Abstract
We ask when the CM (Castelnuovo–Mumford) regularity of a vector bundle on a projective variety X is numerical, and address the case when X is an abelian variety. We show that the continuous CM-regularity of a semihomogeneous vector bundle on an abelian variety X is a piecewise-constant function of Chern data, and we also use generic vanishing theory to obtain a sharp upper bound for the continuous CM-regularity of any vector bundle on X. From these results we conclude that the continuous CM-regularity of many semihomogeneous bundles — including many Verlinde bundles when X is a Jacobian — is both numerical and extremal.
Communicated by: I. Coskun
Acknowledgements
This research started while the authors participated in the Combinatorial Algebraic Geometry Major Thematic Program at the Fields Institute; we thank its organizers for the motivating atmosphere. We would like to thank Montserrat Teixidor i Bigas for discussions which led us to focus on semihomogeneous bundles, Mihnea Popa for useful discussions and helpful comments on an earlier draft, and the referee for suggesting some improvements.
References
[1] M. A. Barja, R. Pardini, L. Stoppino, Linear Systems on Irregular Varieties. Preprint 2018, arXiv:1606.03290 [math.AG]10.1017/S1474748019000069Search in Google Scholar
[2] C. Birkenhake, H. Lange, Complex abelian varieties. Springer 2004. MR2062673 Zbl 1056.1406310.1007/978-3-662-06307-1Search in Google Scholar
[3] O. Debarre, On coverings of simple abelian varieties. Bull. Soc. Math. France134 (2006), 253–260.MR2233707 Zbl 1109.1401710.24033/bsmf.2508Search in Google Scholar
[4] M. Green, R. Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville. Invent. Math. 90 (1987), 389–407. MR910207 Zbl 0659.1400710.1007/BF01388711Search in Google Scholar
[5] N. Grieve, Index conditions and cup-product maps on Abelian varieties. Internat. J. Math. 25 (2014), 1450036, 31. MR3195553 Zbl 1297.1404910.1142/S0129167X14500360Search in Google Scholar
[6] M. G. Gulbrandsen, Fourier-Mukai transforms of line bundles on derived equivalent abelian varieties. Matematiche (Catania)63 (2008), 123–137. MR2512513 Zbl 1189.14049Search in Google Scholar
[7] C. D. Hacon, A derived category approach to generic vanishing. J. Reine Angew. Math. 575 (2004), 173–187. MR2097552 Zbl 1137.1401210.1515/crll.2004.078Search in Google Scholar
[8] J. Kollär, Singularities of pairs. In: Algebraic geometry—Santa Cruz 1995, volume 62, Part 1, of Proc. Sympos. Pure Math., 221–287, Amer. Math. Soc. 1997. MR1492525 Zbl 0905.1400210.1090/pspum/062.1/1492525Search in Google Scholar
[9] A. Küronya, V. Lozovanu, A Reider-type theorem for higher syzygies on abelian surfaces. Preprint 2017, arXiv:1509.0862110.14231/AG-2019-025Search in Google Scholar
[10] S. Mukai, Semi-homogeneous vector bundles on an Abelian variety. J. Math. Kyoto Univ. 18 (1978), 239–272. MR0498572 Zbl 0417.1402910.1215/kjm/1250522574Search in Google Scholar
[11] S. Mukai, Duality between D(X) and D(X̂) with its application to Picard sheaves. Nagoya Math. J. 81 (1981), 153–175. MR607081 Zbl 0417.1403610.1017/S002776300001922XSearch in Google Scholar
[12] D. Mumford, Abelian varieties. Oxford Univ. Press 1970. MR0282985 Zbl 0223.14022Search in Google Scholar
[13] Y. Mustopa, Castelnuovo–Mumford Regularity and GV-Sheaves on Irregular Varieties. Preprint 2016, arXiv:1607.06550 [math.AG], revised version in preparation.Search in Google Scholar
[14] D. Oprea, The Verlinde bundles and the semihomogeneous Wirtinger duality. J. Reine Angew. Math. 654 (2011), 181–217. MR2795755 Zbl 1223.1403310.1515/crelle.2011.032Search in Google Scholar
[15] G. Pareschi, M. Popa, Regularity on abelian varieties. I. J. Amer. Math. Soc. 16 (2003), 285–302. MR1949161 Zbl 1022.1401210.1090/S0894-0347-02-00414-9Search in Google Scholar
[16] G. Pareschi, M. Popa, GV-sheaves, Fourier-Mukai transform, and generic vanishing. Amer. J. Math. 133 (2011), 235–271. MR2752940 Zbl 1208.1401510.1353/ajm.2011.0000Search in Google Scholar
[17] G. Pareschi, M. Popa, Regularity on abelian varieties III: relationship with generic vanishing and applications. In: Grassmannians, moduli spaces and vector bundles, volume 14 of Clay Math. Proc., 141–167, Amer. Math. Soc. 2011. MR2807853 Zbl 1236.14020Search in Google Scholar
[18] M. Popa, Verlinde bundles and generalized theta linear series. Trans. Amer. Math. Soc. 354 (2002), 1869–1898. MR1881021 Zbl 0996.1401510.1090/S0002-9947-01-02923-3Search in Google Scholar
[19] J.-H. Yang, Holomorphic vector bundles over complex tori. J. Korean Math. Soc. 26 (1989), 117–142. MR1005874 Zbl 0683.32019Search in Google Scholar
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