Classification, Reduction, and Stability of Toric Principal Bundles

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 ℰ over X and the morphisms between them. When G is connected and reductive, we characterize the equivariant automorphism group Aut_T(ℰ ) of ℰ as the intersection of certain parabolic subgroups of G that arise naturally from the T -action on ℰ . We then give a criterion for the equivariant reduction of the structure group of ℰ to a Levi subgroup of G in terms of Aut_T(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 .