Classification, Reduction, and Stability of Toric Principal Bundles

Abstract

Let X be a complex toric variety equipped with the action of an algebraic torus T, and let G be a complex linear algebraic group. We classify all T-equivariant principal G-bundles \mathcal {E} over X and the morphisms between them. When G is connected and reductive, we characterize the equivariant automorphism group \text {Aut}_T(\mathcal {E} ) of \mathcal {E} as the intersection of certain parabolic subgroups of G that arise naturally from the T-action on \mathcal {E}. We then give a criterion for the equivariant reduction of the structure group of \mathcal {E} to a Levi subgroup of G in terms of \text {Aut}_T(\mathcal {E} ). We use it to prove a principal bundle analogue of Kaneyama’s theorem on equivariant splitting of torus equivariant vector bundles of small rank over a projective space. When X is projective and G is connected and reductive, we show that the notions of stability and equivariant stability are equivalent for any T-equivariant principal G-bundle over X.