Now showing 1 - 10 of 11
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    TANNAKIAN CLASSIFICATION OF EQUIVARIANT PRINCIPAL BUNDLES ON TORIC VARIETIES
    (01-12-2020)
    Biswas, Indranil
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    Poddar, Mainak
    Let X be a complete toric variety equipped with the action of a torus T, and G a reductive algebraic group, defined over an algebraically closed field K. We introduce the notion of a compatible ∑-filtered algebra associated to X, generalizing the notion of a compatible ∑-filtered vector space due to Klyachko, where ∑ denotes the fan of X. We combine Klyachko's classification of T-equivariant vector bundles on X with Nori's Tannakian approach to principal G-bundles, to give an equivalence of categories between T-equivariant principal G-bundles on X and certain compatible ∑-filtered algebras associated to X, when the characteristic of K is 0.
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    On equivariant Serre problem for principal bundles
    (01-08-2018)
    Biswas, Indranil
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    Poddar, Mainak
    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)].
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    A classification of equivariant principal bundles over nonsingular toric varieties
    (01-12-2016)
    Biswas, Indranil
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    Poddar, Mainak
    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.
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    Toric co-higgs bundles on toric varieties
    (01-01-2021)
    Biswas, Indranil
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    Poddar, Mainak
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    Rayan, Steven
    Starting from the data of a nonsingular complex projective toric variety, we define an associated notion of toric co-Higgs bundle. We provide a Lie-theoretic classification of these objects by studying the interaction between Klyachko’s fan filtration and the fiber of the co-Higgs bundle at a closed point in the open orbit of the torus action. This can be interpreted, under certain conditions, as the construction of a coarse moduli scheme of toric co-Higgs bundles of any rank and with any total equivariant Chern class.
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    SYZ duality for parabolic Higgs moduli spaces
    (01-09-2012)
    Biswas, Indranil
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    We prove the SYZ (Strominger-Yau-Zaslow) duality for the moduli space of full flag parabolic Higgs bundles over a compact Riemann surface. In Hausel and Thaddeus (2003) [12], the SYZ duality was proved for moduli spaces of Higgs vector bundles over a compact Riemann surface. © 2012 Elsevier B.V..
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    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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    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.
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    On stability of tangent bundle of toric varieties
    (01-10-2021)
    Biswas, Indranil
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    Genc, Ozhan
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    Poddar, Mainak
    Let X be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle TX. In particular, a complete answer is given when X is a Fano toric variety of dimension four with Picard number at most two, complementing the earlier work of Nakagawa (Tohoku. Math. J.45 (1993) 297–310; 46 (1994) 125–133). We also give an infinite set of examples of Fano toric varieties for which TX is unstable; the dimensions of this collection of varieties are unbounded. Our method is based on the equivariant approach initiated by Klyachko (Izv. Akad. Nauk. SSSR Ser. Mat.53 (1989) 1001–1039, 1135) and developed further by Perling (Math. Nachr. 263/264 (2004) 181–197) and Kool (Moduli spaces of sheaves on toric varieties, Ph.D. thesis (2010) (University of Oxford); Adv. Math. 227 (2011) 1700–1755).
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    Polystable parabolic principal g-bundles and hermitian-einstein connections
    (18-02-2013)
    Biswas, Indranil
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    We show that there is a bijective correspondence between the polystable parabolic principal G-bundles and solutions of the Hermitian-Einstein equation. © Canadian Mathematical Society 2011.
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    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.