Cohomology of line bundles on horospherical varieties
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Mathematische Zeitschrift
Cohomology of line bundles on horospherical varieties Narasimha Chary Bonala1,2 · Benoît Dejoncheere3 Received: 10 May 2019 / Accepted: 24 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete horospherical varieties. The main tool in this article is the machinery of Grothendieck–Cousin complexes, and we also prove a Künneth-like formula for local cohomology. Keywords Grothendieck–Cousin complexes · Horospherical varieties · Local cohomology Mathematics Subject Classification 14M27 · 14L30
1 Introduction Let G be a connected reductive algebraic group over the field of complex numbers C and let H be a closed subgroup of G. A homogeneous space G/H is said to be horospherical if H contains the unipotent radical of a Borel subgroup of G, or equivalently, G/H is isomorphic to a torus bundle over a flag variety G/P. A normal G-variety is called horospherical if it contains a dense open G-orbit isomorphic to a horospherical homogeneous space G/H . Toric varieties and flag varieties are horospherical varieties, and horospherical varieties form an interesting class of spherical varieties generalizing both of these. A spherical variety is a normal G-variety such that a Borel subgroup B of G acts with a dense open orbit. One of the main importance of horosperical varieties is that, any spherical variety has a degeneration to a horospherical variety, such a degeneration being called a horospherical contraction (see for instance [29, Proposition 7.10]). The advantage of degenerating to a horospherical one is
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Benoît Dejoncheere [email protected] Narasimha Chary Bonala [email protected]
1
Max Planck Institute for Mathematics, Vivatsgasse 7, Bonn, Germany
2
Present Address: Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany
3
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada
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N. C. Bonala, B. Dejoncheere
that their underlying combinatorial description is easier to understand as compared to general spherical varieties. Let L be a line bundle on a normal G-variety X . Up to replacing G by a finite cover, we can assume that G = C ×[G, G], where C is a torus and [G, G] is a simply-connected semisimple group, hence one can assume that L is G-linearized. If X is complete, then the cohomology groups H i (X , L) are finite dimensional representations of G (see [16, Theorem 11.6] and [5, page 589]). If X is a flag variety, then Borel–Weil–Bott Theorem describes these cohomology groups (see [1, Section 10] or [9]). If X is a toric variety, then the cohomology groups can be described using combinatorics of its associated fan (proved by Demazure in [8], see also [7, Chapter 9] for more details). For spherical varieties, in [5], Brion has given a bound on t
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