Proceedings of the Conference on Complex Analysis: Minneapolis 1964

315 Pages·1965·11.077 MB·English

by  K. Stein (auth.), Professor Dr. Alfred Aeppli, Professor Dr. Eugenio Calabi, Professor Dr. Helmut Röhrl (eds.)

Proceedings of the Conference on Complex Analysis
This Conference was supported by the United
States Air Force Office of Scientific Research
Proceedings of the
Conference on Complex Analysis
Minneapolis 1964
Edited by
A. Aeppli . E. Calabi . H. Rohrl
Springer-Verlag
Berlin . Heidelberg . N ew York
1965
Professor Dr. ALFRED AEPPLI
School of Mathematics, Institnte of Technology, University of Minnesota, Minneapolis
Professor Dr. EUGENIO CALABI
Department of Mathematics, University of Pennsylvania, Philadelphia
Professor Dr. HELMUT ROHRL
Department of Mathematics, University of California at Ran Diego, La Jolla
All rights, especially that of translation into foreign langnages, reserved.
It is also forbidden to reprodnce this book, either whole or in part, by photomechanical
means (photostat, microfilm and I or microcard) or by other procedure without written
permission from Springer-Verlag
ISBN-13: 978-3-642-48018-8 e-ISBN-13: 978-3-642-48016-4
DOl: 10.1007/978-3-642-48016-4
© by Springer-Verlag Berlin' Heidelberg 1965
Softcover repring of the hardcover 1s t edition 1965
Library of Congress Catalog Card Nnmber 65 -14 592
Title-No. 1257
Preface
This volume contains the articles contributed to the Minnesota Con
ference on Complex Analysis (COCA). The Conference was held March
16-21, 1964, at the University of Minnesota, under the sponsorship
of the U. S. Air Force Office of Scientific Research with thirty-one
invited participants attending. Of these, nineteen presented their papers
in person in the form of one-hour lectures. In addition, this volume con
tains papers contributed by other attending participants as well as by
participants who, after having planned to attend, were unable to do so.
The list of particip ants, as well as the contributions to these Proceed
ings, clearly do not represent a complete coverage of the activities in all
fields of complex analysis. It is hoped, however, that these limitations
stemming from the partly deliberate selections will allow a fairly com
prehensive account of the current research in some of those areas of
complex analysis that, in the editors' belief, have rapidly developed
during the past decade and may remain as active in the foreseeable
future as they are at the present time.
In conclusion, the editors wish to thank, first of all, the participants
and contributors to these Proceedings for their enthusiastic cooperation
and encouragement. Our thanks are due also to the University of Min
nesota, for offering the physical facilities for the Conference, and to
Springer-Verlag for publishing these proceedings.
Finally, a word of gratitude is due to the U.S. Air Force Office of
Scientific Research, and in particular to Dr. R. G. POHRER, for the
financial help and the efficient administrative procedure which enabled
us to go all the way from the original proposal of the conference, in
November 1963 to the appearance of these Proceedings in such a short
time.
A. AEPPLI
E. CALABI
Minneapolis, in December 1964 H. ROHRL
Contents
Page
STEIN, K.: On Factorization of Holomorphic Mappings. . . . . . . . .. 1
BUNGART, L.: Cauchy Integral Formulas and Boundary Kernel Functions in
Several Complex Variables. . . . . . . . . . . . . . . . . . . .. 7
POHL, W. F.: Extrinsic Complex Projective Geometry. With 1 Figure . .. 18
BERGMAN, S.: Some Properties of Pseudo-conformal Images of Circular Do-
mains in the Theory of Two Complex Variables. With 2 Diagrams . .. 30
SATAKE, 1.: Holomorphic Imbeddings of Symmetric Domains into a Siegel
Space ....................... 40
AEPPLI, A.: On Determining Sets in a Stein Manifold . . . . 48
AEPPLI, A.: On the Cohomology Structure of Stein Manifolds 58
STOLL, W.: Normal Families of Non-negative Divisors . . 70
KOHN, J. J.: Boundaries of Complex Manifolds . . . . . . . 81
HOLMANN, H.: Local Properties of Holomorphic Mappings . . 94
BERs, L.: Automorphic Forms and General Teichmilller Spaces 109
GRIFFITHS, PH. A.: The Extension Problem for Compact Submanifolds of Com-
plex Manifolds I (the Case of a Trivial Normal Bundle) . . . . . . . . 113
KURANISHI, M.: New Proof for the Existence of Locally Complete Families
of Complex Structures . . . . . . . . . . . . . . . . . . . . . . 142
KUHLMANN, N.: Algebraic Function Fields on Complex Analytic Spaces . . 155
ROYDEN, H. L.: Riemann Surfaces with the Absolute AB-maximum Principle 172
ANDREOTTI, A., and E. VESENTINI: A Remark on Non-compact Quotients of
Bounded Symmetric Domains . . . . . . . . . . . . . . . . . . . 175
BREMERMANN, H. J.: Pseudo-convex Domains in Linear Topological Spaces. 182
GUNNING, R. C.: Connections for a Class of Pseudogroup Structures . . . . 186
HIRONAKA, H.: A Fundamental Lemma on Point Modifications . . . . . . 194
ROHRL, H.: Transmission Problems for Holomorphic Fiber Bundles. With
2 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
ROSSI, H.: Attaching Analytic Spaces to an Analytic Space Along a Pseudo-
concave Boundary. . . . . . .. .............. 242
MORIMOTO, A.: Non-compact Complex Lie Groups without Non-constant
Holomorphic Functions. . . . . . 256
BISHOP, E.: Uniform Algebras . . .. .............. 272
MASKIT, B.: Construction of Kleinian Groups. . . . . . . . . . . . . . 281
AHLFORS, L. V.: The Modular Function and Geometric Properties of Quasi-
conformal Mappings. With 2 Figures . . . . . . . . . . . 296
KALLIN, E.: Polynomial Convexity: The Three Spheres Problem 301
Appendix: Problems Submitted. . . . . . . . . . . . . . . 305
List of Participants
ABHYANKAR, S. HIRONAKA, H. MURAKAMI, H.
AEPPLI, A. HOLMANN, H. POHL, W. F.
AHLFORS, L. V. ISE, M. PUMPLUN, D.
BERGMAN, S. KALLIN, E. M. ROHRL, H.
BERs, L. KOHN, J.J. ROSSI, H.
BISHOP, E. A. KUHLMANN, N. ROYDEN, H. L.
BUNGART, L. KURANISHI, M. SATAKE,I.
CALABI, E. MAOLANE, S. SPALLEK, K. H.
GRIFFITHS, P. A. lVlASKIT, B. STEIN, K.
GUGGENHEIMER, H. W. MATSUMURA, H. STOLL, W.
GUNNING, R. C.
On Factorization of Holomorphic Mappings*
By
K. STEIN
Introduction
Let X, Y be reduced complex spaces, r: X --+ Y a holomorphic map
ping, denote by R the equivalence relation in X defined by the level sets
(i.e. the connected components of the fibres) of r. If the level sets are
compact then by a theorem of H. CARTAN [1] the quotient space X/R
carries naturally the structure of a complex space and the natural pro
jection e:X --+ X/R is a proper holomorphic mapping; thus r admits a
factorization r = r* 0 e where r*:X/R --+ Y is a nowhere degenerate
holomorphic mapping.
It is known that things become more complicated if the assumption
on the compactness of the level sets of r is dropped. In this case the
quotient space X/R need not be Hausdorff, not even if the local rank
of r is constant. Consider the following example: Put X: = C2 (ZI, Z2) -
- {(ZI,Z2):!ZI! ;:;:;: 1, !Z2! = I}, Y:= Cl(ZI), let r:X --+ Y be the holo
morphic mapping defined by X:3 (ZI, Z2) --+ ZI E Y. The fibres r-1 (ZI)
consist of two connected components if ! ZI! ;:;:;: 1 and otherwise they are
connected. Obviously two points ofthe quotient space X/R which corre
spond to the two connected components of a fibre r-1 (Zl) with! Zl! = 1
do not satisfy the Hausdorff separation axiom. But X/R is a non Haus
dorff manifold, moreover a complex structure can be introduced on X/R
in an evident manner such that X/ R becomes a non Hausdorff Riemann
surface and that the mappings e:X --+X/R and r*:X/R --+ Y become
holomorphic mappings. Thus there is a factorization r = r* 0 e as in
the case of compact level sets but with the restriction that X/R is not
a complex space in the usual sense.
Now this statement is a special case of a more general proposition. In
this paper we consider holomorphic mappings r: X --+ Y of constant local
rank where X is a complex manifold. We introduce the notion of quasi
complex space (§ 1); it turns out that the space X/ R carries the structure
of a quasi complex space (§ 2). We then get a factorization theorem (§ 3)
which corresponds to an earlier statement on factorizations of proper
*
Received June 8, 1964.
Conference on Complex Analysis 1
2 K. STEIN
holomorphic mappings ([5]). The proofs of the statements given below
are sketched.
Notations. By a complex space we always mean a reduced complex
space i.e. a complex space in the sense of J. P. SERRE ([2], [7]). For basic
notions connected with complex spaces and more generally with ringed
spaces compare [1], [3]; for equivalence relations in ringed spaces see
[1]. - TJ;le local rank of an holomorphic mapping .:X -'»- Y in a point
x E Xis denoted by rkx., the global rank ofds rk.: = sup rkx• [5].
XEX
1. Quasi complex spaces
Let Xl = (Xl, Ol), X2 = (X2' ( 2) be puredimensional complex
spaces of equal dimension, a:XI -'»-X2 a proper surjective nowhere
degenerate holomorphic mapping (thus (Xl, a, X2) is an analytic cove
ring). Consider an equivalence relation RI in Xl such that
(a) RI is open (i.e. the saturation of any open set is open)
(b) a is constant on each Rl-equivalence class.
Let XI/RI = Z be the quotient space of Xl with respect to RI and put
0= DI/RI (compare [1]). Then the ringed space (Z, 0) is called a
quasi analytic covering space.
We say that the ringed space (X, D) = X is a quasi complex space if X
is locally isomorphic to quasi analytic covering spaces, i. e. if each point
I
x E X has an open neighbourhood U such that (U, 0 U) can be bi
morphically mapped onto a quasi analytic covering space (Z, 0).
Clearly a quasi complex space is a locally quasi compact Tl-space.l
With respect to quasi complex spaces one has the usual notions of
complex analysis like holomorphic and meromorphic junction, holomorphic
mapping, nowhere degenerate holomorphic mapping, local analytic and
analytic subset, normality etc. If A is a local analytic subset of the quasi
complex space X then the ringed structure of X induces a ringed struc
ture on A, and it is easily seen that A supplied with this ringed structure
is a quasi complex space again, therefore A can be called a quasi complex
subspace of X.
Every quasi complex space X contains an open dense subset M such
that M has locally the structure of a complex manifold. Furthermore X
can be reduced in each point Xo E X to a germ of a complex space in the
following sense: There is an open neighbourhood U of Xo and a nowhere
degenerate holomorphic mapping 7:: U -'»- Y onto a complex space
Y = (Y, 5) such that the induced homomorphism *7:xo: 5T(xo) -'»- Oxo
1 In [4] H. HOLMANN introduced the notion of pseudo complex spaces; these
spaces are locally complex spaces and in particular locally Hausdorff. Any pseudo
complex space is quasi complex, but a quasi complex space need not to be locally
Hausdorff.
On Factorization of Holomorphic Mappings 3
is an isomorphism. This statement is a conclusion of the following Pro
position l.
We define: Let X be a quasi complex space, X' a complex space.
A holomorphic mapping (!: X --? X' is called a c-reduction of X if
(i) (! is nowhere degenerate and surjective,
(ii) (! majorizes each mapping cp:X --? Y of X into a complex space Y
(i. e. there is a holomorphic mapping ft: X' --? Y such that cp
=ftO(!).
It is immediate that two c-reductions (!:X --? X' and (!l:X --? Xl are
holomorphically equivalent: There exists a biholomorphic mapping
IX:X' --? X~ such that (!l = IX 0 (!.
Now one has the
Proposition 1. Let X be a quasi complex space. Assume that a nowhere
degenerate holomorphic mapping '1jJ: X --? Y of X into a complex space Y
exists. Then each of the following conditions is sufficient for the existence of a
c-reduction of X:
(1) X is normal,
(2) '1jJ is proper.
In case (1) the reduction space X' is normal.
To prove this one shows first that under both conditions '1jJ can be
assumed as a surjective mapping. Then one considers the class (l; of all
nowhere degenerate holomorphic mappings of X onto complex spaces;
(l; is ordered in a natural manner. By means of Zorn's lemma one sees
that a maximal element exists in (l; and this element gives a c-reduction
e: e
X --? X' of X. If X is normal then X' must be normal too, otherwise
could be lifted to a holomorphic mapping into the normalization of X'
and this would contradict (ii).
Remark. There are quasi complex spaces which have no c-reduction.
Example: Take the unit disk U: = {z: z < 1} in the complex plane,
J J
choose a real irrational number ~ and a real number 8 > 0 such that the
domains Dl = {z: 1/2 < JzJ < 1, -8 < cp < 8} (z = JzJo e2"'i'P) and
D2 = {z:1/2 < JzJ < 1, -8 + ~ < cp < 8 + n are disjoint. Consider
the mapping {J: U --? U defined by
z --? Z if z E U - (Dl U D2),
z --? z . e2"'i~ if z E Dl ,
Z --?z 0 e-2nig if z ED2,
and identify points of U which are related by {J. This gives a quasi
complex space X (which is a non Hausdorff Riemann surface) and one
has a natural holom-orphic mapping ft: U --? X. Now iff is a mero-morphic
function in X then f: = f 0 ft is a meromorphic function in U, and f satisfies
1*