On equivariant Serre problem for principal bundles

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Biswas I., Dey A., Poddar M.

INTERNATIONAL JOURNAL OF MATHEMATICS, vol.29, no.9, 2018 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 29 Issue: 9
  • Publication Date: 2018
  • Doi Number: 10.1142/s0129167x18500544
  • Keywords: Principal bundle, equivariant bundle, Serre problem, toric variety, OKA PRINCIPLE, VECTOR-BUNDLES, MODULES


Let E-G be a Gamma-equivariant algebraic principal G-bundle over a normal complex affine variety X equipped with an action of Gamma, where G and Gamma are complex linear algebraic groups. Suppose X is contractible as a topological Gamma-space with a dense orbit, and x(0) is an element of X is a Gamma-fixed point. We show that if Gamma is reductive, then E-G admits a Gamma-equivariant isomorphism with the product principal G-bundle X x rho E-G(x(0)), where rho : Gamma -> G is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal G-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal G-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)].