The Geometry of Moduli Spaces of Sheaves

by Daniel Huybrechts , Manfred Lehn

• ISBN13: 9780521134200
• ISBN10: 052113420X
• Edition: 2nd
• Format: Paperback
• Copyright: 2010-07-05
• Publisher: Cambridge University Press

Summary

Now back in print, this highly regarded book has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces, which include moduli spaces in positive characteristic, moduli spaces of principal bundles and of complexes, Hilbert schemes of points on surfaces, derived categories of coherent sheaves, and moduli spaces of sheaves on Calabi-Yau threefolds. The authors review changes in the field since the publication of the original edition in 1997 and point the reader towards further literature. References have been brought up to date and errors removed. Developed from the authors' lectures, this book is ideal as a text for graduate students as well as a valuable resource for any mathematician with a background in algebraic geometry who wants to learn more about Grothendieck's approach.

Table of Contents

Preface to the second edition	p. ix
Preface to the first edition	p. xi
Introduction	p. xiii
General Theory	p. 1
Preliminaries	p. 3
Some Homological Algebra	p. 3
Semistable Sheaves	p. 10
The Harder-Narasimhan Filtration	p. 15
An Example	p. 20
Jordan-Hölder Filtration and S-Equivalence	p. 23
¿-Semistability	p. 25
Boundedness I	p. 29
Families of Sheaves	p. 34
Flat Families and Determinants	p. 34
Grothendieck's Quot-Scheme	p. 40
The Relative Harder-Narasimhan Filtration	p. 48
Appendix
Flag-Schemes and Deformation Theory	p. 51
A Result of Langton	p. 58
Further comments (second edition)	p. 61
The Grauert-Mülich Theorem	p. 63
Statement and Proof	p. 64
Finite Coverings and Tensor Products	p. 69
Boundedness II	p. 75
The Bogomolov Inequality	p. 79
Appendix
e-Stability and Some Estimates	p. 82
Further comments (second edition)	p. 87
Moduli Spaces	p. 89
The Moduli Functor	p. 90
Group Actions	p. 91
The Construction---Results	p. 98
The Construction---Proofs	p. 104
Local Properties and Dimension Estimates	p. 112
Universal Families	p. 117
Appendix
Gieseker's Construction	p. 121
Decorated Sheaves	p. 122
Change of Polarization	p. 126
Further comments (second edition)	p. 132
Sheaves on Surfaces	p. 141
Construction Methods	p. 143
The Serre Correspondence	p. 145
Elementary Transformations	p. 151
Examples of Moduli Spaces	p. 154
Appendix
Further comments (second edition)	p. 164
Moduli Spaces on K3 Surfaces	p. 166
Low-Dimensional...	p. 167
...and Higher-Dimensional Moduli Spaces	p. 175
Appendix
The Irreducibility of the Quot-scheme	p. 184
Further comments (second edition)	p. 187
Restriction of Sheaves to Curves	p. 193
Flenner's Theorem	p. 193
The Theorems of Mehta and Ramanathan	p. 197
Bogomolov's Theorems	p. 204
Appendix
Further comments (second edition)	p. 212
Line Bundles on the Moduli Space	p. 213
Construction of Determinant Line Bundles	p. 213
A Moduli Space for ¿-Semistable Sheaves	p. 220
The Canonical Class of the Moduli Space	p. 232
Further comments (second edition)	p. 236
Irreducibility and Smoothness	p. 239
Preparations	p. 239
The Boundary	p. 241
Generic Smoothness	p. 242
Irreducibility	p. 243
Proof of Theorem 9.2.2	p. 245
Proof of Theorem 9.3.2	p. 251
Symplectic Structures	p. 255
Trace Map, Atiyah Class and Kodaira-Spencer Map	p. 256
The Tangent Bundle	p. 262
Forms on the Moduli Space	p. 264
Non-Degeneracy of Two-Forms	p. 267
Appendix
Further comments (second edition)	p. 271
Birational properties	p. 272
Kodaira Dimension of Moduli Spaces	p. 272
More Results	p. 277
Examples	p. 281
Appendix
Further comments (second edition)	p. 287
References	p. 290
Glossary of Notations	p. 316
Index	p. 321
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