Current Topics in Complex Algebraic Geometry

✍ Scribed by Herbert Clemens (editor), Janos Kollár (editor)

• Publisher: Cambridge University Press
• Year: 1996
• Tongue: English
• Leaves: 158
• Series: Mathematical Sciences Research Institute Publications; 28
• Edition: 1
• Category: Library

✦ Synopsis

This volume collects a series of survey articles on complex algebraic geometry, which in the early 1990s was undergoing a major change. Algebraic geometry has opened up to ideas and connections from other fields that have traditionally been far away. This book gives a good idea of the intellectual content of the change of direction and branching out witnessed by algebraic geometry in the past few years.

✦ Table of Contents

Preface
Fundamental Groups of Smooth Projective Varieties
1. Positive results
2. Simple obstructions
3. Groups with more than one end
4. Rational homotopy
5. Representation varieties
6. Lattices in Lie groups
7. Maps to curves
8. Complex manifolds
Vector Bundles on Curves and Generalized Theta Functions: Recent Results and Open Problems
Introduction
1. The moduli space
2. The determinant bundle
3. Base points
4. Rank 2
5. The Verlinde formula
6. The Verlinde formula: finite-dimensional proofs
7. The Verlinde formula: infinite-dimensional proofs
8. The strange duality
9. The projective connection
10. Are there generalized theta functions?
Recent Results in Higher-Dimensional Birational Geometry
Contents
1. Introduction
2. Notation, Minimal Models, etc.
3. Semistable Flips
4. Birational theory of Mori fibrations
5. Log abundance
6. Effective base point freeness
The Schottky Problem: An Update
Introduction
1. The analytical approach
2. The geometrical approach
Spectral Covers
1. Introduction
2. Hitchin's system
3. Some related systems
4. Decomposition of spectral Picards
4.1. The question
4.2. Decomposition of spectral covers.
4.3. Decomposition of spectral Picards.
4.4. The distinguished Prym.
5. Abelianization
5.1. Abstract versus K-valued objects
5.2. The regular semisimple case: the shift
5.3. The regular case: the twist along the ramification
5.4. Adding values and representations.
6. Symplectic and Poisson structures
7. Some applications and problems
Adjoint Linear Systems
Torelli Groups and Geometry of Moduli Spaces of Curves
1. Introduction
2. Mapping class groups and moduli
3. The Johnson homomorphism
4. A second definition of the Johnson homomorphism
5. Picard groups
6. Normal functions
7. Extending normal functions
8. Normal functions over M(L)
9. Technical results on variations over M_g
10. Normal functions and cycles mod algebraic equivalence
11. The Harris-Pulte theorem
12. The Franchetta conjecture for curves with a level
13. The monodromy of roots of the canonical bundle
14. Heights of Cycles defined over M_g(L)
15. Results for Abelian Varieties
Vector Bundles and Brill-Noether Theory
1. Line bundles
2. Brill-Noether theory
3. Vector bundles on a curve
4. Toward a Brill-Noether type theory for two-bundles