Equivariant Vector Bundles on T-Varieties (bibtex)

by Nathan Owen Ilten, Hendrik Süß

Abstract:

Let X be a T-variety, where T is an algebraic torus. We describe a fully faithful functor from the category of T-equivariant vector bundles on X to a certain category of filtered vector bundles on a suitable quotient of X by T. We show that if X is factorial, this functor gives an equivalence of categories. This generalizes Klyachko's description of equivariant vector bun dles on toric varieties. We apply our machinery to show that vector bundles of low rank on projective space which are equivariant with respect to special subtori of the maximal torus must s plit, generalizing a theorem of Kaneyama. Further applications include descriptions of global vector fields on T-varieties, and a study of equivariant deformations of equivariant vector bun dles.

Reference:

Equivariant Vector Bundles on T-Varieties (Nathan Owen Ilten, Hendrik Süß), Transformation Groups, volume 20, 2015.

Bibtex Entry:

@article{tvb, author={{Ilten}, Nathan Owen and {S{\"u}{\ss}}, Hendrik}, title="{Equivariant Vector Bundles on T-Varieties}", eprint = {1305.0024}, year =2015, month=dec, volume=20, issue=4, pages={1043-1073}, url= {http://arxiv.org/abs/1305.0024}, journal ={Transformation Groups}, doi = {10.1007/s00031-015-9312-2}, pages={1-31}, gsid={1642701722204989888}, abstract = { Let X be a T-variety, where T is an algebraic torus. We describe a fully faithful functor from the category of T-equivariant vector bundles on X to a certain category of filtered vector bundles on a suitable quotient of X by T. We show that if X is factorial, this functor gives an equivalence of categories. This generalizes Klyachko's description of equivariant vector bun dles on toric varieties. We apply our machinery to show that vector bundles of low rank on projective space which are equivariant with respect to special subtori of the maximal torus must s plit, generalizing a theorem of Kaneyama. Further applications include descriptions of global vector fields on T-varieties, and a study of equivariant deformations of equivariant vector bun dles. }, }

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