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ISSN 1088-9485(e) ISSN 0273-0979(p)

Book Review

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Book Information

Author(s): Alexander Polishchuk
Title: Abelian varieties, theta functions and the Fourier transform
Additional book information: Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (UK), 2003, xvi+292 pp., £48.00, ISBN 0-521-80804-9

References:

1.: A. Beilinson, A. Polishchuk, Torelli theorem via Fourier-Mukai transform. Moduli of abelian varieties (Texel Island, 1999), 127-132, Progr. Math. 195, Birkhäuser, Basel, 2001. MR 1827017 (2002i:14013)
2.: C. Birkenhake, H. Lange Complex abelian varieties, 2nd edition. Grund. Math. Wiss. 302. Springer-Verlag, Berlin (2004). MR 2062673 (2005c:14001)
3.: P. Cartier, Quantum mechanical commutation relations and theta functions. Algebraic Groups and Discontinuous Subgroups, pp. 361-383; Proc. Sympos. Pure Math. 9, Amer. Math. Soc., Providence (1966). MR 0216825 (35:7654)
4.: P. Griffiths, J. Harris, Principles of algebraic geometry. Wiley, New York (1978). MR 1288523 (95d:14001)
5.: C. Houzel, Fonctions elliptiques et intégrales abéliennes. Abrégé d'histoire des mathématiques 1700-1900, pp. 1-113. Hermann, Paris (1978). MR 0504183 (80k:01002b)
6.: S. Lefschetz, On certain numerical invariants of algebraic varieties with application to abelian varieties, Trans. Amer. Math. Soc. 22 (1921), 327-482. MR 1501178
7.: S. Mukai, Duality between and $D(\hat X)$ with its application to Picard sheaves. Nagoya Math. J. 81 (1981), 153-175. MR 0607081 (82f:14036)
8.: D. Mumford, Abelian varieties. Oxford University Press, London (1970). MR 0282985 (44:219)
9.: D. Mumford, On the equations defining abelian varieties, I, II, III. Invent. Math. 1 (1966) 287-354; 3 (1967) 75-135 and 215-244. MR 0204427 (34:4269), MR 0219541 (36:2621), MR 0219542 (36:2622)
10.: D. Orlov, Equivalences of derived categories and K3 surfaces, Algebraic geometry, 7. J. Math. Sci. (New York) 84 (1997), 1361-1381. MR 1465519 (99a:14054)
11.: E. Picard, Sur les intégrales de différentielles totales algébriques de première espèce, J. de Math. Pures et Appliquées (4) 1 (1885), 281-346.
12.: E. Picard, H. Poincaré, Sur un théorème de Riemann relatif aux fonctions de variables indépendantes admettant systèmes de périodes, C.R. Acad. Sci. Paris 97 (1883), 1284-1287.
13.: B. Riemann, Theorie der Abel'schen Functionen, J. reine angew. Math. 54 (1857), 115-155.
14.: G. Scorza, Intorno alla teoria generale delle matrici di Riemann e ad alcune sue applicazioni, Rend. del Circolo Mat. di Palermo 41 (1916), 263-380.
15.: A. Weil, Variétés abéliennes et courbes algébriques. Hermann, Paris (1948). MR 0029522 (10:621d)
16.: A. Weil, Sur certains groupes d'opérateurs unitaires. Acta Math. 111 (1964), 143-211. MR 0165033 (29:2324)

Additional Information:

Reviewer(s):
Arnaud Beauville
Affiliation: Université de Nice
Email: beauville@math.unice.fr

Review Information:
Journal: Bull. Amer. Math. Soc. 42 (2005), 529-533.

MSC (2000): Primary 14K25; Secondary 14H42
PII: S 0273-0979(05)01064-5
Posted: April 7, 2005
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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