arXiv subjects

Pritthijit Biswas

Publications and source records attributed to Pritthijit Biswas.

5 recordsLinked to original sources

arXiv2026

Parabolic Lie algebroid connections on parabolic principal bundles over curves

Let $X$ be a compact connected Riemann surface and $S\,\subset\, X$ a finite subset. We consider parabolic principal $G$--bundles $\mathcal{E}_{G}$ on $X$ with parabolic structure on $S$, where $G$ is a connected complex reductive affine algebraic group. Let $P\, \subset\, G$ be a parabolic subgroup and $\mathcal{E}_{P}\, \subset\, \mathcal{E}_{G}$ a reduction of structure group of $\mathcal{E}_{G}$ to $P$. We give a criterion for the existence of a parabolic Lie algebroid connection on $\mathcal{E}_{P}$ for any given parabolic Lie algebroid on $(X,\,S)$ whose anchor map is not surjective. More precisely, $\mathcal{E}_{P}$ admits a parabolic Lie algebroid connection if the reduction $\mathcal{E}_{P}\, \subset\, \mathcal{E}_{G}$ is parabolically infinitesimally rigid. In particular, the Harder--Narasimhan reduction of $\mathcal{E}_{G}$ admits a parabolic Lie algebroid connection.

math.AG↗

arXiv2025

A note on the Brill-Noether loci of small codimension in moduli space of stable bundles

Let $X$ be a smooth projective curve of genus $g$ over the field $\mathbb{C}$. Let $M_{X}(2,L)$ denote the moduli space of stable rank $2$ vector bundles on $X$ with fixed determinant $L$ of degree $2g-1$. Consider the Brill-Noether subvariety $W^{1}_{X}(2,L)$ of $M_{X}(2,L)$ which parametrises stable vector bundles having at least two linearly independent global sections. In this article, for generic $X$ and $L$, we show that $W^{1}_{X}(2,L)$ is stably-rational when $g=3$, unirational when $g=4$, and rationally chain connected by Hecke curves, when $g\geq 5$. We also show triviality of low dimensional rational Chow groups of an associated Brill-Noether hypersurface.

math.AG↗

arXiv2024

Cohomology of flag bundles over compact Hermitian locally symmetric spaces

Let $E\to B$ be a complex analytic fiber bundle with fiber $F$, a flag variety over a compact complex manifold $B$. We shall obtain a description of the cohomology of $E$ when $B=X_Γ:=Γ\backslash X, E=Y_Γ:=Γ\backslash Y$ and $F=K/H$, a flag variety, where $Y=G/H$ and $X=G/K$, a Hermitian globally symmetric space of non-compact type with $G$ being a real, connected, non-compact, semisimple linear Lie group with no compact factors and simply connected complexification, $K\subset G$, a maximal compact subgroup, $H=Z_K(S)$, the centralizer in $K$ of a toral subgroup $S\subseteq K$ containing $Z(K)$, the center of $K$ and $Γ\subset G$, a uniform and torsionless lattice in $G$. We also obtain a description of the Picard group of $Y_Γ$ and $X_Γ$, for which the complexification of $G$ need not be simply connected. Moreover when $G$ is simple, we obtain the values of $ q$ for which $H^{p,q}(X_Γ)$ vanishes when $p=0,1$. This extends the results of R. Parthasarathy from $1980$, who considered (partially) the case $p=0$.

math.DG↗

arXiv2022

Homological methods in certain Picard group computations

Let $G$ be a connected complex semisimple Lie group, $Γ$ be a cocompact, irreducible and torsionless lattice in $G$ and $K$ be a maximal compact subgroup of $G$. Assume $Γ$ acts by left multiplication and $K$ acts by right multiplication on $G$. Let $M_Γ= Γ\backslash G$, $X=G/K$ and $X_Γ=Γ\backslash X$. In this article we prove that for any $n\geq0$, the composition $H^{n}(X_Γ,\mathbb{C})\rightarrow H^{n}(M_Γ,\mathbb{C})\rightarrow H^{n}(M_Γ,\mathcal{O}_{M_Γ})$ is an isomorphism. As an application when $G$ is simply connected, we compute the Picard group of $M_Γ$ for the cases rank($G$) $=1,2$. More precisely we show that if rank($G$) $=1$, $Pic(M_Γ)=(\mathbb{C}^{r}/\mathbb{Z}^{r})\oplus A$ and if rank($G$) $=2$, then $Pic(M_Γ)\cong A$ via the first Chern class map, where $A$ is the torsion subgroup of $H^{2}(M_Γ,\mathbb{Z})$ and $r$ is the rank of $Γ/[Γ,Γ]$.

math.CV↗

arXiv2022

Picard groups of certain compact complex parallelizable manifolds and related spaces

Let $G$ be a complex simply connected semisimple Lie group and let $Γ$ be a torsionless uniform irreducible lattice in $G$. Then $Γ\backslash G$ is a compact complex non-Kähler manifold whose tangent bundle is holomorphically trivial. In this note we compute the Picard group of $Γ\backslash G$ when $\rank(G)\geq 3$. When $\rank(G)\lneq 3$, we determine the group $Pic^0(Γ\backslash G)\subset Pic(Γ\backslash G)$ of topologically trivial holomorphic line bundles. When $\rank(G)\ge 2$, we also show that $Pic^0(P_Γ)$ is isomorphic to $Pic^0(Y)$ where $P_Γ$ is a $Γ\backslash G$-bundle associated to a principal $G$-bundle over a compact connected complex manifold $Y$, and, when $\rank(G)\ge 3$, we show that $Pic(Y)\to Pic(P_Γ)$ is injective with finite cokernel.

math.DG↗