Arijit Dey

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.

We show that each of the irreducible components of moduli of rank 2 torsion-free sheaves with odd Euler characteristic over a reducible nodal curve is rational.

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)].

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.

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.

We correct the proof of [2, Proposition 3.1.1].

Let M(2,w̲,χ) be the moduli space of rank 2 torsion-free sheaves of fixed determinant and odd Euler characteristic over a reducible nodal curve with each irreducible component having utmost two nodal singularities. We show that in each irreducible component of M(2,w̲,χ), the closure of rank 2 vector bundles is rational.

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..

We compute Betti numbers of both the components of the moduli space of rank 2 semi-stable torsion-free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.

Let X be any scheme defined over a Dedekind scheme S with a given section x∈X(S). We prove the existence of a pro-finite S-group scheme ℵ(X,x) and a universal ℵ(X,x)-torsor dominating all the pro-finite pointed torsors over X. Though ℵ(X,x) may not be unique in general it still can provide useful information in order to better understand X. In a similar way we prove the existence of a pro-algebraic S-group scheme ℵalg(X,x) and a ℵalg(X,x)-torsor dominating all the pro-algebraic and affine pointed torsors over X. The case where X→S has no sections is also considered.