Book Review
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Book Information:
Author:
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.,
ISBN 0-521-80804-9,
£48.00
A. Beilinson and A. Polishchuk, Torelli theorem via Fourier-Mukai transform, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 127–132. MR 1827017
Christina Birkenhake and Herbert Lange, Complex abelian varieties, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004. MR 2062673, DOI 10.1007/978-3-662-06307-1
Pierre Cartier, Quantum mechanical commutation relations and theta functions, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 361–383. MR 0216825
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523, DOI 10.1002/9781118032527
Abrégé d’histoire des mathématiques 1700–1900. Tome I, Hermann, Paris, 1978 (French). Algèbre, analyse classique, théorie des nombres; Édité par Jean Dieudonné. MR 504182
Solomon Lefschetz, On certain numerical invariants of algebraic varieties with application to abelian varieties, Trans. Amer. Math. Soc. 22 (1921), no. 3, 327–406. MR 1501178, DOI 10.1090/S0002-9947-1921-1501178-3
Shigeru Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175. MR 607081
David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
D. Mumford, On the equations defining abelian varieties. I, Invent. Math. 1 (1966), 287–354. MR 204427, DOI 10.1007/BF01389737
D. O. Orlov, Equivalences of derived categories and $K3$ surfaces, J. Math. Sci. (New York) 84 (1997), no. 5, 1361–1381. Algebraic geometry, 7. MR 1465519, DOI 10.1007/BF02399195
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.
André Weil, Variétés abéliennes et courbes algébriques, Publ. Inst. Math. Univ. Strasbourg, vol. 8, Hermann & Cie, Paris, 1948 (French). MR 0029522
André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 165033, DOI 10.1007/BF02391012
- 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
- 2.
- C. Birkenhake, H. Lange Complex abelian varieties, 2nd edition. Grund. Math. Wiss. 302. Springer-Verlag, Berlin (2004). MR 2062673
- 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
- 4.
- P. Griffiths, J. Harris, Principles of algebraic geometry. Wiley, New York (1978). MR 1288523
- 5.
- C. Houzel, Fonctions elliptiques et intégrales abéliennes. Abrégé d'histoire des mathématiques 1700-1900, pp. 1-113. Hermann, Paris (1978). MR 0504182
- 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 
with its application to Picard sheaves. Nagoya Math. J. 81 (1981), 153-175. MR 0607081
- 8.
- D. Mumford, Abelian varieties. Oxford University Press, London (1970). MR 0282985
- 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, 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
- 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
- 16.
- A. Weil, Sur certains groupes d'opérateurs unitaires. Acta Math. 111 (1964), 143-211. MR 0165033
Review Information:
Reviewer:
Arnaud Beauville
Affiliation:
Université de Nice
Email:
beauville@math.unice.fr
Journal:
Bull. Amer. Math. Soc.
42 (2005), 529-533
Published electronically:
April 7, 2005
Review copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
