> D
D
The complex affine structures for which
morphism, hence for which M = D/p(r),
is not necessarily the full
is a nonsingular complex analytic
hence a local homeomorphism, it is not clear that mapping.
r
which is mapped onto itself by the affine transformations
p(r) C A2(e) ; but the group
2
T e p.
f(Tp) = p(T)
.
f
need always be a covering: T2
D =
and
f
is a home-_-
can be characterized as the complete
affine structures, in the differential-geometric sense; there is an extensive literature on such manifolds, in the real or complex case, [12, 24, and further references cited there], but completeness is too restrictive an assumption in
The classification of all compact complex
{be complex analytic case. Surfaces of the form
T2/
F ,
F
where
is a properly discontinuous group
of affine transformations with no fixed points, was carried out by T. Suwa in [40] 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 on
surface M
K12 = 1.
of type (vi)
If
Km = 1
for some integer m > 0
for a
then some unbranched m-sheeted covering surface
M of M will have a trivial canonical bundle and trivial second Chern
class, hence M must be a surface of type (ii) or (iii); but M cannot be a Kaehler manifold, since
Thus any surface M
M
is not, so that M must be of type (iii).
of type (vi) for which
Km = 1
for some integer
m > 0 must be the quotient of a Kodaira surface M by a finite group of automorphisms of M having no fixed points; and arguing as in the proof of Theorem 39 in [32], it follows that M must have an affine structure. all these surfaces for which
for some m > 1
Km = 1
Since
can be taken as fairly
Well known it is really sufficient to limit further consideration here generally to those surfaces having
c1 = 0, c2 = 0, but
all these surfaces are of type (vi) of course.
m > 0;
Km # 1
for any
It is convenient to
introduce here the following simple observations about the complex analytic tangent bundle
T
Lemma 4.
c2 = 0 then
T
and its dual bundle
T*
for such surfaces.
On a compact complex surface M
for which
cl = 0,
and K # 1 for any integer m > 0 , if dim F(M, (9'(T e T*)) > 1 can be represented by coordinate transformations of the forms
-112-
0
as
-1 13
(143)
-1
,
or
Ka 0
Tas
with as 2 = Ka 1
dim r(M,6 (KV ® T ® T*)) > 0
while if
for some integer
v # 0
then
T
can be represented by
where (144)
T
aP
aap = Kay-1 , or
=
b
a0
i
1
a compact complex surface for which it is only assumed that
if
c2 = 0, a =
(145)
}
'a13
dim r(M, & a 1 ® T)) > 0 then
c1 = 0
and
for some complex analytic line bundle
can be represented by
T
Ta13 =
aa
bap
0
daiB
)
where
as , da
= Ka 1
;
and furthermore
(146)
ba
Proof.
= 0
unless either
Any element
complex analytic functions
a2 = 1
or
a2 K
f e r(M,C9'(KV (S T (9 T*))
fal J
= 1
.
is represented by
in the coordinate neighborhoods
Ua ,
and
-113-
bey satisfy
i
v
in
Ua 0 U, ; or viewing
functions in
k
' KaP Tapk fB,, TBaj
faj
2 X 2
as
Fa =
matrices of complex analytic
equivalently
U. ,
v
Fa= Kap Tap FO 113 a Ua 0 U. .
The traces and determinants of these matrices thus satisfy
tr Fa = Ka in Ua 0 U, .
If
tr F. , det F. = Ka2V
V
Kv
then the line bundles
v # 0
det F and
K2V
have trivial
Chern classes but are not analytically trivial, hence can have no nontrivial complex analytic sections; and therefore Pa # 0
there must exist matrices
coordinate neighborhoods
Ua
H.
det Ha = 1
Ha Ta o
the bundle
Hp F8 H01
T
;
Ho1
and since
det Fa = 0
.
If
and
1\
/0
Ha Fa Ha
The matrices
and
of complex analytic functions in the
such that
-1 _
tr Fa = 0
0
0
1
are also coordinate transformations which represent
H. Fa
Hal
Hp1
Ha Ta 6
Ka8
1 Ha Ta 8 H;
it follows readily that
'aa6
1 _
(147)
(
Ha Ta B HS
o
ba13\ da p
*erl' da B = as p Ka 6 Furthermore Ka 9 = let Ta 6 = as 8 da S , so %at a2 = Ka -1 and da6 = Ka 1 The functions as 8 describe a .
6
6
8
-114-
complex analytic line bundle analytic vector bundle line bundle
d
;
T
contained as a subbundle in the complex
a
and the functions
,
da
describe the quotient 0
so (147) represents the vector bundle
analytic extension of the line subbundle
T
as a complex
by the line bundle
a
d.
It is
well known that the set of all complex analytic equivalence classes of such extensions is parametrized by the cohomology group
ba {act-
}
Hl(M,(9(a d-1)),
with
the cocycle describing the extension (147), [23]; so in this
13
particular case the possible extensions (147) are described by the cohomology group
H1(M, (9(K V )).
line bundle
The Chern classes of the manifold M
and of the
are all zero, so by the Riemann-Roch-Hirzebruch formula
K-v
[23] as extended to arbitrary complex manifolds by Atiyah and Singer [1]
it
follows that
dim H1(M,a(K-V)) = dim F(M, L9-(K-V)) + dim 112(M, (9-(K-V))
and by the Serre duality theorem
dim H2(M, O(K_V)) = dim I'(M, (9(KV+1 Thus if
v
0
-1
then
Hl ( M ,
thus it can be supposed that (144).
If
v = -1
6'(
ba P
K V )) = 0
and the extension (147) is trivial;
and that gives the first part of
= 0,
the extension (147) may not be trivial; but at least the
second part of (144) holds.
can only be asserted that
If
V = 0
tr Fa and
then at the beginning of the proof i*
det Fa
there is always one nontrivial element of by the identity matrix
I
are constants.
F(M, 0(i (F z*)),
in each coordinate neighborhood
another linearly independent element
{Fa}
then
{Fa + c I}
Of course that represented
Um.
If there
is a nontrivial
-115-
element for any complex constant
c
so if
;
is evidently always at least one nontrivial element There exist matrices
det Fa = 0.
Ua
coordinate neighborhoods
a = Ka 1
,
a
and
da
or
Next if
fa
0/
0
In the first case arguing as before
c.
H 1
Ha Ta
and
O
In the second case arguing
ba
have the form (147) with
6
= 0
which is the first part of (143). is a nontrivial element, for some
f E r(M, 0'(a 1 (9 T))
complex analytic line bundle functions
(c
0.
which is the second part of (143).
as 6 Ka 6 ,
and
have the form (147) with da 0 = as
HS 1
similarly the matrices
for which
1)
CO
for some nonzero complex constant
Ha T
det H. = 1
=
0
the matrices
{Fa}
of complex analytic functions in the
Ha
such that
-1
Ha Fa Ha
there
dim r(M, C9-(T ( T*)) > 1
a, then
is represented by complex analytic
f
and these functions
Ua
in the coordinate neighborhoods
satisfy fa In Ua n U..
aa6 Tapk
=
Ua
.
fat and
ytic section.
have no
ga represent a nontrivial complex analytic
election of some complex line bundle
X
f
Indeed if these functions have a nontrivial common
ga then the functions
idlern class of
6
It is easy to see that the functions
¢eommon zeros in
divisor
i
_
X
is trivial, and hence
over M; X
but since
b2 = 0
the
cannot have any nontrivial complex
Thus the common zeros of f and fa
hence represent the second Chern class of the bundle
are isolated, and
a 1 E T
,
[32]; but
-116-
since
fa
a 1
and
r
can have no common zeros at all.
Ua
and
Ha of
There thus exist matrices
complex analytic functions in the coordinate neighborhoods det Ha = 1
fa
have trivial Chern classes it follows that
such that
and
fa
1
Ha
=
fa
The coordinate transformations the form (147) with
as
B
da
0
Ha ',C ,d
H13 1
= Ka
are then easily seen to be of
The possible extensions (147) are
parametrized by the cohomology group H1(M, 9-(a2 K))
;
and by the Riemann-
Roch-Hirzebruch theorem and the Serre duality theorem
dim H1(M,
Thus
dim H1(M, &-(a2K)) = 0
dim r(M, 6-(a2K)) + dim P(M, C9(.2))
unless
a2 = 1
these two cases it can be assumed that
ba
or
= 0
a2K = 1 .
;
and except for
That suffice to complete
i3
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 equivalently, if it can be represented by coordinate transformations of the form Caa
ba6
(148)
0
da s
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,
-117-
if it can be represented by coordinate transformations of the form (148)
With ba bindle
= 0 ; and otherwise the bundle is called indecomposable.
6
is reducible then
T
r(M, 6-(a 1 O T))# 0
g ; and it follows from Lemma 4 that if r(M,(9-(a 1 (9 T)) # 0
Whenever bindle
T
c1 = 0
If the
for the line subbundle
and
c2 = 0
then conversely
for some complex analytic line bundle
is reducible and contains
a
is of course true for the dual bundle
as a subbundle.
T
;
a
the
The same result
indeed on a two-dimensional
manifold M (i49)
S
as is obvious since for
*
= K
T
,
2 X 2 matrices
1
C1
a/
adlbc l -a
d/
Ca If M
_
is a two-dimensional compact complex manifold for which
0, c2 = 0,
Km # 1
and
for any integer m > 0, and if m has a
Complex analytic affine connection, then the curvature tensor
r = {raj
}
1J 2J 3
is a complex analytic tensor field on M 32, 33 ; thus
r
and is skew-symmetric in the indices
can be viewed ass/ a section
r e r(M,G (K (S T (S T*))
It now follows from Lemma 4 that if
r
.
is nontrivial then
T
can be
represented by coordinate transformations of the form
a6
=
(aa6 0
0
a.,Kaf3
)
where
(aaKa6)2 = 1
.
-118-
Thus either
a ® K = 1,
covering surface M trivial.
over
The bundle
bundle of M ;
or after passing to an unbranched two-sheeted
T
M
the bundle on M induced by a(& K
induces over M
hence either M
is
the complex analytic tangent
or the two-sheeted covering surface M
will have a complex analytic tangent bundle
r
that can be represented by
coordinate transformations of the form
Caa
a Ti The dual bundle
0) 1
0
must then have the same form; and consequently either
T
T*
M
or M has a nontrivial complex analytic section of the bundle
nontrivial complex analytic one-form.
However since M
,
a
is a surface of
type (vi) in the list given in Table 1 then as noted in that table r(M, 0-l'o) = 0. cl = 0
and
On the other hand the covering surface
c2 = 0,
also has
and no power of the canonical bundle on M
trivial either; consequently M P(M, (9'1'0) = 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 Ht(M,
cohomology group
0,(p(T))
when the tangent bundle and of the bundle
Note first that the Chern classes of M trivial, since the Chern classes of
T
T
is reducible. p(T)
are all
are trivial, so it follows from the
Riemann-Roch-Hirzebruch theorem [23] that
-119-
dim Hl(M,(9(p(T))) = dim r(M, 6(p(1))) + dim 1 ?(M, C3(p(1)))
prom the Serre duality theorem [23] note that
H 2(M, O{p(%)))
s r(M, 6{K ® p(T)*)) ; and since by (149) necessarily a p(T)
.
K ® p(T)* = K ® p(T*)
it follows that
,
dim H '(M, (9"(p(T))) = 2 dim r(M, &(p(T)))
(150)
If the tangent bundle
.
is reducible, so can be represented by coordinate
T
transformations of the form (148), then as noted in the proof of Lemma 4 the obstruction to reducing the entries
g(M, 6-(ad-1) )
to
0
lies in the group
dim Hl(M, B(ad-1) ) = dim r(M, 6-(ad-1) ) +
and
,
ba
+ dim r(M,61(Kda 1)); so it can be assumed that ba0 = 0 unless either a = d or a = K d . ba
P
# 0 ,
if ba
P
Leaving aside for the moment the cases in which then it is clear that an element
= 0
is described by complex analytic functions neighborhoods
and
Ua ,
-1
1
1
-2
2
Since the line bundles
2
with
b2 = 0
d = 1 T
or
a
2
fa12
and
j1
or
is symmetric in
-1
2
and
-2
1
1
f a22 = aaSda3 fP22 2
aaPf 12
-1
2
f022 - daP fP22
0
r(M,
unless either
furthermore, since
d = a2 ; a
1
have trivial Chern classes on any surface
d
it follows that a = d2
-1
fa12 - daf1312
P
fall = aaP daPfP11
bundle
ajlJ2
and satisfy
J2
1
or
in the coordinate
f 1
and that these functions are symmetric in the indices
fall = aaP f011
11
f e r(M, 0(p(T)))
d
when
b = 0 ,
a d = K 1
really
a = 1 and the
r(M, &(p(T))) = 0
-120-
a = 1, d = K-1
unless either b = 0
then
T
a3 = K-1
or
d = a2 .
,
If
a = 1, d = K-1,
is also reducible and contains 1 as a subbundle, so that
F(M, 6(T*)) = r(M, C9-1'0) # 0 ; but that is impossible for a surface of type (vi), as observed in Table 1.
Therefore if
T
is reducible and
ba 6 = 0 then dim I'(M, 9-(p(T))) =
a = d ,
Next if
1
if a3=K1 and d=a2
0
otherwise
jI
a2 = K 1
so that
then no matter what
,
ba
may be,
fall = aa1 fill,
for any section f e r(M, G(p(T))) it follows first that
hence that fa 1 1 = 0 ; then using this it further follows that 2 hence that fa12 =0
2 as.1 f012 ,
2
fa1 2 =
f = 0.
it is finally concluded that hence
a = K d ,
a2 = 1
and
and so on, consecutively, until
Finally consider the case that
K-2,
d2 =
;
and ba
cannot be reduced to 6
zero {
the last assertion is equivalent to the condition that the cocycle
;
0}
is not cohomologous to zero in
a S ba
11j(M, 8(a d-1))
= Hh(M, 19'(K))
hence to the condition that there cannot exist holomorphic functions the coordinate neighborhoods Ua 0 Uf3
.
For any section
fa 1 1 = da f6 1 1 ,
f e r(M,L4'(p(T)))
fa
hence that
that fa22 = acxl fs22
.
If
so that it can be assumed that
1
a
1
as
that fa12 = fP12 is a constant. so if
a # 1
then as usual
the constant multiple
fa
2 fa 2 2 12
2 2 =
bP
a
in
= Ka 1 ga - g in
ba
such that
Ua
ga
it follows first that
= 0 ; but then using this it follows
necessarily fa22 = 0 ; but if a = 1, = 1 ,
then it is only possible to conclude
Next
ff22 = 2 fa12 bsa + f022 dal
0,
but if
a = 1
of the cocycle
bf3
this asserts that
a
is cohomologous
-121-
{o zero, and by hypothesis that is only the case if both cases it follows that
and
fa 1 2 = 0
gimilar argument shows that fa 1 1 =
0
Thus in
fa12 = 0.
and using this, a
fa 2 2 = 0 ;
fa 1 2 = 0, fa 1 2 = 0
,
therefore, assuming that the complex tangent bundle
.
In summary
is reducible and hence
T
can be represented by coordinate transformations of the form (148), then
1 (151)
a3 = K-1
2
if
0
in all other cases.
d = a2,
,
and
b = 0,
dim H (M, 6{p(t)))
Thus except for the first case in (151) there exists a complex analytic
gffine connection on M ; and with the same exception, recalling the corollary to Theorem 2, that affine structure must be unique. in (151) the complex tangent bundle direct sum of the two line bundles
is analytically equivalent to the
2
and
a
In the exceptional case
d ; and since all line bundles
are analytically equivalent to flat line bundles on the manifolds considered here, it follows that bundle.
T
M
being
is analytically equivalent to a flat
Therefore there must exist a not necessarily symmetric complex analytic
affine connection
1 s
;
and the symmetric part of
s
1
will then be a
ai1i2 .Complex analytic affine connection in the sense used here. There remains the question whether there are any such manifolds having an irreducible complex analytic tangent bundle; but that is a question Which I have not yet been able to answer.
The only known surfaces of this
Category are those studied and classified by Inoue [24]; these surfaces have 'teducible
(but not decomposable) complex analytic tangent bundles, and hence '.have unique complex affine structures.
Actually of course these surfaces were
`described bby Inoue as quotients of subdomains of
1 -
by properly discontinuous
-122-
groups of affine transformations, and that exhibits quite explicitly the affine structures.
It is perhaps worth noting that for any one of the Inoue
surfaces having a nontrivial but necessarily finite fundamental group the universal covering surface is a simply-connected surface of the same type,
hence can be represented as the quotient of a subdomain of
T2
by a
properly discontinuous group of affine transformations; and the uniqueness of the affine structures shows that the covering translation group must also consist of affine transformations, leading directly to the corresponding representation for the original surface.
It is perhaps also worth remarking
that the arguments of Kodaira and Inoue use quite strongly the flat complex analytic covariant derivative.
Finally the results described in this
section can be summarized as follows. Theorem 7. c7 = 0 ,
Km = 1
and
The compact complex surfaces M for some integer
for which
c1 = 0,
m > 0, all admit complex affine
structures.
The compact complex surfaces M
but
for any integer m > 0, are of type (vi) in Table 1; for such
Km # 1
for which
c1 = 0, c2 = 0,
surfaces all complex analytic affine connections are integrable; and there exist complex analytic affine connections whenever the complex analytic tangent bundle unless
t
t = a e a2
is reducible, indeed there exists a unique such connection where
a
is a line bundle such that
a3 = K-1
.
-123-
,j§12.
Complex projective structures on surfaces Complex projective structures, the general structures in the case
,of one-dimensional manifolds, have been the least investigated of the
Turning now to the
Structures considered here in the two-dimensional case.
discussion of these structures, recall from Theorem 5 that on a two-dimensional
+.nifold which admits a complex analytic projective connection the Chern classes must satisfy 2 1 c2=3c1
(152)
,The first stage of the discussion is to determine in general terms which
two-dimensional compact complex manifolds M tion (152).
satisfy this topological restric-
Unlike the conditions considered previously this topological
restriction does not preclude the possibility that the surface M contains exceptional curves of the first kind, so that possibility must be taken into account.
Thus the surface M
I* by the application of
v
can be viewed as arising from a minimal surface quadratic transforms for some integer
and the numerical invariants of the surfaces M and M*
v > 0
are then related
as follows, [5] 2
(153)
bl = bl ,
b2 = b2
+ V,
c1 = (c1)
c2=c2 +v .
- v
Irom (152) and (153) it then follows that 2
3 c2 = (cl) - 4v
(154)
It is convenient to consider separately three possibilities for 2
the minimal surface M*
.
(i)
First suppose that
(cl) < 0.
It follows
;
-124-
from Theorem 55 of [32] that M
is either an algebraic ruled surface or a
surface of Kodaira's class VIIo
.
(The complex projective plane has
* 2
Recall that a ruled surface of genus
(c1) = 9. )
is a complex analytic
g
fibre bundle over a nonsingular algebraic curve, with the complex projective line as fibre and the group of linear fractional transformations as structure group; and that the numerical invariants of such a surface are as follows, [32, page 1052]
:
b1 = 2g,
(155)
b2 = 2,
(c1 )2
= 8(1-g),
c2 = 4(1-g)
Comparing (154) and (155), it follows immediately that g > 1
course
since
v = g-l ; and of
On the other hand for a surface
c1 = 9(1-g) < 0.
M*
2
of Kodaira's class
c2
VIIo it follows as in [32] that
Comparing (154) and (155) it follows immediately that
-(cl
=
(c *)2 = v
;
but since
* 2 (cl )
< 0
by assumption while
(c1) = 0.
Next suppose that
v > 0
this case really cannot occur.
If actually cl = 0
then M
(ii)
is one of the
surfaces listed in Table 1 so that, referring to that table, necessarily
c2 > 0
;
and therefore from (154) it follows that
c 2 = v = 0.
Thus
M = N*
must be one of the surfaces already considered in the investigation of surfaces having complex affine structures.
Leaving these surfaces aside, it
follows from Theorems 21 and 55 of [32] that an algebraic ruled surface of genus 1 ,
M*
must otherwise be either
an elliptic surface (of Kodaira's
classes IV0 or VI ), or a surface of Kodaira's class VII0 0
verify that if M*
;
It is easy to
is either a ruled surface of genus 1 or a surface of
Kodaira's class VIIo then cl = 0
.
c.
= v = C ,
so that M = M* ,
these cases too can be left aside.
If M*
and moreover
is an elliptic surface
-12')-
*
then as in §12 of [31] it follows that
c2
is the sum of the Euler
numbers of all the singular fibres, hence that
(154) necessarily c2 = v = 0 to have Euler number
and M = M* .
c2
> 0
so again from
;
For an elliptic surface M
all the singular fibres must also have Euler
c2 = 0
number zero, hence by [31] the singular fibres can only be multiples of
For any elliptic surface M having only such
nonsingular elliptic curves.
singularities there is a finite-sheeted covering M along fibres, such that
M
of M, branched only
is an elliptic surface with no singular fibres
at all, [31]; and since the branch curves are elliptic curves it follows from the natural extension of the Riemann-Hurwitz relation that the Euler
number of M
is a multiple of the Euler number of M.
However M
is a
differential fibre bundle having a complex torus as fibre, hence M must (c*
have zero Euler number.
(iii)
as in [32] and [43] that M*
Finally suppose that
2
l) > 0.
It follows
is an algebraic surface of Kodaira's class Vo,
an algebraic surface of general type.
It has been conjectured [43] that the
Chern numbers of such surfaces satisfy the inequality
(156)
*2 * c2 >31 (cl)
That inequality was demonstrated for algebraic surfaces with an ample .Canonical bundle by S.-T. Yau in [49], and for arbitrary algebraic surfaces of general type by Y. Miyaoka (On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225-237).
hence that M = M* . established.
It then follows from (154) that
In summary then, the following result has been
V = O ,
-126-
Theorem 8.
The compact complex surfaces
M
for which the Chern
numbers satisfy the equality
c2 = 31 c12
(152)
are the following: (i)
surfaces arising from ruled surfaces of genus of
(ii)
g-l
g > 1 by the application
quadratic transformations, for which surfaces
surfaces for which
c1 = 0
and
ci = 9(1-g) < 0
c2 = 0
(iii) minimal elliptic surfaces having as singular fibres only multiples of nonsingular elliptic curves and having
c1 # 0 , for which surfaces
2
c1=0 (iv)
minimal algebraic surfaces of general type with Chern numbers which satisfy (152), for which surfaces
ci > 0
Turning next to the question of which of these surfaces admit complex analytic projective connections, the surfaces of type (i) in the preceding list are a somewhat special class, since they all contain exceptional curves of the first kind.
It is easy to see that no surface with exceptional curves
of the first kind can admit complex analytic projective connections. Indeed
suppose that M V > 0
arises from a minimal surface M* by the application of
quadratic transforms, and that M has a complex analytic projective
connection.
This connection induces a complex analytic projective connection
on the complement of a finite number of points on the surface M* . of local coordinates
(z1, z2)
In terms
in an open neighborhood of one of these points
this induced complex analytic projective connection is described by complex
-127-
analytic functions
fl
jlj2
(z)
in the complement of a point; but these
functions extend to analytic functions even at the exceptional point, by the extended Riemann removable singularities theorem for functions of several complex variables, so the connection extends to a complex analytic projective
However the Chern numbers of M* must then satisfy
connection on M* .
both (152) and (154), which is impossible since it was assumed that
and that contradiction shows that M
v > 0 ;
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 connec-
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 connections.
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
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 Riemann surface
0 of genus
g, the fibre being a nonsingular elliptic curve
F
and the
structural group being the group of complex analytic automorphisms of [31].
When
F
is represented as the quotient of the complex line
M
F
by
-128-
a lattice subgroup
the automorphisms of
of
linear transformations z -> X z+ Tl of such that
A c`
=
where
ct ,
for a general torus
;
are all represented by
F
X
is a complex number
the only possibilities are
F
A = ± 1, but for those special tori with complex multiplication it is possible that
A
is a complex number with
X4
= 1
or
that the structural group of the bundle M
A6
= 1.
Suppose further then
can be reduced to the
over
subgroup of translations, so that only those automorphisms of the form
z -> z + q
V = {Ua}
,
That means that
appear.
where
admits a coordinate covering
M
is the product of a coordinate neighborhood
Ua
the Riemann surface 0 with local coordinate
hood Ua on the torus representation
with local coordinate
F
F = Tlo
za 1
;
on
Ua
and a coordinate neighbor-
za 2
induced from the
and the coordinate transitions are of the form
zal = fa6 (zI3 l) za2 = zB2+1)m8 (zs1) where
are the coordinate transition functions of the induced coordinate
{fa
= {Ua}
covering
of
and ria
S
:
UU n Ut , 3
-> Ct are some complex
Actually the values taken on by the functions
analytic mappings.
really be viewed as only determined modulo
,
so that
}
{TIa
can
T),,,
can be
13
considered as describing a cohomology class
T)
E H1(A ,
is the sheaf of germs of holomorphic mappings from A the functions TIa p
varies over
6'(F)) into
F;
where
6(F)
but when
are viewed as complex-valued functions then as the point B
Ua n U ft
rjoy(p) -
y ,
which can be assumed connected, clearly
TIa0,Y
is a fixed element of o , and these
-129-
elements describe a cohomology class c(ry) = {Tla0'Y} E H2(A, Z ) _
.
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
2g+2
if c(r)) =0,
2g + 1
otherwise
(158)
Using (158) and the facts that
ci = c2 = 0, the other numerical invariants
can be readily determined to be the following:
of the surface M
TABLE 2 pg
q
b2
c(r)) = 0
g
g+l
g+l
2g+2
4g+2
(ii)
c(H)
g
g+l
g
2g+1
4g
c(1) # 0.
If
0
and
c(H) = 0
a cohomology class
is nonalgebraic when
then the mappings
{rya
0
}
can be viewed as describing
, but for the projective line
E H1( , 0-)
so that it can been be assumed that
r
= 0
bl = 1, b2 = 0,
VIIo ; but in that case been considered. ftmctions
fa
If
M
so that c
1
H1(,n,
,
£9-
= 0 ,
and hence that M = ]p1 x F.
This surface has the obvious projective structure. then
bl
f4 1'0)
(i)
It is clear from this table that M g = 0
dim r(M,
If
g = 0
and
c(r))
0
is one of the surfaces of Kodaira's class
= 0, so this can be rejected here as having already
g = 1
and
c(ry) = 0
then the coordinate transition
can be assumed to be affine mappings, and the mappings %
Which represent an element in
H1(zI, m-)
can be assumed to be constant; thus
-130-
the given coordinate covering of
on M, indeed M
already exhibits an affine structure
M
is a complex torus.
If
g = 1
and
0
c(ry)
then M
is a nonalgebraic minimal elliptic surface with bI = 3, b2 = 4 ; thus is a Kodaira surface, hence admits an affine structure.
ing the trivial case M = ]P1 X F can further be assumed that
Thus after eliminat-
and those surfaces for which
g > 1.
M
It then follows that
that M necessarily has a nontrivial canonical bundle and
c1 = 0
it
pg > 1, hence cl # 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
clzai
-
(159)
(zI3l) (faiB rGs (Z,31)
0\ 1
J
and 2z (160) z _j
s1 sj l sj 2
z
1
fa (z01) + 82 S1 S 2% '(zp1)
Oj2
so recalling (104) a complex analytic projective connection on M is described by complex analytic functions
on M such that
(162)
in the coordinate neighborhoods
s 1
ld2
sal j + sat j = 0 for all j and that in
s612(zo) - sa12(za) - fa6 (z61)-1
Ua n US
(zs1) sa2 2(za) = 0'
Ua
-131-
(163)
s611(z6) - fa6 (z61) sa11(Za) - 2-a6 (z61) sa12(Za) -
(165)
sP12(z6)
2
1
1
sa12(za) + fa6(z61)-1na6(z61)2 sa22(za) 2
2
fa6 (z61) sa12(za) - cx6(z61) sa22(za) - 3 fa6 (z61)(166)
fa6 (z61)
s011(z6) + fa6 (z61) "a6 (z61) sall(za) + 2 a (z61)2 sa12(za) + fa6 (z61)-1 TIa6 (z61)3 sa22(za) - fa6 (261)2 sall(za)
-TG6 (z61) fa6 (z61)-1 fa6 (z61) + rah (z61) Equation (161) shows that
s
1 a2 2 is independent of za2
.
and induces a complex
analytic tangent vector field over A , that is, is a complex analytic section of the inverse of the canonical bundle over A ; and since n is of genus g > 1
it follows that that sa 1 2
0. Upon substituting this into (162) it then follows is a constant, independent of a, say sa 1 2 = C. Next upon sa 2 2 =
substituting both of these results into (163) that equation reduces to
(167)
so i l(z6) - fa6 (z61) sa11(za) =
2Cr
6 (z61) + 3 fa6 (z61)-lfa6(z61)
.
-132-
Now (167) implies that
sa
is independent of
1 1
za 2
sa
and that
1 1
induces a one-cochain on n exhibiting the fact that the cocycle
{2 C
B
+ fa S /3 fa 0}
is cohomologous to zero in H1(l, LP-(K0))
where
,
is the canonical bundle of A and is described by the coordinate trans-
K 0
fa
formations
in
complex analytic affine connection ; so since
H
satisfying (167) when the cocycle
the cocycle {fa
is not cohomologous to zero
{rla
it is
Returning to the definition of the functions %,3,
readily verified that under the canonical isomorphism the cohomology class represented by
belonging to the lattice subgroup
c(ry) # 0
sa
and the constant
C
0))
and that this number in turn can
C C
[31].
satisfying (167) when
1 1
Hn(,n,, &'(K
represents a complex number
{r
be identified with c(,)) e H(A, functions
fa
is chosen so that {f''
t , &(K)) , and the constant
-6c
g > 1
Therefore there can only exist functions
is not cohomologous to zero.
in
and as in
{e(j2£a )} is cohomologous to zero only when A admits a
§7 the cocycle
sa 1 1
H1(A, 6(Ko))
However fd/fag =
P(A, 6-) =
H1(A,
here of course
Ua fl U. ;
c(q) # 0.
Thus there can only exist On the other hand if
is chosen in the uniquely appropriate way then
the coordinates on n can be chosen in such a manner that
_
fa p/ fa p
_ -6c rj&S
and in that case (167) reduces to the assertion that
;
independent of
za 2
and that
sa
1 1(za 1)dza 1
There not only always exist functions
sa l
1 1
sa 1 2'
and
sa
c(r)) # 0
is
1 1
is an Abelian differential on satisfying (167), but the
space of all such functions is a complex vector space of dimension tinning under the assumption that
sa
and that the functions
g. sa
Con2 2'
have been chosen as above, (164) reduces to the condition 1 1
-133-
is also constant; indeed of course
sa22
reduces to the negative of (163), and of course
Sa 1 2
that
sa 2 2
=
Next (165)
-C.
2
1
- Sa l l
'
Finally
using all these results (166) reduces after a judicious bit of simplification to
(168)
faO(zs1)2sa11(za)
sO2
1 e*(j3fas(z.1)) - 31 a(Z61) sP11Nl)
= 6C
The two expressions on the right-hand side of (168) are separately cocyles representing classes in the cohomology group sa
implies that on
1 1
is independent of
H1(A, &1_(KO 2))
;
and (168)
za 2 and induces a one-cochain
exhibiting the fact that the right-hand side is cohomologous to zero.
However
Hi(x, 15-(Ko)) = H (A
always exist such functions
,
sa
(3-(K-1)) = 0
1 1
a complex linear space of dimension
since
g > 1 ; so there
and the set of all such functions form ,
3g-3.
Altogether then the surface
M admits complex analytic projective connections precisely when
c(11) # 0
and the set of all such connections if nonempty is a complex linear space
4g-3.
of dimension
Turning next to the question of the integrability of these complex analytic projective connections, using the explicit descriptions of the functions
just obtained and substituting these descriptions into (108),
s 1
ai1j2 (112), and (113) successively, it follows after a simple calculation that
xa121 = xa112 = ` sall(zal)2 + 2c2sa11(zal)
xa221 = xa122
2
1
-C sall(zal)
-134-
thus the expressions
are always symmetric, and the connections xal j j 1 2 3 are therefore always integrable. In summary the results obtained for this special subclass of surfaces are the following.
M be a complex analytic fibre bundle over a
Theorem 9. Let
compact Riemann surface elliptic curve
F
translations of
d
of genus
g > 1, the fibre being a nonsingular
and the structural group being merely the group of This surface has
F.
cl # 0, c1 = c2 = 0, and the other
numerical invariants are as in Table 2, depending on the structural invariant
c(ry) e Hl(d, K).
If c(q) = 0 there are no complex analytic projective
connections on M.
If
c(TI)
0
the nonalgebraic surface M
has complex
analytic projective connections parametrized by a complex vector space of
dimension
4g-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. of complex flat canonical structures the topological condition
For the case c1 = 0
is
both necessary and sufficient for the existence of those structures, by Theorem 6; while for the case of complex affine structures the topological conditions
cl = 0
and
c2 = 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.
topological condition
For the case of complex projective structures the c2 = 1 ci
is necessary but not sufficient for the
-135-
existence 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 M -> M, branched only along fibres, so that
M
is a minimal elliptic surface having
no singular fibres; so M can be represented as the quotient of M by a finite group P
of complex analytic automorphisms of Z,
[31].
Then M
is a complex analytic fibre bundle over a compact Riemann surface 0 of genus
g, the fibre being a nonsingular elliptic curve
group being the group translations of
F
G
of complex automorphisms of
is a subgroup of finite index in
P finite unbranched covering
->
F
and the structural
F. The group of so by passing to a
G;
the structural group of the induced
fibre bundle can be reduced to the subgroup of translations of
there is a finite unbranched covering M -> M
so that M
F.
Thus
is one of the
restricted subclass of surfaces of type (iii) just considered; and M can be represented as the quotient of M
by a finite group F
of complex analytic
p
automorphisms of M.
The complex analytic projective connections on M
are related to the r- invariant complex analytic projective connections on
x
M; and that reduces the problem to an analysis of the behavior of the complex
-136-
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, [49].
He has shown that those surfaces that
also have an ample canonical bundle can be represented as quotient spaces of the unit ball in
M2
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).
[6].
Examples of surfaces of this type were considered earlier by A. Borel,
-137-
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Library of Congress Cataloging in Publication Data
Gunning, Robert Clifford, 1931On uniformization of complex manifolds. (Mathematical notes ; 2.) Bibliography: p. 1. Complex manifolds. 2. Connections (Mathematics) 3. Pseudogroups. I. Title. II. Series: Mathematical notes (Princeton, N. J.) ; 22. 1978 78-55535 QA331.G785 515'.7 ISBN o-691-08176-x
