Fourier-Mukai transforms in algebraic geometry

Fourier-Mukai transforms in algebraic geometry / D. Huybrechts.

This work is based on a course given at the Institut de Mathematiques de Jussieu, on the derived category of coherent sheaves on a smooth projective variety. It is aimed at students with a basic knowledge of algebraic geometry and contains full proofs and exercises that aid the reader.

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Bibliographic Details
Main Author: Huybrechts, Daniel (Author)
Format: eBook
Language:English
Published: Oxford : Clarendon Press, 2006.
Series:Oxford mathematical monographs.
Subjects:
Fourier transformations.
Geometry, Algebraic.
MATHEMATICS > Geometry > Algebraic.
Fourier transformations
Geometry, Algebraic
Online Access:Click for online access
Click for online access
Table of Contents:
  • Contents
  • 1 Triangulated categories
  • 1.1 Additive categories and functors
  • 1.2 Triangulated categories and exact functors
  • 1.3 Equivalences of triangulated categories
  • 1.4 Exceptional sequences and orthogonal decompositions
  • 2 Derived categories: a quick tour
  • 2.1 Derived category of an abelian category
  • 2.2 Derived functors
  • 2.3 Spectral sequences
  • 3 Derived categories of coherent sheaves
  • 3.1 Basic structure
  • 3.2 Spanning classes in the derived category
  • 3.3 Derived functors in algebraic geometry
  • 3.4 Grothendieckâ€?Verdier duality
  • 4 Derived category and canonical bundle â€? I4.1 Ample (anti- )canonical bundle
  • 4.2 Autoequivalences for ample (anti- )canonical bundle
  • 4.3 Ample sequences in derived categories
  • 5 Fourierâ€?Mukai transforms
  • 5.1 What it is and Orlovâ€?s result
  • 5.2 Passage to cohomology
  • 6 Derived category and canonical bundle â€? II
  • 6.1 Kodaira dimension under derived equivalence
  • 6.2 Geometrical aspects of the Fourierâ€?Mukai kernel
  • 6.3 Nefness under derived equivalence
  • 6.4 Derived equivalence versus birationality
  • 6.5 Recap: Kodaira dimension, canonical ring, etc. 7 Equivalence criteria for Fourierâ€?Mukai transforms
  • 7.1 Fully faithful
  • 7.2 Equivalences
  • 7.3 Canonical quotients
  • 8 Spherical and exceptional objects
  • 8.1 Autoequivalences induced by spherical objects
  • 8.2 Braid group actions
  • 8.3 Beilinson spectral sequence
  • 8.4 They go together
  • 9 Abelian varieties
  • 9.1 Basic definitions and facts
  • 9.2 The Poincaré bundle as a Fourierâ€?Mukai kernel
  • 9.3 Sl[Sub(2)]-action
  • 9.4 Derived equivalences of abelian varieties
  • ""9.5 Autoequivalences of abelian varieties""""10 K3 surfaces""; ""10.1 Recap: K3 surfaces""; ""10.2 Derived equivalence of K3 surfaces""; ""10.3 Recap: Moduli spaces of sheaves""; ""11 Flips and flops""; ""11.1 Preparations: Closed embeddings and blow-ups""; ""11.2 Derived categories under blow-up""; ""11.3 The standard flip""; ""11.4 The Mukai flop""; ""12 Derived categories of surfaces""; ""12.1 Recap: Enriques classification of algebraic surfaces""; ""12.2 Minimal surfaces with kod = â€?∞, 2""; ""12.3 Surfaces with torsion canonical bundle""; ""12.4 Properly elliptic surfaces""
  • 13 Where to go from here13.1 McKay correspondence for derived categories
  • 13.2 Homological mirror symmetry
  • 13.3 D-branes and stability conditions
  • 13.4 Twisted derived categories
  • References
  • Index
  • A
  • B
  • C
  • D
  • E
  • F
  • G
  • H
  • I
  • K
  • L
  • M
  • O
  • P
  • Q
  • R
  • S
  • T
  • Y

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