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    Publication
    Classification, Reduction, and Stability of Toric Principal Bundles
    (01-01-2023)
    Dasgupta, Jyoti
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    Khan, Bivas
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    Biswas, Indranil
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    Poddar, Mainak
    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.
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    Publication
    Equivariant principal bundles and logarithmic connections on toric varieties
    (01-01-2016)
    Biswas, Indranil
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    Poddar, Mainak
    Let M be a smooth complex projective toric variety equipped with an action of a torus T, such that the complement D of the open T-orbit in M is a simple normal crossing divisor. Let G be a complex reductive affine algebraic group. We prove that an algebraic principal G-bundle EG → M admits a T-equivariant structure if and only if EG admits a logarithmic connection singular over D. If EH → M is a T-equivariant algebraic principal H-bundle, where H is any complex affine algebraic group, then EH in fact has a canonical integrable logarithmic connection singular over D.