Derived Categories in Algebraic Geometry [electronic resource] : Tokyo 2011 / Yujiro Kawamata

Categorical representability and intermediate Jacobians of Fano threefolds / Marcello Bernardara, Michele Bolognesi -- Fourier–Mukai functors: a survey / Alberto Canonaco, Paolo Stellari -- Flops and about: a guide / Sabin Cautis -- A note on derived categories of Fermat varieties / Akira Ishii, Kazushi Ueda -- Homology of infinite loop spaces / Dmitry Kaledin -- Cluster algebras and derived categories / Bernhard Keller -- Some derived equivalences between noncommutative schemes and algebras / Izuru Mori -- Lagrangian-invariant sheaves and functors for abelian varieties / Alexander Polishchuk -- Generic vanishing filtrations and perverse objects in derived categories of coherent sheaves / Mihnea Popa -- The fundamental group is not a derived invariant / Christian Schnell -- Introduction and open problems of Donaldson–Thomas theory / Yukinobu Toda -- Notes on formal deformations of abelian categories / Michel Van den Bergh.

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The study of derived categories is a subject that attracts increasingly many young mathematicians from various fields of mathematics, including abstract algebra, algebraic geometry, representation theory and mathematical physics. The concept of the derived category of sheaves was invented by Grothendieck and Verdier in the 1960s as a tool to express important results in algebraic geometry such as the duality theorem. In the 1970s, Beilinson, Gelfand and Gelfand discovered that a derived category of an algebraic variety may be equivalent to that of a finite dimensional non-commutative algebra, and Mukai found that there are non-isomorphic algebraic varieties that have equivalent derived categories. In this way the derived category provides a new concept that has many incarnations. In the 1990s, Bondal and Orlov uncovered an unexpected parallelism between derived categories and birational geometry. Kontsevich’s homological mirror symmetry provided further motivation for the study of derived categories. This book is the proceedings of a conference held at the University of Tokyo in January 2011 on the current status of the research on derived categories related to algebraic geometry. Most articles are survey papers on this rapidly developing field. The book is suitable for young mathematicians who want to enter this exciting field. Some basic knowledge of algebraic geometry is assumed.