We classify holomorphic as well as algebraic torus equivariant principal G-bundles over a nonsingular toric variety X, where G is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results. © 2016 World Scientific Publishing Company.

Arijit Dey

Department of Mathematics

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A classification of equivariant principal bundles over nonsingular toric varieties

International Journal of Mathematics

Let EG be a-equivariant algebraic principal G-bundle over a normal complex affine variety X equipped with an action of , where G and are complex linear algebraic groups. Suppose X is contractible as a topological -space with a dense orbit, and x0 X is a -fixed point. We show that if is reductive, then EG admits a -equivariant isomorphism with the product principal G-bundle X × EG(x0), where : 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)]. © 2018 World Scientific Publishing Company.

On equivariant Serre problem for principal bundles

We give a classification of the holomorphic (resp. algebraic) torus equivariant principal G-bundles on a nonsingular toric variety X when G is an Abelian, closed, holomorphic (resp. algebraic) subgroup of the complex general linear group. We prove that any such bundle splits, that is, admits a reduction of structure group to a torus. We give an explicit parametrization of the isomorphism classes of such bundles for a large family of G when X is complete. © 2015 Elsevier Masson SAS

Journal	International Journal of Mathematics
Publisher	World Scientific Publishing Co. Pte Ltd
ISSN	0129167X
Open Access	Yes