Vector Bundles on Complex Projective Spaces: With an Appendix by S. I. Gelfand

248 Pages·1980·2.21 MB·English

by  Christian Okonek

Modern Birkha¨user Classics
Many of the original research and survey monographs in pure and
applied mathematics published by Birkha¨user in recent decades have
beengroundbreakingandhavecometoberegardedasfoundationalto
thesubject.ThroughtheMBCSeries,aselectnumberofthesemodern
classics,entirelyuncorrected,arebeingre-releasedinpaperback(and
as eBooks) to ensure that these treasures remain accessible to new
generationsofstudents, scholars, andresearchers.
Christian Okonek
Michael Schneider
Heinz Spindler
Vector Bundles on Complex
Projective Spaces
With an Appendix by S. I. Gelfand
Corrected reprintofthe1988Edition
Christian Okonek Michael Schneider†
Institut für Mathematik Universität Bayreuth
Universität Zürich Germany
Winterthurerstrasse 190
8057 Zürich
Switzerland
[email protected]
Heinz Spindler
Institut für Mathematik
Universität Osnabrück
Albrechtstraße 28a
49076 Osnabrück
Germany
[email protected]
2010MathematicsSubjectClassification: Primary Classification: 14-02, 32-02
Secondary Classification: 14D20, 14D21, 14D22, 14J60, 14J81, 14N05, 14N25; 32L10,
32J25, 32G13, 32G81
ISBN978-3-0348-0150-8 e-ISBN978-3-0348-0151-5
DOI10.1007/978-3-0348-0151-5
Library of Congress Control Number: 2011930254
(cid:2)c 1980BirkhäuserVerlag
Originallypublishedunderthesametitleasvolume3intheProgress inMathematicsseriesby
Birkha¨userVerlag,Switzerland,ISBN978-0-8176-3000-3
Corrected second printing 1988 by BirkhäuserVerlag,ISBN978-0-8176-3385-1
Corrected reprint2011bySpringerBasel AG
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialiscon-
cerned,specificallytherightsoftranslation,reprinting,re-useofillustrations,broadcasting, reproduction
onmicrofilmsorinotherways,andstorageindatabanks.Foranykindofuse whatsoever, permission
from thecopyrightownermustbeobtained.
Coverdesign:deblik,Berlin
Printedonacid-freepaper
SpringerBaselAGispartofSpringerScience+BusinessMedia
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Introduction
These lecture notes are intended as an introduction to the methods
ofclassificationofholomorphicvectorbundlesoverprojectivealgebraic
manifolds X. To be as concrete as possible we have mostly restricted
ourselves to the case X = P . According to Serre (GAGA) the classifi-
n
cation of holomorphic vector bundles is equivalent to the classification
of algebraic vector bundles. Here we have used almost exclusively the
language of analytic geometry. The book is intended for students who
have a basic knowledge of analytic and (or) algebraic geometry. Some
fundamental results from these fields are summarized at the beginning.
One of the authors gave a survey in the S´eminaire Bourbaki 1978
on the current state of the classification of holomorphic vector bundles
over P . This lecture then served as the basis for a course of lectures
n
in Go¨ttingen in the Winter Semester 78/79. The present work is an
extended and up-dated exposition of that course. Because of the in-
troductory nature of this book we have had to leave out some difficult
topics such as the restriction theorem of Barth. As compensation we
have appended to each section a paragraph in which historical remarks
are made, further results indicated and unsolved problems presented.
The book is divided into two chapters. Each chapter is subdivided
into several sections which in turn are made up of a number of para-
graphs. Each section is preceded by a short description of its contents.
In assembling the list of literature we have done our best to include
allthearticlesaboutvectorbundlesoverP whichareknowntous. On
n
theotherhandwehavenotthoughtitnecessarytoincludeworksabout
theclassificationofholomorphicvectorbundlesovercurves. Thereader
interested in this highly developed theory is recommended to read an
article by Tjurin (Russian Math. Surveys 1974) or the lecture notes of
a course held at Tata Institute by Newstead.
Part of the present interest in holomorphic vector bundles comes
from the connection to physics. The mathematician who is interested
inthisconnectionisrecommendedtoseetheENS-S´eminaireofDouady
and Verdier. In the final paragraph of the present lecture notes he will
also find remarks about that topic and some literature citations.
R. M. Switzerhasnotonlytranslatedthemanuscriptofthesenotes
into English but has also aided us in answering many mathematical
questions. For this assistance we wish to thank him heartily. Further-
more we wish to thank Mrs. M. Schneider for doing such a good job
withtheunpleasanttaskoftypingthesenotesandH.Hoppeforassem-
blingtheindexanddoingthedifficultjobofinsertingthemathematical
symbols.
v
vi
Preface to the second printing
Since the appearance of this book there has been considerable ac-
tivity concerning vector bundles on projective spaces and other model
manifolds.
At first there were mainly two reasons for this:
i) The connection between algebraic bundles on P and subvari-
n
eties X ⊂ P of projective space.
n
ii) The connection between algebraic vector bundles on P and
3
solutions of the Yang–Mills equations (via the Penrose trans-
form) on S4.
Through the spectacular work of Donaldson it became apparent that
for algebraic surfaces there is a very close connection between algebraic
and differential geometry and topology.
Recently Donaldson and Uhlenbeck, Yau solved the conjecture of
Hitchin and Kobayashi (i.e., stable vector bundles admit a Hermite–
Einstein metric). This shows once more how differential geometry has
become more and more important in the theory of vector bundles.
For this new edition we compiled a list of references for the period
1980–1987which(eveniffarfromcomplete)shouldbehelpfulforfuture
research. We also include some older papers which we were not aware
of when the book was first written.
Bayreuth and G¨ottingen, August 1987.
Preface to the MBC-Series Edition
This release is an essentially unchanged edition of the second print-
ing of the book “Vector Bundles on Complex Projective Spaces” origi-
nally published as Volume 3 of Birkh¨auser’s series “Progress in Math-
ematics”. An english translation of S. I. Gelfand’s appendix to the
russian edition has been added.
Dave Benson of the University of Aberdeen retyped the original
manuscriptinAMSLATEXandpolishedR.Zeinstra’senglishtranslation
of Gelfand’s appendix. He also suggested to contact Springer to re-
release this book. We thank him heartily for all his efforts.
Osnabru¨ck and Zu¨rich, August 2010.
Contents
Introduction v
Chapter 1. Holomorphic vector bundles and the geometry of P 1
n
§1. Basic definitions and theorems 1
1.1. Serre duality, the Bott formula, Theorem A and
Theorem B 1
1.2. Chern classes and dual classes 6
§2. The splitting of vector bundles 11
2.1. The theorem of Grothendieck 12
2.2. Jump lines and the first examples 14
2.3. The splitting criterion of Horrocks 21
2.4. Historical remarks 23
§3. Uniform bundles 24
3.1. The standard construction 24
3.2. Uniform r-bundles over P , r < n 27
n
3.3. A non-homogeneous uniform (3n−1)-bundle over P 33
n
3.4. Some historical remarks, further results, and open
questions 37
§4. Examples of indecomposable (n−1)-bundles over P 38
n
4.1. Simple bundles 38
4.2. The null correlation bundle 40
4.3. The example of Tango 42
4.4. Concluding remarks and open questions 46
§5. Holomorphic 2-bundles and codimension 2 locally complete
intersections 46
5.1. Construction of 2-bundles associated to a locally complete
intersection 47
5.2. Examples 52
5.3. Historical remarks 56
§6. Existence of holomorphic structures on topological bundles 56
6.1. Topological classification of bundles over P , n ≤ 6 57
n
vii
viii CONTENTS
6.2. 2-bundles over P 60
2
6.3. 2-bundles over P 63
3
6.4. 3-bundles over P 67
3
6.5. Concluding remarks 70
Chapter 2. Stability and moduli spaces 71
§1. Stable bundles 71
1.1. Some useful results from sheaf theory 71
1.2. Stability: definitions and elementary properties 81
1.3. Examples of stable bundles 91
1.4. Further results and open questions 96
§2. The splitting behavior of stable bundles 97
2.1. Construction of subsheaves 98
2.2. Applications of the theorem of Grauert and Mu¨lich 105
2.3. Historical remarks, further results, and open questions 118
§3. Monads 120
3.1. The theorem of Beilinson 120
3.2. Examples 124
3.3. A stable 2-bundle over P 130
4
3.4. Historical remarks 136
§4. Moduli of stable 2-bundles 137
4.1. Construction of the moduli spaces for stable 2-bundles
over P 137
2
4.2. Irreducibility of M (0,n) 161
P2
4.3. Examples 173
4.4. Historical remarks, further results, and open problems 183
Bibliography 189
Supplemental Bibliography 195
Index 211
Appendix A. Sheaves on P and problems in linear algebra 215
n
§1. The exterior algebra and modules over the exterior algebra 215
§2. Sheaves on P and Λ-modules. Formulation of the main
theorem 219
§3. Supplement to the main theorem 222
§4. Equivalence of derived categories 225
§5. The category A 231
Bibliography for Appendix A 239
CHAPTER 1
Holomorphic vector bundles and the geometry of
P
n
§1. Basic definitions and theorems
In this section we shall establish the notation and assemble the
most important facts about the cohomology of projective spaces with
coefficients in an analytic coherent sheaf. Then we shall recall the
definition of the Chern classes of a vector bundle and for holomorphic
bundles we shall interpret them in some cases as the dual classes of
appropriate submanifolds.
1.1. Serre duality, the Bott formula, Theorem A and The-
orem B. Foran(n+1)-dimensionalcomplexvectorspaceV wedenote
by P(V) the associated projective space of lines in V; in particular we
have
P = P(Cn+1).
n
P has a natural structure as compact complex manifold.
n
Let X be a complex space with structure sheaf O . If F is a
X
coherent analytic sheaf over X and x ∈ X a point, then we denote the
stalk of F at x by F and set
x
F(x) = F /m F = F ⊗ O /m
x x x x OX,x X,x x
where m ⊂ O denotes the maximal ideal of the local ring O .
x X,x X,x
Let E be a holomorphic vector bundle over X. Then we have the
sheafO (E)of germsof holomorphicsectionsin E. O (E)is a locally
X X
free sheaf of rank r = rkE. In what follows we shall not distinguish
betweenavectorbundleE andtheassociatedlocallyfreesheafO (E).
X
With the notation introduced above we then have
E(x) = E /m E
x x x
forthefibreoverxofaholomorphicvectorbundleE. Ahomomorphism
of sheaves φ: E → F between two holomorphic vector bundles defines
for every point x a C-linear map
φ(x): E(x) → F(x)
C. Okonek et al., Vector Bundles on Complex Projective Spaces, Modern Birkhäuser Classics, 1
DOI 10.1007/978-3-0348-0151-5_1, © SSRSSpringer Basel AG 2011