On equivariant Serre problem for principal bundles

Author:

Biswas Indranil1,Dey Arijit2,Poddar Mainak3ORCID

Affiliation:

1. School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India

2. Department of Mathematics, Indian Institute of Technology-Madras, Chennai, India

3. Middle East Technical University, Northern Cyprus Campus, Guzelyurt, Mersin 10, Turkey

Abstract

Let [Formula: see text] be a [Formula: see text]-equivariant algebraic principal [Formula: see text]-bundle over a normal complex affine variety [Formula: see text] equipped with an action of [Formula: see text], where [Formula: see text] and [Formula: see text] are complex linear algebraic groups. Suppose [Formula: see text] is contractible as a topological [Formula: see text]-space with a dense orbit, and [Formula: see text] is a [Formula: see text]-fixed point. We show that if [Formula: see text] is reductive, then [Formula: see text] admits a [Formula: see text]-equivariant isomorphism with the product principal [Formula: see text]-bundle [Formula: see text], where [Formula: see text] is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal [Formula: see text]-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal [Formula: see text]-bundles over any complex toric variety, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math. 27(14) (2016)].

Funder

National Board for Higher Mathematics

Science and Engineering Research Board

METU NCC

Publisher

World Scientific Pub Co Pte Lt

Subject

General Mathematics

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