Fourier-Mukai and Nahm transforms in geometry and mathematical physics
type: Item
instance of: scholarly article
title: Fourier-Mukai and Nahm transforms in geometry and mathematical physics
zbMATH Open document ID: 1186.14001
author: Claudio Bartocci
author: Ugo Bruzzo
author: Daniel Hernandez Ruiperez
published in: Progress in Mathematics
publication date: 2009-09-22T00:00:00Z
review text: The main aim of the book under review is to study a class of functors between derived categories of coherent sheaves of smooth varieties, known as integral (or, in some cases, Fourier-Mukai) functors. Recently, this subject is rapidly developing and the book under review contains a valuable survey of the known results. The book contains 7 chapters and 4 appendices. The authors essentially assume that the reader is familiar with homological algebra at the level of the classical book by \textit{S. Gelfand} and \textit{Yu. I. Manin} [Methods of homological algebra. Berlin: Springer (2003; Zbl 1006.18001)]. But to make the book more self-contained, basic notions and results of homological algebra are quickly recalled in Appendix A written by F. Sancho. Chapter 1 contains definition of integral functors and their first basic properties (also in the equivariant case). Most of Chapter 2 is devoted to the proof of Orlov's theorem describing fully faithful exact functors between derived categories of coherent sheaves on smooth varieties as integral functors. It also contains characterization of Fourier-Mukai functors among all integral functors and Bondal and Orlov's reconstruction theorem for varieties with ample canonical or anticanonical divisor. In Chapter 3 the authors study Fourier-Mukai transforms on abelian varieties (in characteristic zero). Many but not all results from this chapter can be found in \textit{A. Polishchuk}'s book [Abelian varieties, theta functions, and the Fourier transform. Cambridge: Cambridge University Press (2003; Zbl 1018.14016)]. In particular, the authors study in some cases behaviour of slope stability under integral transforms. Chapter 4 is devoted to Fourier-Mukai transforms on complex \(K3\) surfaces. The main part of this chapter is devoted to construction of Fourier-Mukai transforms on reflexive surfaces. In Chapter 5 the authors try to justify the words ``Nahm transforms'' in the title of the book but contrary to other chapters, this …
reviewed by: Adrian Langer
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 53C07
zbMATH DE Number: 5605336
zbMATH Keywords: coherent sheaves
zbMATH Keywords: derived category
zbMATH Keywords: Fourier-Mukai functor
zbMATH Keywords: abelian varieties
zbMATH Keywords: \(K3\) surfaces
zbMATH Keywords: elliptic fibration
zbMATH Keywords: Nahm transform
zbMATH Keywords: stability conditions
MaRDI profile type: Publication
full work available at URL: b11801
OpenAlex ID: W4230631591
DOI: 10.1007/B11801
description: scientific article; zbMATH DE number 5605336