Differential Analysis on Complex Manifolds

314 Pages·1980·1.207 MB·English

by  R. O. Wells Jr. (auth.)

65
Graduate Texts in Mathematics
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K.A.Ribet
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(continued after the subject index)
Raymond O. Wells, Jr.
Differential Analysis
on Complex Manifolds
Third Edition
New Appendix
By Oscar Garcia-Prada
RaymondO. Wells, Jr.
Jacobs University Bremen
Campus Ring 1
28759 Bremen
Germany
Editorial Board
S.Axler K.A.Ribet
Mathematics Department Department of Mathematics
San Francisco State University of California
University at Berkeley
San Francisco,CA 94132 Berkeley,CA 94720-3840
USA USA
[email protected] [email protected]
Mathematics Subject Classification 2000: 58-01,32-01
Library ofCongress Control Number: 2007935275
ISBN:978-0-387-90419-0
Printed on acid-free paper.
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PREFACE TO THE
FIRST EDITION
Thisbookisanoutgrowthandaconsiderableexpansionoflecturesgiven
at Brandeis University in 1967–1968 and at Rice University in 1968–1969.
The first four chapters are an attempt to survey in detail some recent
developments in four somewhat different areas of mathematics: geometry
(manifolds and vector bundles), algebraic topology, differential geometry,
and partial differential equations. In these chapters, I have developed vari-
ous tools that are useful in the study of compact complex manifolds. My
motivation for the choice of topics developed was governed mainly by
the applications anticipated in the last two chapters. Two principal top-
ics developed include Hodge’s theory of harmonic integrals and Kodaira’s
characterization of projective algebraic manifolds.
This book should be suitable for a graduate level course on the general
topic of complex manifolds. I have avoided developing any of the theory of
several complex variables relating to recent developments in Stein manifold
theory because there are several recent texts on the subject (Gunning and
Rossi, Hörmander). The text is relatively self-contained and assumes famil-
iarity with the usual first year graduate courses (including some functional
analysis), but since geometry is one of the major themes of the book, it is
developed from first principles.
Each chapter is prefaced by a general survey of its content. Needless to
say, there are numerous topics whose inclusion in this book would have
been appropriate and useful. However, this book is not a treatise, but
an attempt to follow certain threads that interconnect various fields and
to culminate with certain key results in the theory of compact complex
manifolds. In almost every chapter I give formal statements of theorems
which are understandable in context, but whose proof oftentimes involves
additional machinery not developed here (e.g., the Hirzebruch Riemann-
RochTheorem);hopefully,theinterestedreaderwillbesufficientlyprepared
(and perhaps motivated) to do further reading in the directions indicated.
v
vi Preface to the First Edition
Text references of the type (4.6) refer to the 6th equation (or theorem,
lemma, etc.) in Sec. 4 of the chapter in which the reference appears. If
the reference occurs in a different chapter, then it will be prefixed by the
Roman numeral of that chapter, e.g., (II.4.6.).
Iwouldliketoexpressappreciationandgratitudetomanyofmycolleagues
and friends with whom I have discussed various aspects of the book during
itsdevelopment.InparticularIwouldliketomentionM.F.Atiyah,R.Bott,
S. S. Chern, P. A. Griffiths, R. Harvey, L. Hörmander, R. Palais, J. Polking,
O. Riemenschneider, H. Rossi, and W. Schmid whose comments were all
very useful. The help and enthusiasm of my students at Brandeis and Rice
during the course of my first lectures, had a lot to do with my continuing
the project. M. Cowen and A. Dubson were very helpful with their careful
reading of the first draft. In addition, I would like to thank two of my
students for their considerable help. M. Windham wrote the first three
chapters from my lectures in 1968–69 and read the first draft. Without his
notes,thebookalmostsurelywouldnothavebeenstarted.J.Drouilhetread
the final manuscript and galley proofs with great care and helped eliminate
numerous errors from the text.
IwouldliketothanktheInstituteforAdvancedStudyfortheopportunity
to spend the year 1970–71 at Princeton, during which time I worked on
the book and where a good deal of the typing was done by the excellent
Institute staff. Finally, the staff of the Mathematics Department at Rice
University was extremely helpful during the preparation and editing of the
manuscript for publication.
Houston Raymond O. Wells, Jr.
December 1972
PREFACE TO THE
SECOND EDITION
In this second edition I have added a new section on the classical finite-
dimensional representation theory for sl(2,C). This is then used to give
a natural proof of the Lefschetz decomposition theorem, an observation
first made by S. S. Chern. H. Hecht observed that the Hodge ∗-operator is
essentiallyarepresentationoftheWeylreflectionoperatoractingonsl(2,C)
andthisfactleadstonewproofs(duetoHecht)ofsomeofthebasicKähler
identities which we incorporate into a completely revised Chapter V. The
remainder of the book is generally the same as the first edition, except
that numerous errors in the first edition have been corrected, and various
examples have been added throughout.
I would like to thank my many colleagues who have commented on the
firstedition,whichhelpedagreatdealingettingridoferrors.Also,Iwould
like to thank the graduate students at Rice who went carefully through the
book with me in a seminar. Finally, I am very grateful to David Yingst
and David Johnson who both collated errors, made many suggestions, and
helped greatly with the editing of this second edition.
Houston Raymond O. Wells, Jr.
July 1979
vii
PREFACE TO THE
THIRD EDITION
In the almost four decades since the first edition of this book appeared,
many of the topics treated there have evolved in a variety of interesting
manners.Inboththe1973and1980editionsofthisbook,onefindsthefirst
fourchapters(vectorbundles,sheaftheory,differentialgeometryandelliptic
partial differential equations) being used as fundamental tools for solving
difficult problems in complex differential geometry in the final two chapters
(namely the development of Hodge theory, Kodaira’s embedding theorem,
and Griffiths’ theory of period matrix domains). In this new edition of
the book, I have not changed the contents of these six chapters at all, as
they have proved to be good building blocks for many other mathematical
developments during these past decades.
I have asked my younger colleague Oscar García-Prada to add an
Appendix to this edition which highlights some aspects of mathematical
developments over the past thirty years which depend substantively on the
tools developed in the first six chapters. The title of the Appendix, “Moduli
spaces and geometric structures” and its introduction gives the reader a
good overview to what is covered in this appendix.
Theobjectofthisappendixistoreportonsometopicsincomplexgeome-
trythathavebeendevelopedsincethebook’ssecondeditionappearedabout
25 years ago. During this period there have been many important devel-
opments in complex geometry, which have arisen from the extremely rich
interaction between this subject and different areas of mathematics and
theoretical physics: differential geometry, algebraic geometry, global analy-
sis, topology, gauge theory, string theory, etc. The number of topics that
could be treated here is thus immense, including Calabi-Yau manifolds
and mirror symmetry, almost-complex geometry and symplectic mani-
folds, Gromov-Witten theory, Donaldson and Seiberg-Witten theory, to
mention just a few, providing material for several books (some already
written).
ix
x Preface to the Third Edition
However, since already the original scope of the book was not to be a
treatise, “but an attempt to follow certain threads that interconnect various
fields and to culminate with certain key results in the theory of compact
complex manifolds…”, as I said in the Preface to the first edition, in the
Appendixwehavechosentofocusonaparticularsetoftopicsinthetheory
of moduli spaces and geometric structures on Riemann surfaces. This is
a subject which has played a central role in complex geometry in the last
25 years, and which, very much in the spirit of the book, reflects another
instanceofthepowerfulinteractionbetweendifferentialanalysis(differential
geometryandpartialdifferentialequations),algebraictopologyandcomplex
geometry. In choosing the topic, we have also taken into account that the
book provides much of the background material needed (Chern classes,
theory of connections on Hermitian vector bundles, Sobolev spaces, index
theory, sheaf theory, etc.), making the appendix (in combination with the
book) essentially self-contained.
Itismyhopethatthisbookwillcontinuetobeusefulformathematicians
for some time to come, and I want to express my gratitude to Springer-
Verlag for undertaking this new edition and for their patience in waiting
for our revision and the new Appendix. One note to the reader: the Subject
Index and the Author Index of the book refer to the original six chapters of
this book and not to the new Appendix (which has its own bibliographical
references).
Finally, I want to thank Oscar García-Prada so very much for the
painstaking care and elegance in which he has summarized some of the
most exciting results in the past years concerning the moduli spaces of
vector bundles and Higgs’ fields, their relation to representations of the
fundamental group of a compact Riemann surface (or more generally of a
compact Kähler manifold) in Lie groups, and to the solutions of differen-
tial equations which have their roots in the classical Laplace and Einstein
equations, yielding a type of non-Abelian Hodge theory.
Bremen Raymond O. Wells, Jr.
June 2007
CONTENTS
Chapter I Manifolds and Vector Bundles 1
1. Manifolds 2
2. Vector Bundles 12
3. Almost Complex Manifolds and the ∂¯-Operator 27
Chapter II Sheaf Theory 36
1. Presheaves and Sheaves 36
2. Resolutions of Sheaves 42
3. Cohomology Theory 51
4. Cˇech Cohomology with Coefficients in a Sheaf 63
Chapter III Differential Geometry 65
1. Hermitian Differential Geometry 65
2. The Canonical Connection and Curvature of a Hermitian
Holomorphic Vector Bundle 77
3. Chern Classes of Differentiable Vector Bundles 84
4. Complex Line Bundles 97
xi