<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>We study the property of <italic>continuous Castelnuovo-Mumford regularity</italic>, 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 endo-morphism 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 <italic>Index</italic> and <italic>Generic Vanishing Theory</italic> 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].</p>
</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 endo-morphism 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].

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

Department of Mathematics & Computer Science

4302 Herzberg Laboratories, Carleton University

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

We study the property of continuous Castelnuovo-Mumford regularity, for semihomogeneous vector bundles over a given Abelian variety, which was formulated in A....

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

Published Online: Jan 21, 2022

Page range: 95 - 119

Received: Feb 04, 2021

Accepted: Aug 23, 2021

DOI: https://doi.org/10.2478/aupcsm-2021-0007

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

© 2021 Nathan Grieve, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

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

Published Online: Jan 21, 2022

Page range: 95 - 119

Received: Feb 04, 2021

Accepted: Aug 23, 2021

DOI: https://doi.org/10.2478/aupcsm-2021-0007

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

© 2021 Nathan Grieve, published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

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