Syzygies of abelian varieties

Pareschi, Giuseppe¹

¹ Dipartimento di Matematica, Università di Roma, Tor Vergata V.le della Ricerca Scientifica, I-00133 Roma, Italy

Journal of the American Mathematical Society, Tome 13 (2000) no. 3, pp. 651-664

Résumé

We prove a conjecture of R. Lazarsfeld on the syzygies (of the homogeneous ideal) of abelian varieties embedded in projective space by multiples of an ample line bundle. Specifically, we prove that if $A$ is an ample line on an abelian variety, then $A^{\otimes n}$ satisfies the property $N_{p}$ as soon as $n\ge p+ 3$. The proof uses a criterion for the global generation of vector bundles on abelian varieties (generalizing the classical one for line bundles) and a criterion for the surjectivity of multiplication maps of global sections of two vector bundles in terms of the vanishing of the cohomology of certain twists of their Pontrjagin product.

Détail

DOI : 10.1090/S0894-0347-00-00335-0

Bibliographie

[1] Ein, Lawrence, Lazarsfeld, Robert Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension Invent. Math. 1993 51 67

[2] Green, Mark L. Koszul cohomology and the geometry of projective varieties J. Differential Geom. 1984 125 171

[3] Green, Mark L. Koszul cohomology and geometry 1989 177 200

[4] Kempf, George Toward the inversion of abelian integrals. I Ann. of Math. (2) 1979 243 273

[5] Kempf, George Toward the inversion of abelian integrals. I Ann. of Math. (2) 1979 243 273

[6] Kempf, George R. Multiplication over abelian varieties Amer. J. Math. 1988 765 773

[7] Kempf, George R. Linear systems on abelian varieties Amer. J. Math. 1989 65 94

[8] Kempf, George R. Projective coordinate rings of abelian varieties 1989 225 235

[9] Trjitzinsky, W. J. General theory of singular integral equations with real kernels Trans. Amer. Math. Soc. 1939 202 279

[10] Koizumi, Shoji Theta relations and projective normality of Abelian varieties Amer. J. Math. 1976 865 889

[11] Lazarsfeld, Robert A sampling of vector bundle techniques in the study of linear series 1989 500 559

[12] Mumford, D. On the equations defining abelian varieties. I Invent. Math. 1966 287 354

[13] Mumford, David Varieties defined by quadratic equations 1970 29 100

[14] Mumford, David Abelian varieties 1970

[15] Mukai, Shigeru Duality between 𝐷(𝑋) and 𝐷(𝑋̂) with its application to Picard sheaves Nagoya Math. J. 1981 153 175

[16] Sekiguchi, Tsutomu On the normal generation by a line bundle on an Abelian variety Proc. Japan Acad. Ser. A Math. Sci. 1978 185 188