Current Topics in Complex Algebraic Geometry

✍ Scribed by Herbert Clemens, Janos Kollár

• Publisher: Cambridge University Press
• Year: 1996
• Tongue: English
• Leaves: 158
• Series: Mathematical Sciences Research Institute Publications 28
• 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......Page 2
Fundamental Groups of Smooth Projective Varieties......Page 5
1. Positive results......Page 6
2. Simple obstructions......Page 8
3. Groups with more than one end......Page 9
4. Rational homotopy......Page 11
5. Representation varieties......Page 13
6. Lattices in Lie groups......Page 14
7. Maps to curves......Page 15
8. Complex manifolds......Page 17
Introduction......Page 21
1. The moduli space......Page 22
2. The determinant bundle......Page 23
3. Base points......Page 24
4. Rank 2......Page 25
5. The Verlinde formula......Page 27
6. The Verlinde formula: finite-dimensional proofs......Page 28
7. The Verlinde formula: infinite-dimensional proofs......Page 30
8. The strange duality......Page 32
9. The projective connection......Page 33
10. Are there generalized theta functions?......Page 35
1. Introduction......Page 38
2. Notation, Minimal Models, etc.......Page 39
3. Semistable Flips......Page 43
4. Birational theory of Mori fibrations......Page 49
5. Log abundance......Page 53
6. Effective base point freeness......Page 56
Introduction......Page 60
2. The geometrical approach......Page 61
1. Introduction......Page 68
2. Hitchin's system......Page 69
3. Some related systems......Page 71
4.1. The question......Page 73
4.2. Decomposition of spectral covers.......Page 75
4.3. Decomposition of spectral Picards.......Page 76
4.4. The distinguished Prym.......Page 77
5.1. Abstract versus K-valued objects......Page 78
5.2. The regular semisimple case: the shift......Page 79
5.3. The regular case: the twist along the ramification......Page 81
5.4. Adding values and representations.......Page 83
6. Symplectic and Poisson structures......Page 85
7. Some applications and problems......Page 86
Adjoint Linear Systems......Page 90
1. Introduction......Page 99
2. Mapping class groups and moduli......Page 101
3. The Johnson homomorphism......Page 105
4. A second definition of the Johnson homomorphism......Page 109
5. Picard groups......Page 114
6. Normal functions......Page 118
7. Extending normal functions......Page 120
8. Normal functions over M(L)......Page 121
9. Technical results on variations over M_g......Page 125
10. Normal functions and cycles mod algebraic equivalence......Page 127
11. The Harris-Pulte theorem......Page 129
12. The Franchetta conjecture for curves with a level......Page 130
13. The monodromy of roots of the canonical bundle......Page 132
14. Heights of Cycles defined over M_g(L)......Page 135
15. Results for Abelian Varieties......Page 141
1. Line bundles......Page 145
2. Brill-Noether theory......Page 147
3. Vector bundles on a curve......Page 150
4. Toward a Brill-Noether type theory for two-bundles......Page 153