Positivity in algebraic geometry I

395 Pages·2004·1.762 MB·English

by  Lazarsfeld R.K.

Ergebnisse der Mathematik Volume 48
und ihrer Grenzgebiete
3.Folge
A Series of Modern Surveys
in Mathematics
Editorial Board
M. Gromov, Bures-sur-Yvette
J. Jost, Leipzig
J. Kollár, Princeton
H.W. Lenstra, Jr., Leiden
J.Tits, Paris
D.B. Zagier, Bonn/Paris
G.M. Ziegler, Berlin
Managing Editor R. Remmert, Münster
Robert Lazarsfeld
Positivity
in Algebraic
Geometry I
Classical Setting:
Line Bundles and Linear
Series
1 23
Prof. Robert Lazarsfeld
Department of Mathematics
University of Michigan
Ann Arbor, MI 48109, USA
e-mail: [email protected]
Library of Congress Control Number: 2004109578
Mathematics Subject Classification (2000):
Primary: 14-02
Secondary: 14C20, 14F05, 14F17, 32L10, 32J99, 14J17
ISBN 978-3-540-2258-7 ISBN 978-3-642-18808-4 (eBook)
DOI 10.1007/978-3-642-18808-4
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, bro-
adcasting, reproduction on microfilms or in any other ways, and storage in data banks. Duplicati-
on of this publication or parts thereof is permitted only under the provisions of the German Copy-
right Law of September 9, 1965, in its current version, and permission for use must always be
obtained from Springer. Violations are liable for prosecution under the German Copyright Law.
Springer is a part of Springer Science+Business Media
springeronline.com
© Springer-Verlag Berlin Heidelberg 2004
Originally published by Springer Berlin Heidelberg New York 2004
Softcover reprint of the hardcover 1st edition 2004
Typesetting: Computer to film by the authors’ data
Printed on acid-free paper 41/3142XT - 5 4 3 2 1 0
To Lee Yen, Sarah, and John
Preface
Theobjectofthisbookistogiveacontemporaryaccountofabodyofworkin
complex algebraic geometry loosely centered around the theme of positivity.
Ourfocusliesonanumberofquestionsthatgrewupwiththefieldduring
the period 1950–1975. The sheaf-theoretic methods that revolutionized alge-
braicgeometryinthefifties—notablytheseminalworkofKodaira,Serre,and
Grothendieck — brought into relief the special importance of ample divisors.
By the mid sixties a very satisfying theory of positivity for line bundles was
largelycomplete,andfirststepsweretakentoextendthepicturetobundlesof
higherrank.In a related direction,workof Zariskiand othersled toagreatly
deepenedunderstandingofthebehavioroflinearseriesonalgebraicvarieties.
At the border with topology the classical theorems of Lefschetz were under-
stoodfromnewpointsofview,andextendedinsurprisingways.Hartshorne’s
book[276]andthesurveyarticlesintheArcataproceedings[281]giveagood
picture of the state of affairs as of the mid seventies.
The years since then have seen continued interest and activity in these
matters.Workinitiatedduringtheearlierperiodhasmaturedandfoundnew
applications. More importantly, the flowering of higher dimensional geometry
has led to fresh perspectives and — especially in connection with vanishing
theorems — vast improvements in technology. It seems fair to say that the
current understanding of phenomena surrounding positivity goes fundamen-
tally beyond what it was thirty years ago. However, many of these new ideas
haveremainedscatteredintheliterature,andothersuptonowhavenotbeen
worked out in a systematic fashion. The time seemed ripe to pull together
someofthesedevelopments,andthepresentvolumesrepresentanattemptto
do so.
The book is divided into three parts. The first, which occupies Volume I,
focuses on line bundles and linear series. In the second volume, Part Two
takes up positivity for vector bundles of higher ranks. Part Three deals with
ideas and methods coming from higher-dimensional geometry, in the form of
VIII Preface
multiplier ideals. A brief introduction appears at the beginning of each of the
parts.
I have attempted to aim the presentation at non-specialists. Not conceiv-
ing of this work as a textbook, I haven’t started from a clearly defined set of
prerequisites. But the subject is relatively non-technical in nature, and famil-
iaritywiththecanonicaltexts[280]and[248](combinedwithoccasionalfaith
and effort) is more than sufficient for the bulk of the material. In places —
forexample,Chapter4onvanishingtheorems—ourexpositionisifanything
more elementary than the standard presentations.
I expect that many readers will want to access this material in short seg-
ments rather than sequentially, and I have tried to make the presentation as
friendly as possible for browsing. At least a third of the book is devoted to
concrete examples, applications, and pointers to further developments. The
more substantial of these are often collected together into separate sections.
Othersappearasexamplesorremarks(typicallydistinguishedbythepresence
andabsencerespectivelyofindicationsofproof).Sourcesandattributionsare
generallyindicatedinthebodyofthetext:thesereferencesaresupplemented
by brief sections of notes at the end of each chapter.
We work throughout with algebraic varieties defined over the complex
numbers. Since substantial parts of the book involve applications of vanish-
ing theorems, hypotheses of characteristic zero are often essential. However I
have attempted to flag those foundational discussions that extend with only
minor changes to varieties defined over algebraically closed fields of arbitrary
characteristic. By the same token we often make assumptions of projectivity
wheninrealitypropernesswoulddo.AgainItrytoprovidehintsorreferences
for the more general facts.
Although we use the Hodge decomposition and the hard Lefschetz theo-
rem on several occasions, we say almost nothing about the Hodge-theoretic
consequences of positivity. Happily these are treated in several sources, most
recently in the beautiful book [600], [599] of Voisin. Similarly, the reader will
findrelativelylittlehereaboutthecomplexanalyticsideofthestory.Forthis
we refer to Demailly’s notes [126] or his anticipated book [119].
Concerning matters of organization, each chapter is divided into several
sections,manyofwhicharefurtherpartitionedintosubsections.Areferenceto
Section3.1pointstothefirstsectionofChapter3,whileSection3.1.Brefersto
the second subsection therein. Statements are numbered consecutively within
each section: so for example Theorem 3.1.17 indicates a result from Section
3.1 (which, as it happens, appears in 3.1.B). As an aid to the reader, each
of the two volumes reproduces the table of contents of the other. The index,
glossary,andlistofreferencescoverbothvolumes.LargepartsofVolumeIcan
be read without access to Volume II, but Volume II makes frequent reference
to Volume I.
Preface IX
Acknowledgements. I am grateful to the National Science Foundation, the
Guggenheim Foundation and the University of Michigan for support during
the preparation of these volumes.1
I have benefited from comments and suggestions from many students and
colleagues,including:T.Bauer,G.Bini,F.Bogomolov,A.Bravo,H.Brenner,
A. Chen, D. Cutkosky, M. De Cataldo, T. De Fernex, J.-P. Demailly, I. Dol-
gachev, L. Ein, G. Farkas, R. Friedman, T. Garrity, A. Gibney, C. Hacon,
R. Hartshorne, S. Helmke, M. Hering, M. Hochster, J. Howald, C. Huneke,
P. Jahnke, M. Jonsson, Y. Kawamata, D. Keeler, M. Kim, F. Knudsen,
A. Ku¨ronya, J. M. Landsberg, H. Lee, M. Musta¸tˇa, T. Nevins, M. Nori,
M. Paun, T. Peternell, H. Pinkham, M. Popa, I. Radloff, M. Ramachandran,
M. Reid, J. Ross, M. Roth, H. Schenck, J. Sidman, R. Smith, A. Sommese,
T. Stafford, I. Swanson, T. Szemberg, B. Teissier, Z. Teitler, D. Varolin,
E. Viehweg, P. Vojta, J. Winkelmann, A. Wolfe, and Q. Zhang.
Thisprojecthasprofitedfromcollaborationswithanumberofco-authors,
includingJean-PierreDemailly,MarkGreen,andKarenE.Smith.Jointwork
many years ago with Bill Fulton has helped to shape the content and presen-
tation of Chapter 3 and Part Two. I likewise owe a large mathematical debt
to Lawrence Ein: I either learned from him or worked out together with him
a significant amount of the material in Chapter 5 and Part Three, and our
collaboration has had an influence in many other places as well.
I am grateful to several individuals for making particularly valuable con-
tributions to the preparation of these volumes. Ja´nos Koll´ar convinced me to
startthebookinthefirstplace,andBillFultoninsisted(onseveraloccasions)
thatIfinishit.Besidestheirencouragement,theycontributeddetailedsugges-
tionsfromcarefulreadingsofdraftsofseveralchapters.Ialsoreceivedcopious
comments on different parts of a preliminary draft from Thomas Eckl, Jun-
Muk Hwang, Steve Kleiman, and Karen Smith. Olivier Debarre and Dano
Kim read through the draft in its entirety, and provided a vast number of
corrections and improvements.
Likemanyfirst-timeauthors,Icouldn’thaveimaginedwhenIbeganwrit-
ing how long and consuming this undertaking would become. I’d like to take
thisopportunitytoexpressmyprofoundappreciationtothefriendsandfamily
members that offered support, encouragement, and patience along the way.
Ann Arbor Robert Lazarsfeld
May 2004
1 Specifically,thisworkwaspartiallysupportedbyNSFGrantsDMS97-13149and
DMS 01-39713.
Contents
Notation and Conventions ..................................... 1
Part One: Ample Line Bundles and Linear Series
Introduction to Part One ...................................... 5
1 Ample and Nef Line Bundles .............................. 7
1.1 Preliminaries: Divisors, Line Bundles, and Linear Series ...... 7
1.1.A Divisors and Line Bundles ......................... 8
1.1.B Linear Series ..................................... 12
1.1.C Intersection Numbers and Numerical Equivalence ..... 15
1.1.D Riemann–Roch ................................... 20
1.2 The Classical Theory .................................... 24
1.2.A Cohomological Properties .......................... 25
1.2.B Numerical Properties.............................. 33
1.2.C Metric Characterizations of Amplitude .............. 39
1.3 Q-Divisors and R-Divisors................................ 44
1.3.A Definitions for Q-Divisors.......................... 44
1.3.B R-Divisors and Their Amplitude.................... 48
1.4 Nef Line Bundles and Divisors ............................ 50
1.4.A Definitions and Formal Properties................... 51
1.4.B Kleiman’s Theorem ............................... 53
1.4.C Cones ........................................... 59
1.4.D Fujita’s Vanishing Theorem ........................ 65
1.5 Examples and Complements .............................. 70
1.5.A Ruled Surfaces ................................... 70
1.5.B Products of Curves................................ 73
1.5.C Abelian Varieties ................................. 79
1.5.D Other Varieties ................................... 80
1.5.E Local Structure of the Nef Cone .................... 82
XII Contents
1.5.F The Cone Theorem ............................... 86
1.6 Inequalities ............................................. 88
1.6.A Global Results.................................... 88
1.6.B Mixed Multiplicities............................... 91
1.7 Amplitude for a Mapping................................. 94
1.8 Castelnuovo–Mumford Regularity.......................... 98
1.8.A Definitions, Formal Properties, and Variants.......... 99
1.8.B Regularity and Complexity.........................107
1.8.C Regularity Bounds ................................110
1.8.D Syzygies of Algebraic Varieties......................115
Notes.......................................................119
2 Linear Series ..............................................121
2.1 Asymptotic Theory ......................................121
2.1.A Basic Definitions..................................122
2.1.B Semiample Line Bundles...........................128
2.1.C Iitaka Fibration...................................133
2.2 Big Line Bundles and Divisors ............................139
2.2.A Basic Properties of Big Divisors ....................139
2.2.B Pseudoeffective and Big Cones......................145
2.2.C Volume of a Big Divisor ...........................148
2.3 Examples and Complements ..............................157
2.3.A Zariski’s Construction .............................158
2.3.B Cutkosky’s Construction...........................159
2.3.C Base Loci of Nef and Big Linear Series ..............164
2.3.D The Theorem of Campana and Peternell .............166
2.3.E Zariski Decompositions ............................167
2.4 Graded Linear Series and Families of Ideals.................172
2.4.A Graded Linear Series ..............................172
2.4.B Graded Families of Ideals ..........................176
Notes.......................................................183
3 Geometric Manifestations of Positivity ....................185
3.1 The Lefschetz Theorems..................................185
3.1.A Topology of Affine Varieties ........................186
3.1.B The Theorem on Hyperplane Sections ...............192
3.1.C Hard Lefschetz Theorem...........................199
3.2 Projective Subvarieties of Small Codimension ...............201
3.2.A Barth’s Theorem..................................201
3.2.B Hartshorne’s Conjectures ..........................204
3.3 Connectedness Theorems .................................207
3.3.A Bertini Theorems .................................207
3.3.B Theorem of Fulton and Hansen .....................210
3.3.C Grothendieck’s Connectedness Theorem .............212
3.4 Applications of the Fulton–Hansen Theorem ................213
3.4.A Singularities of Mappings ..........................214