Fourier-Mukai and Nahm transforms in geometry and mathematical physics
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 one uses analytic methods and it does not contain any proofs. Chapters 6 studies relative Fourier-Mukai functors and their application to (usually relative) moduli spaces of semistable sheaves on (usually Weierstrass) elliptic fibrations. Chapter 7 contains other applications of Fourier-Mukai transforms: classification of Fourier-Mukai partners of complex projective surfaces, Bridgeland's moduli space interpretation for smooth 3-dimensional flops and the derived version of McKay correspondence between the equivariant derived category and the derived category of a crepant resolution of a quotient. Appendices A, B and C contain auxiliary results. The last appendix by E. Macri contains a survey on Bridgeland's stability conditions for derived categories with special emphasis on stability conditions on \(K3\) surfaces. Each chapter finishes with useful comments containing history and indicating further development. There is another book of \textit{D. Huybrechts} [Fourier-Mukai transforms in algebraic geometry. Oxford: Clarendon Press (2006; Zbl 1095.14002)] on a very similar subject. When compared to it, the book under review requires more prerequisities and it is more technical. But to reward the reader it contains some difficult proofs that are skipped in [loc. cit.] (e.g., Kawamata's proof of Orlov's representability theorem). Unlike Huybrecht's book this one contains many results concerning moduli spaces of semistable sheaves, especially related to elliptic fibrations. When reading the book it is safer to assume that the base field has characteristic zero. Sometimes this assumption is hidden so well that it is difficult to spot (e.g., in Chapter 3 it appears at the beginning of 3.2 and lasts for the rest of the chapter) and sometimes it is necessary although the authors do not make this assumption (e.g., in positive characteristic Theorem C.6 is no longer true as stated). As every long book the book under review still contains some misprints (e.g., on p. 66 the inequality between numerical and usual Kodaira dimensions is in the opposite direction) but there are few of them and usually they are not so disturbing as the one mentioned above. In spite of these small omissions the book is very well written and it will certainly be very useful to researchers in algebraic geometry and mathematical physics.
scientific article; zbMATH DE number 5605336