Geodesic
Derived categories of coherent sheaves on Abelian varieties and equivalences between them
Izvestiya. Mathematics, Tome 66 (2002) no. 3, pp. 569-594 
D. O. Orlov. Derived categories of coherent sheaves on Abelian varieties and equivalences between them. Izvestiya. Mathematics, Tome 66 (2002) no. 3, pp. 569-594. http://geodesic.mathdoc.fr/item/IM2_2002_66_3_a4/
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     title = {Derived categories of coherent sheaves on {Abelian} varieties and equivalences between them},
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Voir la notice de l'article provenant de la source Math-Net.Ru

We study derived categories of coherent sheaves on Abelian varieties. We give a criterion for the equivalence of the derived categories on two Abelian varieties and describe the autoequivalence group for the derived category of coherent sheaves of an Abelian variety.

[1] Bondal A., Orlov D., Semiorthogonal decomposition for algebraic varieties, Preprint MPI/95-15

[2] Bondal A., Orlov D., “Reconstruction of a variety from the derived category and groups of autoequivalences”, Compositio Mathematica, 125:3 (2001), 327–344 | DOI | MR | Zbl

[3] Gelfand S. I., Manin Yu. I., Metody gomologicheskoi algebry. Vvedenie v teoriyu kogomologii i proizvodnykh kategorii, Nauka, M., 1988 | MR

[4] Gruson L., Raynaud M., “Critères de platitude et de projectivité”, Inventiones Math., 13 (1971), 1–89 | DOI | MR | Zbl

[5] Hartshorne R., Residues and duality, Lecture Notes in Math, 20, Springer-Verlag, N.Y.–L.–B., 1966 | MR | Zbl

[6] Lenstra H. W., Oort Jr. F., Zarhin Yu. G., “Abelian subvarieties”, J. Algebra, 180 (1996), 513–516 | DOI | MR | Zbl

[7] Lion G., Vergne M., The Weil representation, Maslov index and theta series, Birkhäuser, Basel–Stuttgart, 1980 | MR | Zbl

[8] Miyanishi M., “Some remarks on algebraic homogeneous vector bundles”, Number Theory. Alg. Geom. and Comm. Algebra, Kinokuniya, Tokyo, 1973, 71–93 | MR

[9] Mukai S., “Duality between $D(X)$ and $D(\widehat X)$ with its application to Picard sheaves”, Nagoya Math. J., 81 (1981), 153–175 | MR | Zbl

[10] Mukai S., “Semi-homogeneous vector bundles on an abelian variety”, J. Math. Kyoto Univ., 18:2 (1978), 239–272 | MR | Zbl

[11] Mumford D., Abelian varieties, 2-nd edition, Oxford Univ. Press, Oxford, 1974 | MR | Zbl

[12] Orlov D., “Equivalences of derived categories and $K3$ surfaces”, J. of Math. Sciences. Alg. geom.-5., 85:5 (1997), 1361–1381 | MR

[13] Polishchuk A., “Symplectic biextensions and generalization of the Fourier–Mukai transforms”, Math. Research Letters, 3 (1996), 813–828 | MR | Zbl

[14] Polishchuk A., Biextensions, Weil representations on derived categories, and theta-functions, Ph. D. Thesis, Harvard University, 1996

[15] Verdier J. L., Categories derivées SGA $4\frac12$, Lecture Notes in Math., 569, Springer-Verlag, N. Y., 1977 | MR | Zbl