Syzygies of abelian varieties

Author:
Giuseppe Pareschi

Journal:
J. Amer. Math. Soc. **13** (2000), 651-664

MSC (2000):
Primary 14K05; Secondary 14F05

Published electronically:
April 10, 2000

MathSciNet review:
1758758

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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 is an ample line on an abelian variety, then satisfies the property as soon as . 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.

**[EL]**Lawrence Ein and Robert Lazarsfeld,*Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension*, Invent. Math.**111**(1993), no. 1, 51–67. MR**1193597**, 10.1007/BF01231279**[G1]**Mark L. Green,*Koszul cohomology and the geometry of projective varieties*, J. Differential Geom.**19**(1984), no. 1, 125–171. MR**739785****[G2]**Mark L. Green,*Koszul cohomology and geometry*, Lectures on Riemann surfaces (Trieste, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 177–200. MR**1082354****[K1]**George Kempf,*Toward the inversion of abelian integrals. I*, Ann. of Math. (2)**110**(1979), no. 2, 243–273. MR**549489**, 10.2307/1971261**[K2]**George Kempf,*Toward the inversion of abelian integrals. II*, Amer. J. Math.**101**(1979), no. 1, 184–202. MR**527831**, 10.2307/2373944**[K3]**George R. Kempf,*Multiplication over abelian varieties*, Amer. J. Math.**110**(1988), no. 4, 765–773. MR**955296**, 10.2307/2374649**[K4]**George R. Kempf,*Linear systems on abelian varieties*, Amer. J. Math.**111**(1989), no. 1, 65–94. MR**980300**, 10.2307/2374480**[K5]**George R. Kempf,*Projective coordinate rings of abelian varieties*, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 225–235. MR**1463704****[K6]**Kempf, G.:*Complex abelian varieties and theta functions*, Springer-Verlag, 1990. MR**92h:1402****[Ko]**Shoji Koizumi,*Theta relations and projective normality of Abelian varieties*, Amer. J. Math.**98**(1976), no. 4, 865–889. MR**0480543****[L]**Robert Lazarsfeld,*A sampling of vector bundle techniques in the study of linear series*, Lectures on Riemann surfaces (Trieste, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 500–559. MR**1082360****[M1]**D. Mumford,*On the equations defining abelian varieties. I*, Invent. Math.**1**(1966), 287–354. MR**0204427****[M2]**David Mumford,*Varieties defined by quadratic equations*, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) Edizioni Cremonese, Rome, 1970, pp. 29–100. MR**0282975****[M3]**David Mumford,*Abelian varieties*, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. MR**0282985****[Mu]**Shigeru Mukai,*Duality between 𝐷(𝑋) and 𝐷(𝑋) with its application to Picard sheaves*, Nagoya Math. J.**81**(1981), 153–175. MR**607081****[S]**Tsutomu Sekiguchi,*On the normal generation by a line bundle on an Abelian variety*, Proc. Japan Acad. Ser. A Math. Sci.**54**(1978), no. 7, 185–188. MR**510946**

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2000):
14K05,
14F05

Retrieve articles in all journals with MSC (2000): 14K05, 14F05

Additional Information

**Giuseppe Pareschi**

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

Email:
pareschi@mat.uniroma2.it

DOI:
http://dx.doi.org/10.1090/S0894-0347-00-00335-0

Keywords:
Homogeneous ideal,
Pontrjagin product,
vector bundles

Received by editor(s):
August 24, 1998

Received by editor(s) in revised form:
March 8, 2000

Published electronically:
April 10, 2000

Article copyright:
© Copyright 2000
American Mathematical Society