arXiv subjects
Publications and source records attributed to Arijit Dey.
Let $M(2,\textbf{\underline{w}},χ)$ be the moduli space of rank $2$ torsion-free sheaves over a reducible nodal curve with each component having utmost two nodal singularities. We show that in each component of $M(2,\textbf{\underline{w}},χ)$, the closure of rank $2$ vector bundles with fixed determinant is rational.
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.
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 $M$ 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 $E_G\to M$ admits a $T$--equivariant structure if and only if $E_G$ admits a logarithmic connection singular over $D$. If $E_H\to M$ is a $T$-equivariant algebraic principal $H$--bundle, where $H$ is any complex affine algebraic group, then $E_H$ in fact has a canonical integrable logarithmic connection singular over $D$.
We give a classification of the equivariant principal $G$-bundles on a nonsingular toric variety when $G$ is a closed Abelian subgroup of $GL_k(\mathbb{C})$. We prove that any such bundle splits, that is, admits a reduction of structure group to the intersection of $G$ with a torus. We give an explicit parametrization of the isomorphism classes of such bundles for a large family of $G$ when $X$ is complete.
We prove the SYZ (Strominger-Yau-Zaslow) duality for the moduli space of full flag parabolic Higgs bundles over a compact Riemann surface.The SYZ duality was proved for moduli spaces of Higgs vector bundles over a compact Riemann surface by Hausel and Thaddeus.
Let ${\mathcal P}{\mathcal M}^α_s$ be a moduli space of stable parabolic vector bundles of rank $n \geq 2$ and fixed determinant of degree $d$ over a compact connected Riemann surface $X$ of genus $g(X) \geq 2$. If $g(X) = 2$, then we assume that $n > 2$. Let $m$ denote the greatest common divisor of $d$, $n$ and the dimensions of all the successive quotients of the quasi-parabolic filtrations. We prove that the cohomological Brauer group ${\rm Br}({\mathcal P}{\mathcal M}^α_s)$ is isomorphic to the cyclic group ${\mathbb Z}/ m{\mathbb Z}$. We also show that ${\rm Br}({\mathcal P}{\mathcal M}^α_s)$ is generated by the Brauer class of the projective bundle over ${\mathcal P}{\mathcal M}^α_s$ obtained by restricting the universal projective bundle over $X\times {\mathcal P}{\mathcal M}^α_s$. We also prove that there is a universal vector bundle over $X\times {\mathcal P}{\mathcal M}^α_s$ if and only if $m=1$.
We construct a full, strongly exceptional collection of line bundles on the variety X that is the blow up of the projectivization of the vector bundle O_{P^{n-1}}\oplus O_{P^{n-1}}(b) along a linear space of dimension n-2, where b is a non-negative integer.
The existence and uniqueness of H-N reduction for the Higgs principal bundles over nonsingular projective variety is shown. We also extend the notion of H-N reduction for (Γ, G)-bundles and ramified G-bundles over a smooth curve.