## Algebra & Number Theory

### Local positivity, multiplier ideals, and syzygies of abelian varieties

#### Abstract

We use the language of multiplier ideals in order to relate the syzygies of an abelian variety in a suitable embedding with the local positivity of the line bundle inducing that embedding. This extends to higher syzygies a result of Hwang and To on projective normality.

#### Article information

Source
Algebra Number Theory, Volume 5, Number 2 (2011), 185-196.

Dates
Revised: 23 April 2010
Accepted: 22 May 2010
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.ant/1513882119

Digital Object Identifier
doi:10.2140/ant.2011.5.185

Mathematical Reviews number (MathSciNet)
MR2833789

Zentralblatt MATH identifier
1239.14035

Subjects
Primary: 14K05: Algebraic theory

#### Citation

Lazarsfeld, Robert; Pareschi, Giuseppe; Popa, Mihnea. Local positivity, multiplier ideals, and syzygies of abelian varieties. Algebra Number Theory 5 (2011), no. 2, 185--196. doi:10.2140/ant.2011.5.185. https://projecteuclid.org/euclid.ant/1513882119

#### References

• T. Bauer, “Seshadri constants and periods of polarized abelian varieties”, Math. Ann. 312:4 (1998), 607–623.
• A. Bertram, L. Ein, and R. Lazarsfeld, “Vanishing theorems, a theorem of Severi, and the equations defining projective varieties”, J. Amer. Math. Soc. 4:3 (1991), 587–602.
• M. Brion and S. Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Mathematics 231, Birkhäuser, Boston, 2005.
• P. Buser and P. Sarnak, “On the period matrix of a Riemann surface of large genus”, Invent. Math. 117:1 (1994), 27–56.
• C. Ciliberto, J. Harris, and R. Miranda, “On the surjectivity of the Wahl map”, Duke Math. J. 57:3 (1988), 829–858.
• E. Colombo, P. Frediani, and G. Pareschi, “Hyperplane sections of abelian surfaces”, 2011. To appear in J. Alg. Geom.
• D. Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics 229, Springer, New York, 2005.
• L. Fuentes García, “Some results about the projective normality of abelian varieties”, Arch. Math. $($Basel$)$ 85:5 (2005), 409–418.
• M. L. Green, “Koszul cohomology and the geometry of projective varieties, II”, J. Differential Geom. 20:1 (1984), 279–289.
• M. Green and R. Lazarsfeld, “Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville”, Invent. Math. 90:2 (1987), 389–407.
• J.-M. Hwang and W.-K. To, “Buser–Sarnak invariant and projective normality of abelian varieties”, preprint, 2010.
• S. P. Inamdar, “On syzygies of projective varieties”, Pacific J. Math. 177:1 (1997), 71–76.
• S. P. Inamdar and V. B. Mehta, “Frobenius splitting of Schubert varieties and linear syzygies”, Amer. J. Math. 116:6 (1994), 1569–1586.
• J. N. Iyer, “Projective normality of abelian varieties”, Trans. Amer. Math. Soc. 355:8 (2003), 3209–3216.
• R. Lazarsfeld, “Lengths of periods and Seshadri constants of abelian varieties”, Math. Res. Lett. 3:4 (1996), 439–447.
• R. Lazarsfeld, Positivity in algebraic geometry, I, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 48, Springer, Berlin, 2004.
• L. Li, “Wonderful compactification of an arrangement of subvarieties”, Michigan Math. J. 58:2 (2009), 535–563.
• S. Mukai, “Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves”, Nagoya Math. J. 81 (1981), 153–175.
• D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics 5, Oxford University Press, London, 1970.
• G. Pareschi, “Syzygies of abelian varieties”, J. Amer. Math. Soc. 13:3 (2000), 651–664.
• G. Pareschi and M. Popa, “Regularity on abelian varieties, I”, J. Amer. Math. Soc. 16:2 (2003), 285–302.
• G. Pareschi and M. Popa, “Regularity on abelian varieties, II: Basic results on linear series and defining equations”, J. Algebraic Geom. 13:1 (2004), 167–193. http://www.ams.org/mathscinet-getitem?mr=2005a:14059MR2005a:14059
• J. Wahl, “Introduction to Gaussian maps on an algebraic curve”, pp. 304–323 in Complex projective geometry (Trieste/Bergen, 1989), edited by G. Ellingsrud et al., London Math. Soc. Lecture Note Ser. 179, Cambridge Univ. Press, London, 1992.