zbMATH — the first resource for mathematics

Regularity on abelian varieties I. (English) Zbl 1022.14012
The authors give a criterion for tensor products of sheaves on a abelian variety to be globally generated in terms of the Mukai regularity (M-regularity). In addition they give several applications of their main result. They use the following notations: \(X\) an abelian variety of dimension \(g\) over an algebraically closed field, \(\hat X\) its dual, {\(\mathcal P\)} a Poincaré bundle on \(X \times \hat X\), \(p_X\) and \(p_{\hat X}\) the two projections from \(X \times \hat X\). A coherent sheaf \(F\) on \(X\) is called M-regular, if the codimension of the support of \(R^ip_{\hat X*}({\mathcal P} \otimes p_X^*F)\) is greater than \(i\) for all \(i=1,\ldots g\).
The main result (theorem 2.4) of the article is the M-regularity criterion which says, that the tensor product \(F \otimes \iota_*L\) of two M-regular sheaves \(F\) and \(\iota_*L\) is globally generated for \(L\) an invertible sheaf on a closed subvariety \(\iota:Y \to X\).
The proof of this theorem uses the Fourier-Mukai transform between the derived categories \(D(X)\) and \(D(\hat X)\) and the concept of continuous global generation. The applications are results on base point freeness, very ampleness, vanishing results, and dimensions of spaces of global sections.

14K12 Subvarieties of abelian varieties
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H40 Jacobians, Prym varieties
14K05 Algebraic theory of abelian varieties
Full Text: DOI arXiv
[1] 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. · Zbl 0846.13017
[2] 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. · Zbl 0762.14012
[3] J.A. Chen and C.D. Hacon, Effective generation of adjoint linear series of irregular varieties, preprint (2001).
[4] 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. · Zbl 0659.14007
[5] E. Izadi, Deforming curves representing multiples of the minimal class in Jacobians to non-Jacobians I, preprint mathAG/0103204.
[6] Y. Kawamata, Semipositivity, vanishing and applications, Lectures at the ICTP Summer School on Vanishing Theorems, April-May 2000.
[7] Yujiro Kawamata, On effective non-vanishing and base-point-freeness, Asian J. Math. 4 (2000), no. 1, 173 – 181. Kodaira’s issue. · Zbl 1060.14505
[8] George Kempf, On the geometry of a theorem of Riemann, Ann. of Math. (2) 98 (1973), 178 – 185. · Zbl 0275.14023
[9] 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. · Zbl 0785.14025
[10] George R. Kempf, Complex abelian varieties and theta functions, Universitext, Springer-Verlag, Berlin, 1991. · Zbl 0752.14040
[11] János Kollár, Higher direct images of dualizing sheaves. I, Ann. of Math. (2) 123 (1986), no. 1, 11 – 42. · Zbl 0598.14015
[12] János Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995. · Zbl 0871.14015
[13] Herbert Lange and Christina Birkenhake, Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992. · Zbl 0779.14012
[14] 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.
[15] Robert Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces, Duke Math. J. 55 (1987), no. 2, 423 – 429. · Zbl 0646.14005
[16] Shigeru Mukai, Duality between \?(\?) and \?(\?) with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153 – 175. · Zbl 0417.14036
[17] 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.
[18] 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. · Zbl 0223.14022
[19] D. Mumford, On the equations defining abelian varieties. I, Invent. Math. 1 (1966), 287 – 354. · Zbl 0219.14024
[20] 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. · Zbl 0187.42701
[21] 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. Akira Ohbuchi, A note on the projective normality of special line bundles on abelian varieties, Tsukuba J. Math. 12 (1988), no. 2, 341 – 352. · Zbl 0699.14022
[22] William Oxbury and Christian Pauly, Heisenberg invariant quartics and \?\?_{\?}(2) for a curve of genus four, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 2, 295 – 319. · Zbl 0990.14009
[23] Giuseppe Pareschi, Syzygies of abelian varieties, J. Amer. Math. Soc. 13 (2000), no. 3, 651 – 664. · Zbl 0956.14035
[24] G. Pareschi and M. Popa, Regularity on abelian varieties II: basic results on linear series and defining equations, preprint math.AG/0110004. · Zbl 1073.14061
[25] G. Pareschi and M. Popa, in preparation.
[26] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. · Zbl 0797.18001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.