Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles

Authors

  • Nathan Grieve Department of Mathematics & Computer Science, Royal Military College of Canada; School of Mathematics and Statistics, 4302 Herzberg Laboratories, Carleton University; Département de Mathématiques, Université du Québec á Montréal

Keywords:

Abelian varieties, Mukai regularity, continuous Castelnuovo-Mumford regularity, semihomogeneous vector bundles, Generic Vanishing Theory

Abstract

We study the property of continuous Castelnuovo-Mumford regularity, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in A. Küronya and Y. Mustopa [Adv. Geom. 20 (2020), no. 3, 401-412]. Our main result gives a novel description thereof. It is expressed in terms of certain normalized polynomial functions that are obtained via the Wedderburn decomposition of the Abelian variety’s endomorphism algebra. This result builds on earlier work of Mumford and Kempf and applies the form of the Riemann-Roch Theorem that was established in N. Grieve [New York J. Math. 23 (2017), 1087-1110]. In a complementary direction, we explain how these topics pertain to the Index and Generic Vanishing Theory conditions for simple semihomogeneous vector bundles. In doing so, we refine results from M. Gulbrandsen [Matematiche (Catania) 63 (2008), no. 1, 123–137], N. Grieve [Internat. J. Math. 25 (2014), no. 4, 1450036, 31] and D. Mumford [Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Edizioni Cremonese, Rome, 1970, pp. 29-100].

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Published

2021-10-21

How to Cite

Grieve, N. (2021). Wedderburn components, the index theorem and continuous Castelnuovo-Mumford regularity for semihomogeneous vector bundles. Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica, 20, 95–119. Retrieved from https://studmath.up.krakow.pl/article/view/8634

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