Fourier-Mukai Transforms in Algebraic Geometry

by Huybrechts, Daniel

• ISBN13: 9780199296866
• ISBN10: 0199296863
• Format: Hardcover
• Copyright: 2006-06-29
• Publisher: Clarendon Press

Summary

This seminal text on Fourier-Mukai Transforms in Algebraic Geometry by a leading researcher and expositor is based on a course given at the Institut de Mathematiques de Jussieu in 2004 and 2005. Aimed at postgraduate students with a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. Including notions from other areas, e.g. singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs are given and exercises aid the reader throughout.

Table of Contents

Triangulated categories	1	(26)
Additive categories and functors	1	(10)
Triangulated categories and exact functors	11	(8)
Equivalences of triangulated categories	19	(5)
Exceptional sequences and orthogonal decompositions	24	(3)
Derived categories: a quick tour	27	(35)
Derived category of an abelian category	27	(16)
Derived functors	43	(9)
Spectral sequences	52	(10)
Derived categories of coherent sheaves	62	(28)
Basic structure	62	(7)
Spanning classes in the derived category	69	(2)
Derived functors in algebraic geometry	71	(15)
Grothendieck--Verdier duality	86	(4)
Derived category and canonical bundle -- I	90	(23)
Ample (anti-)canonical bundle	91	(8)
Autoequivalences for ample (anti-)canonical bundle	99	(2)
Ample sequences in derived categories	101	(12)
Fourier--Mukai transforms	113	(23)
What it is and Orlov's result	113	(10)
Passage to cohomology	123	(13)
Derived category and canonical bundle - II	136	(18)
Kodaira dimension under derived equivalence	136	(5)
Geometrical aspects of the Fourier--Mukai kernel	141	(4)
Nefness under derived equivalence	145	(2)
Derived equivalence versus birationality	147	(4)
Recap: Kodaira dimension, canonical ring, etc.	151	(3)
Equivalence criteria for Fourier--Mukai transforms	154	(12)
Fully faithful	154	(5)
Equivalences	159	(3)
Canonical quotients	162	(4)
Spherical and exceptional objects	166	(26)
Autoequivalences induced by spherical objects	166	(9)
Braid group actions	175	(4)
Beilinson spectral sequence	179	(6)
They go together	185	(7)
Abelian varieties	192	(36)
Basic definitions and facts	192	(9)
The Poincare bundle as a Fourier-Mukai kernel	201	(3)
Sl2-action	204	(5)
Derived equivalences of abelian varieties	209	(14)
Autoequivalences of abelian varieties	223	(5)
K3 surfaces	228	(18)
Recap: K3 surfaces	228	(4)
Derived equivalence of K3 surfaces	232	(8)
Recap: Moduli spaces of sheaves	240	(6)
Flips and flops	246	(26)
Preparations: Closed embeddings and blow-ups	246	(8)
Derived categories under blow-up	254	(4)
The standard flip	258	(5)
The Mukai flop	263	(9)
Derived categories of surfaces	272	(14)
Recap: Enriques classification of algebraic surfaces	273	(5)
Minimal surfaces with kod = ∞	278	(2)
Surfaces with torsion canonical bundle	280	(1)
Properly elliptic surfaces	281	(5)
Where to go from here	286	(13)
McKay correspondence for derived categories	286	(3)
Homological mirror symmetry	289	(2)
D-branes and stability conditions	291	(4)
Twisted derived categories	295	(4)
References	299	(6)
Index	305