American Mathematical SocietyInstitute for Advanced Study

IAS/PARK CITYMATHEMATICS SERIES

Volume 17

Analytic and Algebraic Geometry

Common Problems, Different Methods

Jeffery McNealMircea Mustata

Editors

Analytic and Algebraic Geometry

Common Problems, Different Methods

American Mathematical SocietyInstitute for Advanced Study

IAS/PARK CITYMATHEMATICS SERIES

Volume 17

Analytic and Algebraic Geometry

Common Problems, Different Methods

Jeffery McNeal Mircea Mustata

Editors

https://doi.org/10.1090//pcms/017

John C. Polking, Series EditorEzra Miller, Volume Editor

Victor Reiner, Volume EditorBernd Sturmfels, Volume Editor

IAS/Park City Mathematics Institute runs mathematics education programs that bringtogether high school mathematics teachers, researchers in mathematics and mathematicseducation, undergraduate mathematics faculty, graduate students, and undergraduates toparticipate in distinct but overlapping programs of research and education. This volumecontains the lecture notes from the Graduate Summer School program on Analytic andAlgebraic Geometry: Common Problems, Different Methods held in Park City, Utah inthe summer of 2008.

2000 Mathematics Subject Classification. Primary 14E15, 14E30, 14F18, 32W05, 53C21.

Library of Congress Cataloging-in-Publication Data

Analytic and algebraic geometry : common problems, different methods / Jeffery McNeal, MirceaMustata, editors.

p. cm. — (IAS/Park City mathematics series, ISSN 1079-5634 ; v. 17)Includes bibliographical references.ISBN 978-0-8218-4908-8 (alk. paper)1. Geometry, Analytic. 2. Geometry, Algebraic. I. McNeal, Jeffery, D. II. Mustata, Mircea,

1971–

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10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10

Contents

Preface

Jeffery McNeal and Mircea MustataIntroduction 1

Bo BerndtssonAn Introduction to Things ∂ 7

Introduction 9

Lecture 1. The one-dimensional case 111.1. The ∂-equation in one variable 111.2. An alternative proof of the basic identity 141.3. An application: Inequalities of Brunn-Minkowski type 141.4. Regularity — a disclaimer 16

Lecture 2. Functional analytic interlude 192.1. Dual formulation of the ∂-problem 19

Lecture 3. The ∂-equation on a complex manifold 253.1. Metrics 253.2. Norms of forms 273.3. Line bundles 293.4. Calculation of the adjoint and the basic identity 333.5. The main existence theorem and L2-estimate for compact manifolds 353.6. Complete Kahler manifolds 37

Lecture 4. The Bergman kernel 434.1. Generalities 434.2. Bergman kernels associated to complex line bundles 46

Lecture 5. Singular metrics and the Kawamata-Viehweg vanishing theorem 515.1. The Demailly-Nadel vanishing theorem 515.2. The Kodaira embedding theorem 545.3. The Kawamata-Viehweg vanishing theorem 55

Lecture 6. Adjunction and extension from divisors 596.1. Adjunction and the currents defined by divisors 596.2. The Ohsawa-Takegoshi extension theorem 62

Lecture 7. Deformational invariance of plurigenera 717.1. Extension of pluricanonical forms 71

Bibliography 75

v

xiii

vi CONTENTS

John P. D’AngeloReal and Complex Geometry meet the Cauchy-Riemann Equations 77

Preface 79

Lecture 1. Background material 811. Complex linear algebra 812. Differential forms 823. Solving the Cauchy-Riemann equations 84

Lecture 2. Complex varieties in real hypersurfaces 871. Degenerate critical points of smooth functions 872. Hermitian symmetry and polarization 893. Holomorphic decomposition 904. Real analytic hypersurfaces and subvarieties 985. Complex varieties, local algebra, and multiplicities 99

Lecture 3. Pseudoconvexity, the Levi form, and points of finite type 1051. Euclidean convexity 1052. The Levi form 1073. Higher order commutators 1114. Points of finite type 1135. Commutative algebra 1166. A return to finite type 1217. The set of finite type points is open 126

Lecture 4. Kohn’s algorithm for subelliptic multipliers 1291. Introduction 1292. Subelliptic estimates 1303. Kohn’s algorithm 1334. Kohn’s algorithm for holomorphic and formal germs 1345. Failure of effectiveness for Kohn’s algorithm 1396. Triangular systems 1407. Additional remarks 144

Lecture 5. Connections with partial differential equations 1471. Finite type conditions 1472. Local regularity for ∂ 1493. Hypoellipticity, global regularity, and compactness 1504. An introduction to L2-estimates 152

Lecture 6. Positivity conditions 1571. Introduction 1572. The classes Pk 1583. Intermediate conditions 1594. The global Cauchy-Schwarz inequality 1615. A complicated example 1646. Stabilization in the bihomogeneous polynomial case 1667. Squared norms and proper mappings between balls 1718. Holomorphic line bundles 173

CONTENTS vii

Lecture 7. Some open problems 175

Bibliography 177

Dror VarolinThree Variations on a Theme in Complex Analytic Geometry 183

Lecture 0. Basic notions in complex geometry 1891. Complex manifolds 1892. Connections 1993. Curvature 2074. Holomorphic line bundles 210

Lecture 1. The Hormander theorem 2171. Functional analysis 2182. The Bochner-Kodaira identity 2193. Manifolds with boundary 2264. Density of smooth forms in the graph norm 2295. Hormander’s theorem 2346. Singular Hermitian metrics for line bundles 2367. Application: Kodaira embedding theorem 2398. Multiplier ideal sheaves and Nadel’s Theorems 2429. Exercises 248

Lecture 2. The L2 extension theorem 2511. L2 extension 2512. The deformation invariance of plurigenera 2593. Pluricanonical extension on projective manifolds 2654. Exercises 275

Lecture 3. The Skoda division theorem 2771. Statement of the division theorem 2772. Proof of the division theorem 2783. Global generation of multiplier ideal sheaves 2864. Exercises 290

Bibliography 293

Jean-Pierre DemaillyStructure Theorems for Projective and Kahler Varieties 295

0. Introduction 297

1. Numerically effective and pseudo-effective (1,1) classes 2981.A. Pseudo-effective line bundles and metrics with minimal singularities 2981.B. Nef line bundles 3001.C. Description of the positive cones 3021.D. The Kawamata-Viehweg vanishing theorem 3061.E. A uniform global generation property due to Y.T. Siu 3081.F. Hard Lefschetz theorem with multiplier ideal sheaves 309

viii CONTENTS

2. Holomorphic Morse inequalities 310

3. Approximation of closed positive (1,1)-currents by divisors 3123.A. Local approximation theorem through Bergman kernels 3123.B. Global approximation of closed (1,1)-currents on a compact complex

manifold 3143.C. Global approximation by divisors 3203.D. Singularity exponents and log canonical thresholds 326

4. Subadditivity of multiplier ideals and Fujita’s approximate Zariski329

5. Numerical characterization of the Kahler cone 3345.A. Positive classes in intermediate (p, p) bidegrees 3345.B. Numerically positive classes of type (1,1) 3355.C. Deformations of compact Kahler manifolds 341

6. Structure of the pseudo-effective cone and mobile intersection theory 3436.A. Classes of mobile curves and of mobile (n− 1, n− 1)-currents 3436.B. Zariski decomposition and mobile intersections 3466.C. The orthogonality estimate 3526.D. Dual of the pseudo-effective cone 354

7. Super-canonical metrics and abundance 3577.A. Construction of super-canonical metrics 3577.B. Invariance of plurigenera and positivity of curvature of super-canonical

metrics 3637.C. Tsuji’s strategy for studying abundance 364

8. Siu’s analytic approach and Paun’s non vanishing theorem 365

Bibliography 367

Mihai PaunLecture Notes on Rational Polytopes and Finite Generation 371

0. Introduction 373

1. Basic definitions and notations 374

2. Proof of (i) 3762.1. The case nd({KX + Yτ0 +A}) = 0 3772.2. The “x method” for sequences 3782.3. The induced polytope and its properties 382

3. Proof of (ii) 3903.1. The first step 3933.2. Iteration scheme 399

References 402

decomposition theorem

CONTENTS ix

Mircea MustataIntroduction to Resolution of Singularities 405

Lecture 1. Resolutions and principalizations 4091.1. The main theorems 4091.2. Strengthenings of Theorem 1.3 4101.3. Historical comments 414

Lecture 2. Marked ideals 4152.1. Marked ideals 4152.2. Derived ideals 419

Lecture 3. Hypersurfaces of maximal contact and coefficient ideals 4233.1. Hypersurfaces of maximal contact 4233.2. The coefficient ideal 424

Lecture 4. Homogenized ideals 4314.1. Basics of homogenized ideals 4314.2. Comparing hypersurfaces of maximal contact: formal equivalence 4334.3. Comparing hypersurfaces of maximal contact: etale equivalence 434

Lecture 5. Proof of principalization 4375.1. The statements 4375.2. Part I: the maximal order case 4385.3. Part II: the general case 4425.4. Proof of principalization 445

Bibliography 449

Robert LazarsfeldA Short Course on Multiplier Ideals 451

Introduction 453

Lecture 1. Construction and examples of multiplier ideals 455Definition of multiplier ideals 455Monomial ideals 459Invariants defined by multiplier ideals 460

Lecture 2. Vanishing theorems for multiplier ideals 463The Kawamata–Viehweg–Nadel vanishing theorem 463Singularities of plane curves and projective hypersurfaces 465Singularities of theta divisors 467Uniform global generation 468

Lecture 3. Local properties of multiplier ideals 471Adjoint ideals and the restriction theorem 471The subadditivity theorem 474Skoda’s theorem 475

x CONTENTS

Lecture 4. Asymptotic constructions 479A ymptotic multiplier idealsVariants 482Etale multiplicativity of plurigenera 484A comparison theorem for symbolic powers 485

Lecture 5. Extension theorems and deformation invariance of plurigenera 487

Bibliography 493

Janos KollarExercises in the Birational Geometry of Algebraic Varieties 495

1. Birational classification of algebraic surfaces 497

2. Naive minimal models 498

3. The cone of curves 503

4. Singularities 508

5. Flips 513

6. Minimal models 518

Bibliography 523

Christopher D. HaconHigher Dimensional Minimal Model Program for Varieties of LogGeneral Type 525

Introduction 527

Lecture 1. Pl-flips 531

Lecture 2. Multiplier ideal sheaves 535Asymptotic multiplier ideal sheaves 538Extending pluricanonical forms 540

Lecture 3. Finite generation of the restricted algebra 545Rationality of the restricted algebra 545Proof of (1.10) 546

Lecture 4. The minimal model program with scaling 547

Solutions to the exercises 551

Bibliography 555

Alessio Corti, Paul Hacking, Janos Kollar, Robert Lazarsfeld, andMircea MustataLectures on Flips and Minimal Models 557

Lecture 1. Extension theorems 5611.1. Multiplier and adjoint ideals 5611.2. Proof of the Main Lemma 563

s 479

CONTENTS xi

Lecture 2. Existence of flips I 5652.1. The setup 5652.2. Adjoint algebras 5662.3. The Hacon–McKernan extension theorem 5672.4. The restricted algebra as an adjoint algebra 567

Lecture 3. Existence of flips II 571

Lecture 4. Notes on Birkar-Cascini-Hacon-McKernan 5754.1. Comparison of 3 MMP’s 5764.2. MMP with scaling 5784.3. MMP with scaling near �Δ� 5784.4. Bending it like BCHM 5794.5. Finiteness of models 581

Bibliography 583

Preface

The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as partof the “Regional Geometry Institute” initiative of the National Science Foundation.In mid 1993 the program found an institutional home at the Institute for AdvancedStudy (IAS) in Princeton, New Jersey.

The IAS/Park City Mathematics Institute encourages both research and ed-ucation in mathematics and fosters interaction between the two. The three-weeksummer institute offers programs for researchers and postdoctoral scholars, gradu-ate students, undergraduate students, high school teachers, undergraduate faculty,and researchers in mathematics education. One of PCMI’s main goals is to makeall of the participants aware of the total spectrum of activities that occur in math-ematics education and research: we wish to involve professional mathematiciansin education and to bring modern concepts in mathematics to the attention ofeducators. To that end the summer institute features general sessions designedto encourage interaction among the various groups. In-year activities at the sitesaround the country form an integral part of the High School Teachers Program.

Each summer a different topic is chosen as the focus of the Research Programand Graduate Summer School. Activities in the Undergraduate Summer Schooldeal with this topic as well. Lecture notes from the Graduate Summer School arebeing published each year in this series. The first seventeen volumes are:

• Volume 1: Geometry and Quantum Field Theory (1991)• Volume 2: Nonlinear Partial Differential Equations in Differential Geom-etry (1992)

• Volume 3: Complex Algebraic Geometry (1993)• Volume 4: Gauge Theory and the Topology of Four-Manifolds (1994)• Volume 5: Hyperbolic Equations and Frequency Interactions (1995)• Volume 6: Probability Theory and Applications (1996)• Volume 7: Symplectic Geometry and Topology (1997)• Volume 8: Representation Theory of Lie Groups (1998)• Volume 9: Arithmetic Algebraic Geometry (1999)• Volume 10: Computational Complexity Theory (2000)• Volume 11: Quantum Field Theory, Supersymmetry, and EnumerativeGeometry (2001)

• Volume 12: Automorphic Forms and their Applications (2002)• Volume 13: Geometric Combinatorics (2004)• Volume 14: Mathematical Biology (2005)• Volume 15: Low Dimensional Topology (2006)• Volume 16: Statistical Mechanics (2007)• Volume 17: Analytic and Algebraic Geometry: Common Problems, Dif-ferent Methods (2008)

Volumes are in preparation for subsequent years.Some material from the Undergraduate Summer School is published as part

of the Student Mathematical Library series of the American Mathematical Soci-ety. We hope to publish material from other parts of the IAS/PCMI in the future.

xiii

xiv PREFACE

This will include material from the High School Teachers Program and publicationsdocumenting the interactive activities which are a primary focus of the PCMI. Atthe summer institute late afternoons are devoted to seminars of common interestto all participants. Many deal with current issues in education: others treat math-ematical topics at a level which encourages broad participation. The PCMI hasalso spawned interactions between universities and high schools at a local level. Wehope to share these activities with a wider audience in future volumes.

John C. PolkingSeries EditorJuly 2010

Titles in This Series

17 Jeffery McNeal and Mircea Mustata, Editors, Analytic and Algebraic Geometry,2010

16 Scott Sheffield and Thomas Spencer, Editors, Statistical Mechanics, 2009

15 Tomasz S. Mrowka and Peter S. Ozsvath, Editors, Low Dimensional Topology, 2009

14 Mark A. Lewis, Mark A. J. Chaplain, James P. Keener, and Philip K. Maini,Editors, Mathematical Biology, 2009

13 Ezra Miller, Victor Reiner, and Bernd Sturmfels, Editors, GeometricCombinatorics, 2007

12 Peter Sarnak and Freydoon Shahidi, Editors, Automorphic Forms and Applications,2007

11 Daniel S. Freed, David R. Morrison, and Isadore Singer, Editors, Quantum FieldTheory, Supersymmetry, and Enumerative Geometry, 2006

10 Steven Rudich and Avi Wigderson, Editors, Computation Complexity Theory, 2004

9 Brian Conrad and Karl Rubin, Editors, Arithmetic Algebraic Geometry, 2001

8 Jeffrey Adams and David Vogan, Editors, Representation Theory of Lie Groups, 2000

7 Yakov Eliashberg and Lisa Traynor, Editors, Symplectic Geometry and Topology,1999

6 Elton P. Hsu and S. R. S. Varadhan, Editors, Probability Theory and Applications,1999

5 Luis Caffarelli and Weinan E, Editors, Hyperbolic Equations and FrequencyInteractions, 1999

4 Robert Friedman and John W. Morgan, Editors, Gauge Theory and the Topology ofFour-Manifolds, 1998

3 Janos Kollar, Editor, Complex Algebraic Geometry, 1997

2 Robert Hardt and Michael Wolf, Editors, Nonlinear Partial Differential Equations inDifferential Geometry, 1996

1 Daniel S. Freed and Karen K. Uhlenbeck, Editors, Geometry and Quantum FieldTheory, 1995

www.ams.orgAMS on the WebPCMS/17

Analytic and algebraic geometers often study the same geometric structures but bring different methods to bear on them. While this dual approach has been spectacularly successful at solving problems, the language differences between algebra and analysis also represent a difficulty for students and researchers in geometry, particularly complex geometry.

The PCMI program was designed to partially address this language gulf, by presenting some of the active developments in algebraic and analytic geometry in a form suitable for students on the “other side” of the analysis-algebra language divide. One focal point of the summer school was multiplier ideals, a subject of wide current interest in both subjects.

The present volume is based on a series of lectures at the PCMI summer school on analytic and algebraic geometry. The series are designed to give a high-level introduc-tion to the advanced techniques behind some recent developments in algebraic and analytic geometry. The lectures contain many illustrative examples, detailed computa-tions, and new perspectives on the topics presented, in order to enhance access of this material to non-specialists.