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Derived categories of coherent sheaves and equivalences between them. (English. Russian original) Zbl 1118.14021

Russ. Math. Surv. 58, No. 3, 511-591 (2003); translation from Usp. Mat. Nauk 58, No. 3, 89-172 (2003).
The author gives a review on the theory of derived categories in Algebraic Geometry. The main focus lies on the bounded derived categories of coherent sheaves on smooth projective varietes over \(\mathbb{C}\). After some technical background, the proof of the author’s basic theorem [J. Math. Sci., New York 84, 1361–1381 (1997; Zbl 0938.14019)] is given: every fully faithful functor (in the above geometric setting) is represented by an object on the product of the two varieties. This fact is crucial for many applications, among them the possibility to assign a classical correspondence on the levels of cohomology or intersection groups. The final two sections of the article also draw from this theorem substantially.
The next section deals with Orlov’s derived Torelli theorem for \(K3\) surfaces [loc. cit.]. It states that two projective \(K3\) surfaces are derived equivalent (i.e. their derived categories are equivalent as triangulated categories) if and only if their transcendental lattices are Hodge isometric which (by a result on lattices due to V. V. Nikulin [Math. USSR, Izv. 14, 103–167 (1980; Zbl 0427.10014)]) is equivalent to the 24-dimensional full cohomology lattices being Hodge isometric using the Mukai pairing. This should be compared to the classical Torelli theorem for \(K3\) surfaces: two \(K3\) surfaces are isomorphic if and only if their second cohomology lattices are Hodge isometric.
In the final section, the author gives an account of his description of the group of autoequivalences of the derived category of abelian varieties [D. O. Orlov, Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 3, 131–158 (2002; Zbl 1031.18007)]. This group is shown to be an extension of a certain symplectic group by a group of standard autoequivalences, namely the shifts, translations and line bundle twists by line bundles of degree 0. In a rough sense, the symplectic group appearing there is generated by the automorphism group of the abelian variety, line bundle twists for line bundles not of degree 0 and the Fourier-Mukai transform of the Poincar’e bundle (although the latter at first is an equivalence between the derived categories of the abelian variety with its dual abelian variety [S. Mukai, Nagoya Math. J. 81, 153–175 (1981; Zbl 0417.14036)]).

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14A22 Noncommutative algebraic geometry
18E30 Derived categories, triangulated categories (MSC2010)
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