## Syzygies of abelian varieties

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- by Giuseppe Pareschi PDF
- J. Amer. Math. Soc.
**13**(2000), 651-664 Request permission

## 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 $A$ 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.## References

- 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**, DOI 10.1007/BF01231279 - Mark L. Green,
*Koszul cohomology and the geometry of projective varieties*, J. Differential Geom.**19**(1984), no. 1, 125–171. MR**739785** - Mark L. Green,
*Koszul cohomology and geometry*, Lectures on Riemann surfaces (Trieste, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 177–200. MR**1082354** - George Kempf,
*Toward the inversion of abelian integrals. I*, Ann. of Math. (2)**110**(1979), no. 2, 243–273. MR**549489**, DOI 10.2307/1971261 - George Kempf,
*Toward the inversion of abelian integrals. I*, Ann. of Math. (2)**110**(1979), no. 2, 243–273. MR**549489**, DOI 10.2307/1971261 - George R. Kempf,
*Multiplication over abelian varieties*, Amer. J. Math.**110**(1988), no. 4, 765–773. MR**955296**, DOI 10.2307/2374649 - George R. Kempf,
*Linear systems on abelian varieties*, Amer. J. Math.**111**(1989), no. 1, 65–94. MR**980300**, DOI 10.2307/2374480 - 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** - W. J. Trjitzinsky,
*General theory of singular integral equations with real kernels*, Trans. Amer. Math. Soc.**46**(1939), 202–279. MR**92**, DOI 10.1090/S0002-9947-1939-0000092-6 - Shoji Koizumi,
*Theta relations and projective normality of Abelian varieties*, Amer. J. Math.**98**(1976), no. 4, 865–889. MR**480543**, DOI 10.2307/2374034 - 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** - D. Mumford,
*On the equations defining abelian varieties. I*, Invent. Math.**1**(1966), 287–354. MR**204427**, DOI 10.1007/BF01389737 - 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** - 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** - 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** - 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**

## 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
- Received by editor(s): August 24, 1998
- Received by editor(s) in revised form: March 8, 2000
- Published electronically: April 10, 2000
- © Copyright 2000 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**13**(2000), 651-664 - MSC (2000): Primary 14K05; Secondary 14F05
- DOI: https://doi.org/10.1090/S0894-0347-00-00335-0
- MathSciNet review: 1758758