Arijit Dey

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.

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.

Let X be a smooth projective curve of genus g≥1 over a finite field Fq with q elements, such that the function field Fq(X) is a geometric Galois extension of the rational function field Fq(x) with N=#Gal(Fq(X)/Fq(x)). Let ML(2,1) be the moduli space of rank 2 stable vector bundles over X‾=X×FqFq‾ with fixed determinant L, where L is a line bundle on X‾ of degree 1. Let Nq(ML(2,1)) be the cardinality of the set of Fq-rational points of ML(2,1). We give an estimate of Nq(ML(2,1)) in terms of N,q and g. Further we study the fluctuations of the quantity logNq(ML(2,1))−3(g−1)logq as the curve X as well as L varies over a large family of hyperelliptic curves (N=2) of genus g≥2. We find the limiting distribution of Nq(ML(2,1))−3(g−1)log⁡q as g grows and q is fixed, in terms of its characteristic function. When both g and q grow we see that q3/2(log⁡Nq(ML(2,1))−3(g−1)log⁡q) has a standard Gaussian distribution.

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.

We give a classification of the holomorphic (resp. algebraic) torus equivariant principal G-bundles on a nonsingular toric variety X when G is an Abelian, closed, holomorphic (resp. algebraic) subgroup of the complex general linear group. We prove that any such bundle splits, that is, admits a reduction of structure group to a torus. We give an explicit parametrization of the isomorphism classes of such bundles for a large family of G when X is complete.