Regularity on abelian varieties I
Authors:
Giuseppe Pareschi and Mihnea Popa
Journal:
J. Amer. Math. Soc. 16 (2003), 285302
MSC (2000):
Primary 14K05; Secondary 14K12, 14H40, 14E05
Published electronically:
November 27, 2002
MathSciNet review:
1949161
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Abstract: We introduce the notion of Mukai regularity (regularity) for coherent sheaves on abelian varieties. The definition is based on the FourierMukai transform, and in a special case depending on the choice of a polarization it parallels and strengthens the usual CastelnuovoMumford 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.
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 [BEL]
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 [CH]
 J.A. Chen and C.D. Hacon, Effective generation of adjoint linear series of irregular varieties, preprint (2001).
 [GL]
 M. Green and R. Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), 389407. MR 89b:32025
 [Iz]
 E. Izadi, Deforming curves representing multiples of the minimal class in Jacobians to nonJacobians I, preprint mathAG/0103204.
 [Ka1]
 Y. Kawamata, Semipositivity, vanishing and applications, Lectures at the ICTP Summer School on Vanishing Theorems, AprilMay 2000.
 [Ka2]
 Y. Kawamata, On effective nonvanishing and basepointfreeness, Asian J. Math 4 (2000), 173181.MR 2002b:14010
 [Ke1]
 G. Kempf, On the geometry of a theorem of Riemann, Ann. of Math. 98 (1973), 178185.MR 50:2180
 [Ke2]
 G. Kempf, Projective coordinate rings of abelian varieties, in: Algebraic Analysis, Geometry and Number Theory, J.I. Igusa ed., Johns Hopkins Press (1989), 225236.MR 98m:14047
 [Ke3]
 G. Kempf, Complex abelian varieties and theta functions, SpringerVerlag, 1991. MR 92h:14028
 [Ko1]
 J. Kollár, Higher direct images of dualizing sheaves I, Ann. of Math. 123 (1986), 1142. MR 87c:14038
 [Ko2]
 J. Kollár, Shafarevich maps and automorphic forms, Princeton Univ. Press, 1995. MR 96i:14016
 [LB]
 H. Lange and Ch. Birkenhake, Complex abelian varieties, SpringerVerlag, 1992.MR 94j:14001
 [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]
 R. Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces, Duke Math. J. 55 (1987), 423428. MR 89d:14007
 [M1]
 S. Mukai, Duality between and with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153175. MR 82f:14036
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 S. Mukai, Fourier functor and its application to the moduli of bundles on an abelian variety, In: Algebraic Geometry, Sendai 1985, Advanced studies in pure mathematics 10 (1987), 515550.MR 89k:14026
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 D. Mumford, On the equations defining abelian varieties, Invent. Math. 1 (1966), 287354. MR 34:4269
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 D. Mumford, Lectures on curves on an algebraic surface, Princeton University Press, 1966. MR 35:187
 [Oh]
 A. Ohbuchi, A note on the normal generation of ample line bundles on abelian varieties, Proc. Japan Acad. 64 (1988), 119120. MR 90a:14062a
 [OP]
 W. Oxbury and C. Pauly, Heisenberg invariant quartics and for a curve of genus four, Math. Proc. Camb. Phil. Soc. 125 (1999), 295319. MR 99k:14022
 [Pa]
 G. Pareschi, Syzygies of abelian varieties, J. Amer. Math. Soc. 13 (2000), 651664. MR 2001f:14086
 [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]
 Ch. Weibel, An introduction to homological algebra, Cambridge Univ. Press, 1994. MR 95f:18001
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Additional Information
Giuseppe Pareschi
Affiliation:
Dipartamento di Matematica, Università di Roma, Tor Vergata, V.le della Ricerca Scientifica, I00133 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:
http://dx.doi.org/10.1090/S0894034702004149
PII:
S 08940347(02)004149
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
