Fourier-Mukai transforms in algebraic geometry.

*(English)*Zbl 1095.14002
Oxford Mathematical Monographs. Oxford Science Publications. Oxford: Clarendon Press (ISBN 0-19-929686-3/hbk). viii, 307 p. (2006).

The present book is based on a course given by the author at the Institut de Mathématiques de Jussieu in Paris in 2004 and 2005. The aim of the course was to introduce graduate students, with a standard knowledge in algebraic geometry, to the main topics about bounded derived categories of coherent sheaves on smooth projective varieties. The book is a good reference for anyone who wants to deal with the subject. Anyway, a minimal knowledge in algebraic geometry is required.

Derived categories were defined by Verdier in order to give the right framework to any kind of derived functor, but they were initially thought as formal objects and they were not investigated as an interesting invariant of smooth projective varieties. Mukai was the first to show a geometrically motivated equivalence between two derived categories of non isomorphic varieties. In his work he introduced a class of functors which have become one of the most powerful tools in managing derived categories and they are now known as Fourier-Mukai transforms. Moreover, the homological mirror symmetry conjecture, stated by Kontsevich, involves directly derived categories and its statement made the interest in such objects grow outside the algebraic geometry framework.

Right now, many things are known on the subject, but a lot of work has still to be done. This book is essential to know the most important results and their relevance in algebraic geometry; almost every proof is given in full detail and exercises make the reader gain a working knowledge. It is a very good starting point to explore open problems related to derived categories, such as for example moduli space problems and birational classification.

The book is organised as follows:

Chapters 1, 2 and 3 explain the foundational material, introducing additive and triangulated categories, exact functors, triangulated autoequivalences, exceptional objects and semiorthogonal decompositions (chapter 1); the derived category of an abelian category, derived functors and spectral sequences (chapter 2); the derived category of coherent sheaves on a scheme and its connections with algebraic geometry, such as derived functors and various formulas and Grothendieck-Serre duality (chapter 3). A little knowledge in algebraic geometry is required.

Chapters 4 to 8 give the main known results on the subject: autoequivalences of derived categories in the ample (anti-)canonical bundle case, point-like objects and ample sequences (chapter 4); Fourier-Mukai transforms and their passage to \(K\)-theory and cohomology (chapter 5); geometrical aspects of such transforms, Kodaira dimension and nefness under derived equivalence, connections between derived equivalence and birationality (chapter 6); equivalence criteria for Fourier-Mukai transforms (chapter 7); spherical and exceptional objects, autoequivalences and braid group actions (chapter 8). In this part, a deeper knowledge of algebraic geometry could help the reader to see these results in a greater picture, but a standard knowledge is enough to understand them.

Chapters 9 to 12 detail more specific cases in which many results are already known: abelian varieties and their autoequivalences, focusing especially on the Poincaré bundle as a Fourier-Mukai kernel and on the SL\(_2\) action (chapter 9); derived equivalences of \(K3\) surfaces and moduli spaces of sheaves (chapter 10); birational transformations as blow-ups, flips and flops (chapter 11); derived categories of surfaces (chapter 12). Any of these four chapters comes with a brief well referenced introduction on the subject in order to make it almost self-contained.

The last chapter (chapter 13) introduces briefly and gives the most recent results on the topics involving open problems on derived categories: McKay correspondence, homological mirror symmetry, D-branes and stability conditions, twisted derived categories.

Derived categories were defined by Verdier in order to give the right framework to any kind of derived functor, but they were initially thought as formal objects and they were not investigated as an interesting invariant of smooth projective varieties. Mukai was the first to show a geometrically motivated equivalence between two derived categories of non isomorphic varieties. In his work he introduced a class of functors which have become one of the most powerful tools in managing derived categories and they are now known as Fourier-Mukai transforms. Moreover, the homological mirror symmetry conjecture, stated by Kontsevich, involves directly derived categories and its statement made the interest in such objects grow outside the algebraic geometry framework.

Right now, many things are known on the subject, but a lot of work has still to be done. This book is essential to know the most important results and their relevance in algebraic geometry; almost every proof is given in full detail and exercises make the reader gain a working knowledge. It is a very good starting point to explore open problems related to derived categories, such as for example moduli space problems and birational classification.

The book is organised as follows:

Chapters 1, 2 and 3 explain the foundational material, introducing additive and triangulated categories, exact functors, triangulated autoequivalences, exceptional objects and semiorthogonal decompositions (chapter 1); the derived category of an abelian category, derived functors and spectral sequences (chapter 2); the derived category of coherent sheaves on a scheme and its connections with algebraic geometry, such as derived functors and various formulas and Grothendieck-Serre duality (chapter 3). A little knowledge in algebraic geometry is required.

Chapters 4 to 8 give the main known results on the subject: autoequivalences of derived categories in the ample (anti-)canonical bundle case, point-like objects and ample sequences (chapter 4); Fourier-Mukai transforms and their passage to \(K\)-theory and cohomology (chapter 5); geometrical aspects of such transforms, Kodaira dimension and nefness under derived equivalence, connections between derived equivalence and birationality (chapter 6); equivalence criteria for Fourier-Mukai transforms (chapter 7); spherical and exceptional objects, autoequivalences and braid group actions (chapter 8). In this part, a deeper knowledge of algebraic geometry could help the reader to see these results in a greater picture, but a standard knowledge is enough to understand them.

Chapters 9 to 12 detail more specific cases in which many results are already known: abelian varieties and their autoequivalences, focusing especially on the Poincaré bundle as a Fourier-Mukai kernel and on the SL\(_2\) action (chapter 9); derived equivalences of \(K3\) surfaces and moduli spaces of sheaves (chapter 10); birational transformations as blow-ups, flips and flops (chapter 11); derived categories of surfaces (chapter 12). Any of these four chapters comes with a brief well referenced introduction on the subject in order to make it almost self-contained.

The last chapter (chapter 13) introduces briefly and gives the most recent results on the topics involving open problems on derived categories: McKay correspondence, homological mirror symmetry, D-branes and stability conditions, twisted derived categories.

Reviewer: Marcello Bernardara (Nice)

##### MSC:

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14D20 | Algebraic moduli problems, moduli of vector bundles |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

18E30 | Derived categories, triangulated categories (MSC2010) |