Regularity on abelian varieties I

Authors:
Giuseppe Pareschi and Mihnea Popa

Journal:
J. Amer. Math. Soc. **16** (2003), 285-302

MSC (2000):
Primary 14K05; Secondary 14K12, 14H40, 14E05

Published electronically:
November 27, 2002

MathSciNet review:
1949161

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Abstract | References | Similar Articles | Additional Information

Abstract: We introduce the notion of Mukai regularity (-regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the choice of a polarization it parallels and strengthens the usual Castelnuovo-Mumford regularity. Mukai regularity has a large number of applications, ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties, to higher dimensional type statements and to a study of special classes of vector bundles. Some of these applications are explained here, while others are the subject of upcoming sequels.

**[BM]**Dave Bayer and David Mumford,*What can be computed in algebraic geometry?*, Computational algebraic geometry and commutative algebra (Cortona, 1991), Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993, pp. 1–48. MR**1253986****[BEL]**Aaron Bertram, Lawrence Ein, and Robert Lazarsfeld,*Vanishing theorems, a theorem of Severi, and the equations defining projective varieties*, J. Amer. Math. Soc.**4**(1991), no. 3, 587–602. MR**1092845**, 10.1090/S0894-0347-1991-1092845-5**[CH]**J.A. Chen and C.D. Hacon, Effective generation of adjoint linear series of irregular varieties, preprint (2001).**[GL]**Mark Green and Robert Lazarsfeld,*Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville*, Invent. Math.**90**(1987), no. 2, 389–407. MR**910207**, 10.1007/BF01388711**[Iz]**E. Izadi, Deforming curves representing multiples of the minimal class in Jacobians to non-Jacobians I, preprint mathAG/0103204.**[Ka1]**Y. Kawamata, Semipositivity, vanishing and applications, Lectures at the ICTP Summer School on Vanishing Theorems, April-May 2000.**[Ka2]**Yujiro Kawamata,*On effective non-vanishing and base-point-freeness*, Asian J. Math.**4**(2000), no. 1, 173–181. Kodaira’s issue. MR**1802918**, 10.4310/AJM.2000.v4.n1.a11**[Ke1]**George Kempf,*On the geometry of a theorem of Riemann*, Ann. of Math. (2)**98**(1973), 178–185. MR**0349687****[Ke2]**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****[Ke3]**George R. Kempf,*Complex abelian varieties and theta functions*, Universitext, Springer-Verlag, Berlin, 1991. MR**1109495****[Ko1]**János Kollár,*Higher direct images of dualizing sheaves. I*, Ann. of Math. (2)**123**(1986), no. 1, 11–42. MR**825838**, 10.2307/1971351**[Ko2]**János Kollár,*Shafarevich maps and automorphic forms*, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995. MR**1341589****[LB]**Herbert Lange and Christina Birkenhake,*Complex abelian varieties*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992. MR**1217487****[La1]**R. Lazarsfeld,*Multiplier Ideals for Algebraic Geometers*, notes available at www.math.lsa.umich.edu/rlaz, to be included in*Positivity in Algebraic Geometry*, book in preparation.**[La2]**Robert Lazarsfeld,*A sharp Castelnuovo bound for smooth surfaces*, Duke Math. J.**55**(1987), no. 2, 423–429. MR**894589**, 10.1215/S0012-7094-87-05523-2**[M1]**Shigeru Mukai,*Duality between 𝐷(𝑋) and 𝐷(𝑋) with its application to Picard sheaves*, Nagoya Math. J.**81**(1981), 153–175. MR**607081****[M2]**Shigeru Mukai,*Fourier functor and its application to the moduli of bundles on an abelian variety*, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 515–550. MR**946249****[Mu1]**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****[Mu2]**D. Mumford,*On the equations defining abelian varieties. I*, Invent. Math.**1**(1966), 287–354. MR**0204427****[Mu3]**David Mumford,*Lectures on curves on an algebraic surface*, With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. MR**0209285****[Oh]**Akira Ohbuchi,*A note on the normal generation of ample line bundles on abelian varieties*, Proc. Japan Acad. Ser. A Math. Sci.**64**(1988), no. 4, 119–120. MR**966402****[OP]**William Oxbury and Christian Pauly,*Heisenberg invariant quartics and 𝒮𝒰_{𝒞}(2) for a curve of genus four*, Math. Proc. Cambridge Philos. Soc.**125**(1999), no. 2, 295–319. MR**1643798**, 10.1017/S0305004198003028**[Pa]**Giuseppe Pareschi,*Syzygies of abelian varieties*, J. Amer. Math. Soc.**13**(2000), no. 3, 651–664 (electronic). MR**1758758**, 10.1090/S0894-0347-00-00335-0**[PP1]**G. Pareschi and M. Popa, Regularity on abelian varieties II: basic results on linear series and defining equations, preprint math.AG/0110004.**[PP2]**G. Pareschi and M. Popa, in preparation.**[We]**Charles A. Weibel,*An introduction to homological algebra*, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR**1269324**

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Additional Information

**Giuseppe Pareschi**

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

Email:
pareschi@mat.uniroma2.it

**Mihnea Popa**

Affiliation:
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138

Email:
mpopa@math.harvard.edu

DOI:
https://doi.org/10.1090/S0894-0347-02-00414-9

Received by editor(s):
October 22, 2001

Received by editor(s) in revised form:
April 4, 2002

Published electronically:
November 27, 2002

Additional Notes:
The second author was partially supported by a Clay Mathematics Institute Liftoff Fellowship during the preparation of this paper.

Article copyright:
© Copyright 2002
American Mathematical Society