Equivariant vector bundles on \(t\)-varieties
type: Item
instance of: scholarly article
title: Equivariant vector bundles on \(t\)-varieties
zbMATH Open document ID: 1387.14125
author: Hendrik Süss
author: Nathan Owen Ilten
published in: Transformation Groups
publication date: 2015-12-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1305.0024
review text: In the 90's \textit{A. A. Klyachko} [Math. USSR, Izv. 35, No. 2, 337--375 (1990; Zbl 0706.14010); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 1001--1039 (1989)] has published a description of equivariant vector bundles on toric varieties. They can be given by a plain vector space together with a bunch of filtrations -- one for each prime divisor outside the embedded torus. The authors of the paper under review generalize this story to varieties \(X\) admitting a possibly smaller effective torus (\(=T\)) action. While equivariant coherent sheaves on \(X\) just mean coherent sheaves on the stack \([X/T]\), the authors choose a description closer to Klyachko's approach. If \(W\subseteq X\) denotes the locus of points with finite stabilizers (thus \(W=T\) in the toric case), then to an equivariant vector bundle on \(X\) they associate a vector bundle on the stack \(Z=[W/T]\) together with a bunch of filtrations -- one for each prime divisor of \(X\setminus W\). As in the toric case, they fulfill some compatibility relations -- and under certain assumptions this leads to an equivalence of categories. Finally, the authors present applications by calculating global sections, investigating the splitting behavior, and by discussing the special case of the tangent sheaf.
reviewed by: Klaus Altmann
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 6521626
zbMATH Keywords: Torus action
zbMATH Keywords: polyhedral divisor
describes a project that uses: ToricVectorBundles
MaRDI profile type: Publication
OpenAlex ID: W3105192864
arXiv ID: 1305.0024
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DOI: 10.1007/S00031-015-9312-2
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description: scientific article