Symplectic 4-Manifolds and Algebraic Surfaces: Lectures Given at the C. I. M. E. Summer School Held in Cetraro, Italy, September 2-10, 2003

by Auroux, Denis; Catanese, Fabrizio; Manetti, Marco; Seidel, Paul; Siebert, Bernd; Smith, Ivan; Tian, Gang; Catanese, Fabrizio; Tian, Gang

• ISBN13: 9783540782780
• ISBN10: 3540782788
• Edition: 1st
• Format: Paperback
• Copyright: 2008-06-01
• Publisher: Springer Verlag

Summary

Modern approaches to the study of symplectic 4-manifolds and algebraic surfaces combine a wide range of techniques and sources of inspiration. Gauge theory, symplectic geometry, pseudoholomorphic curves, singularity theory, moduli spaces, braid groups, monodromy, in addition to classical topology and algebraic geometry, combine to make this one of the most vibrant and active areas of research in mathematics. It is our hope that the five lectures of the present volume given at the C.I.M.E. Summer School held in Cetraro, Italy, September 2-10, 2003 will be useful to people working in related areas of mathematics and will become standard references on these topics. The volume is a coherent exposition of an active field of current research focusing on the introduction of new methods for the study of moduli spaces of complex structures on algebraic surfaces, and for the investigation of symplectic topology in dimension 4 and higher.

Table of Contents

Lefschetz Pencils, Branched Covers and Symplectic Invariants	p. 1
Introduction and Background	p. 1
Symplectic Manifolds	p. 1
Almost-Complex Structures	p. 3
Pseudo-Holomorphic Curves and Gromov-Witten Invariants	p. 5
Lagrangian Floer Homology	p. 6
The Topology of Symplectic Four-Manifolds	p. 9
Symplectic Lefschetz Fibrations	p. 10
Fibrations and Monodromy	p. 10
Approximately Holomorphic Geometry	p. 17
Symplectic Branched Covers of <$>{\op CP}^2<$>	p. 22
Symplectic Branched Covers	p. 22
Monodromy Invariants for Branched Covers of <$>{\op CP}^2<$>	p. 26
Fundamental Groups of Branch Curve Complements	p. 30
Symplectic Isotopy and Non-Isotopy	p. 33
Symplectic Surfaces from Symmetric Products	p. 35
Symmetric Products	p. 35
Taubes' Theorem	p. 39
Fukaya Categories and Lefschetz Fibrations	p. 42
Matching Paths and Lagrangian Spheres	p. 43
Fukaya Categories of Vanishing Cycles	p. 44
Applications to Mirror Symmetry	p. 48
References	p. 50
Differentiable and Deformation Type of Algebraic Surfaces, Real and Symplectic Structures	p. 55
Introduction	p. 55
Lecture 1: Projective and Kähler Manifolds, the Enriques Classification, Construction Techniques	p. 57
Projective Manifolds, Kähler and Symplectic Structures	p. 57
The Birational Equivalence of Algebraic Varieties	p. 63
The Enriques Classification: An Outline	p. 65
Some Constructions of Projective Varieties	p. 66
Lecture 2: Surfaces of General Type and Their Canonical Models: Deformation Equivalence and Singularities	p. 70
Rational Double Points	p. 70
Canonical Models of Surfaces of General Type	p. 74
Deformation Equivalence of Surfaces	p. 82
Isolated Singularities, Simultaneous Resolution	p. 85
Lecture 3: Deformation and Diffeomorphism, Canonical Symplectic Structure for Surfaces of General Type	p. 91
Deformation Implies Diffeomorphism	p. 92
Symplectic Approximations of Projective Varieties with Isolated Singularities	p. 93
Canonical Symplectic Structure for Varieties with Ample Canonical Class and Canonical Symplectic Structure for Surfaces of General Type	p. 95
Degenerations Preserving the Canonical Symplectic Structure	p. 96
Lecture 4: Irrational Pencils, Orbifold Fundamental Groups, and Surfaces Isogenous to a Product	p. 98
Theorem of Castelnuovo-De Franchis, Irrational Pencils and the Orbifold Fundamental Group	p. 99
Varieties Isogenous to a Product	p. 105
Complex Conjugation and Real Structures	p. 108
Beauville Surfaces	p. 114
Lecture 5: Lefschetz Pencils, Braid and Mapping Class Groups, and Diffeomorphism of ABC-Surfaces	p. 116
Surgeries	p. 116
Braid and Mapping Class Groups	p. 119
Lefschetz Pencils and Lefschetz Fibrations	p. 125
Simply Connected Algebraic Surfaces: Topology Versus Differential Topology	p. 130
ABC Surfaces	p. 134
Epilogue: Deformation, Diffeomorphism and Symplectomorphism Type of Surfaces of General Type	p. 140
Deformations in the Large of ABC Surfaces	p. 141
Manetti Surfaces	p. 145
Deformation and Canonical Symplectomorphism	p. 152
Braid Monodromy and Chisini' Problem	p. 154
References	p. 159
Smoothings of Singularities and Deformation Types of Surfaces	p. 169
Introduction	p. 169
Deformation Equivalence of Surfaces	p. 174
Rational Double Points	p. 174
Quotient Singularities	p. 178
RDP-Deformation Equivalence	p. 181
Relative Canonical Model	p. 182
Automorphisms of Canonical Models	p. 183
The Kodaira Spencer Map	p. 184
Moduli Space for Canonical Surfaces	p. 187
Gieseker's Theorem	p. 188
Constructing Connected Components: Some Strategies	p. 189
Outline of Proof of Gieseker Theorem	p. 190
Smoothings of Normal Surface Singularities	p. 194
The Link of an Isolated Singularity	p. 194
The Milnor Fibre	p. 196
<$>{\op Q}<$>-Gorenstein Singularities and Smoothings	p. 197
T-Deformation Equivalence of Surfaces	p. 201
A Non Trivial Example of T-Deformation Equivalence	p. 203
Double and Multidouble Covers of Normal Surfaces	p. 204
Flat Abelian Covers	p. 204
Flat Double Covers	p. 205
Automorphisms of Generic Flat Double Covers	p. 207
Example: Automorphisms of Simple Iterated Double Covers	p. 209
Flat Multidouble Covers	p. 210
Stability Criteria for Flat Double Covers	p. 213
Restricted Natural Deformations of Double Covers	p. 214
Openess of N (a, b,c)	p. 217
RDP-Degenerations of Double Covers	p. 218
RDP-Degenerations of <$>{\op P}^1 \times {\op P}^1<$>	p. 221
Proof of Theorem 6.1	p. 222
Moduli of Simple Iterated Double Covers	p. 223
References	p. 225
Lectures on Four-Dimensional Dehn Twists	p. 231
Introduction	p. 231
Definition and First Properties	p. 235
Floer and Quantum Homology	p. 249
Pseudo-Holomorphic Sections and Curvature	p. 259
References	p. 265
Lectures on Pseudo-Holomorphic Curves and the Symplectic Isotopy Problem	p. 269
Introduction	p. 269
Pseudo-Holomorphic Curves	p. 270
Almost Complex and Symplectic Geometry	p. 270
Basic Properties of Pseudo-Holomorphic Curves	p. 272
Moduli Spaces	p. 273
Applications	p. 276
Pseudo-Analytic Inequalities	p. 279
Unobstructedness I: Smooth and Nodal Curves	p. 281
Preliminaries on the <$>\overline {\partial}<$>-Equation	p. 281
The Normal <$>\overline {\partial}<$>-Operator	p. 282
Immersed Curves	p. 286
Smoothings of Nodal Curves	p. 287
The Theorem of Micallef and White	p. 288
Statement of Theorem	p. 288
The Case of Tacnodes	p. 289
The General Case	p. 291
Unobstructedness II: The Integrable Case	p. 292
Motivation	p. 292
Special Covers	p. 292
Description of the Deformation Space	p. 294
The Holomorphic Normal Sheaf	p. 296
Computation of the Linearization	p. 299
A Vanishing Theorem	p. 300
The Unobstructedness Theorem	p. 301
Application to Symplectic Topology in Dimension Four	p. 302
Monodromy Representations - Hurwitz Equivalence	p. 303
Hyperelliptic Lefschetz Fibrations	p. 304
Braid Monodromy and the Structure of Hyperelliptic Lefschetz Fibrations	p. 307
Symplectic Noether-Horikawa Surfaces	p. 309
The <$>{\scr C}^0<$>-Compactness Theorem for Pseudo-Holomorphic Curves	p. 311
Statement of Theorem and Conventions	p. 311
The Monotonicity Formula for Pseudo-Holomorphic Maps	p. 312
A Removable Singularities Theorem	p. 315
Proof of the Theorem	p. 316
Second Variation of the <$>\overline {\partial}_J<$>-Equation and Applications	p. 320
Comparisons of First and Second Variations	p. 321
Moduli Spaces of Pseudo-Holomorphic Curves with Prescribed Singularities	p. 323
The Locus of Constant Deficiency	p. 324
Second Variation at Ordinary Cusps	p. 328
The Isotopy Theorem	p. 332
Statement of Theorem and Discussion	p. 332
Pseudo-Holomorphic Techniques for the Isotopy Problem	p. 333
The Isotopy Lemma	p. 334
Sketch of Proof	p. 336
References	p. 339
List of Participants	p. 343
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