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Derived categories of coherent sheaves on abelian varieties and equivalences between them. (English. Russian original) Zbl 1031.18007

Izv. Math. 66, No. 3, 569-594 (2002); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 3, 131-158 (2002).
Let \(A\) be an abelian variety over an algebraically closed field of characteristic zero and let \(\widehat A\) be the dual abelian variety. S. Mukai [Nagoya Math. J. 81, 153-175 (1981; Zbl 0417.14036)] showed that the (bounded) derived categories of coherent sheaves on \(A\) and on \(\widehat{A}\) are equivalent.
The paper under review gives a criterion for equivalence of derived categories of two abelian varieties \(A\) and \(B\). Namely, the derived categories of coherent sheaves \(D^b(A)\) and \(D^b(B)\) are equivalent if and only if there exists an isometric isomorphism \(A\times \widehat{A}\simeq B\times\widehat{B}\). A. Polishchuk [Math. Res. Lett. 3, 813-828 (1996; Zbl 0886.14019)] proved the “if” part of this criterion, but the method of the author is different and uses his theorem saying that every exact equivalence between derived categories can be realized as the Fourier-Mukai transform [see D. O. Orlov, J. Math. Sci., New York 84, 1361-1381 (1997; Zbl 0938.14019)].
In the last part of the paper the author describes the group of autoequivalences of the derived category of an abelian variety.

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
14K05 Algebraic theory of abelian varieties
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