Equivariant vector bundles on \(t\)-varieties
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.
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