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Derived categories of coherent sheaves on algebraic varieties. (English) Zbl 1208.14014
Holm, Thorsten (ed.) et al., Triangulated categories. Based on a workshop, Leeds, UK, August 2006. Cambridge: Cambridge University Press (ISBN 978-0-521-74431-7/pbk). London Mathematical Society Lecture Note Series 375, 408-451 (2010).
This article is a survey on recent developments in the field of derived categories in algebraic geometry, with a particular accent on the theory of stability conditions.
The author recalls the definition of derived categories and Fourier–Mukai functors and the main results in this theory, such as representability of fully faithful functors, and the description of autoequivalences in Section 2. In Section 3 he recalls some result about derived categories and birational geometry, stressing in particular the known cases of invariance of the derived category under birational tranformations. Section 4 is dedicated to non-commutative algebras in the derived categorical setting, for example in the McKay correspondence or in the non-commutative crepant resolutions of singularities.
The second part, which is covered by Section 5, gives a complete and detailed account on Bridgeland’s stability conditions. In particular the author recalls the fundamental results, such as the topological structure of the stability manifold or the construction of stability conditions on \(K3\) surfaces and their conjectural relation with the group of autoequivalences. He gives also an account on the string-theoretical point of view.
For the entire collection see [Zbl 1195.18001].

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)