Techniques of extension of analytic objects

271 Pages·1974·8.977 MB·English

by  Siu, Yum Tong (萧荫堂)

TECHNIQUOEFSE XTENSION
OFA NALYTOIBCJ ECTS
Yum-TgoS niu
DEPARTMENTO F MATHEMATICS
YALEU NIVERSITY
NEW HAVEN,C ONNECTICUT
MARCEL DEKKER, INC. NewYork1 974
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TABLE OF CONTENTS
PREFACE iii
INTRODUCTION 1
CHAPTER 1
EXTENSION OF HOLOMORPHIC AND MEROMORPHIC FUNCTIONS
Levi 's Theorem 13
Rothstein's Theorem 16
APPENDIX
The i-Theorem of Koebe-Bieberbach 31
CHAPTER 2
EXTENSION OF SUBVARIETIES AND HOLOMORPHIC MAPS
The Theorem of Thullen-Remmert-Stein 34
Bishop's Theorem 51
Rothstein's Theorem 67
Ext ens ion across IR n 76
Extension of Holomorphic Maps 85
APPENDIX
Jensen Measures 94
Ranks of Holomorphic Maps 96
Thick Sets 101
Special Analytic Polyhedra 104
A Special Case of the Lemma of Dolbeault
Grothendieck 117
CHAPTER 3
HOMOLOGICAL CODIMENSION, LOCAL COHOMOLOGY, AND
GAP-SHEAVES 123
i
APPENDIX
M-sequences 147
CHAPTER 4
EXTENSION OF COHERENT ANALYTIC SUBSHEAVES 152
CHAPTER 5
EXTENSION OF LOCALLY FREE SHEAVES ON RING DOMAINS 163
APPENDIX
Topology of Sheaf Cohomology Groups 184
Triviality of Holomorphic Vector Bundles 187
CHAPTER 6
EXTENSION OF LOCALLY FREE SHEAVES ON HARTOG' DOMAINS 202
APPENDIX
Duality 229
CHAPTER 7
EXTENSION OF COHERENT ANALYTIC SHEAVES 2.'.35
REFERENCES 250
INDEX 254
INDEX OF SYMBOLS 256
ii
PREFACE
This set of lecture notes is a reproduction (with minor
modifications) of the notes I distributed to the students in
the "troisieme cycle" course "Techniques of Extension of Ana
lytic Objects" I gave at the University of Paris VII in the
second semester of 1971-1972 while I was on a leave of absence
from Yale University.
The results presented here are known. However, some of
the proofs (for example, the proof of the extension theorem
for coherent analytic sheaves from a Hartogs' figure) are new.
Most of the results appear here in book form for the first
time.
The reader is assumed to have some familiarity with the
basic theory of several complex variables, as given, for
example, in Gunning-Rossi's "Analytic Functions of Several
Complex Variables".
The appendices at the end of the chapters contain back
ground material and are mainly for the convenience of the
reader. Some of the material there are slightly different
from the standard formulations usually found in the literature
and some are special cases of well-known results for which
direct proofs are supplied which are much easier than those of
the general cases.
I wish to thank the University of Paris VII and
Professor F. Norguet for their hospitality during my stay in
iii
Paris. I would like also to thank the Alfred P. Sloan Found
ation and the National Science Foundation f·or providing part
of the financial support respectively during my leave of
absence from Yale University and during periods in which this
set of lecture notes was prepared and organized. Finally I
wish to thank Betsy Buslovitz for her excellent typing.
iv
TECHNIQUES OF EXTENSION
OF ANALYTIC OBJECTS