arXiv subjects
Publications and source records attributed to Indranil Biswas.
Let $X$ be a smooth connected complex projective curve of genus $g$, with $g\,\geq\, 3$. Fix an integer $r\geq 2$, a finite subset $D\, \subset\, X$, and a line bundle $L$ on $X$. We compute the Brauer group of the smooth locus of the moduli space of parabolic symplectic stable bundles of rank $r$ on $X$ equipped with a symplectic form taking values in $L(D)$, where $L(D)$ is given the trivial parabolic structure.
Given a holomorphic Lie algebroid on an m-pointed Riemann surface, we define parabolic Lie algebroid connections on any parabolic vector bundle equipped with parabolic structure over the marked points. An analogue of the Atiyah exact sequence for parabolic Lie algebroids is constructed. For any Lie algebroid whose underlying holomorphic vector bundle is stable, we give a complete characterization of all the parabolic vector bundles that admit a parabolic Lie algebroid connection.
We classify all isomorphisms between moduli stacks of vector bundles of fixed determinant on a smooth complex projective of genus at least 4. It is shown that each isomorphism between two different moduli stacks can be described as a composition of a pullback using an isomorphism of curves, dualization of vector bundles and tensoring with the pullback of a line bundle on the curve. We finally compare the 2-group of automorphisms of the moduli stack of vector bundles with the group of automorphisms of the moduli space of semistable vector bundles.
For a $Γ$--equivariant holomorphic Lie algebroid $(V,\, ϕ)$, on a compact Riemann surface $X$ equipped with an action of a finite group $Γ$, we investigate the equivariant holomorphic Lie algebroid connections on holomorphic principal $G$--bundles over $X$, where $G$ is a connected affine complex reductive group. If $(V,\,ϕ)$ is nonsplit, then it is proved that every holomorphic principal $G$--bundle admits an equivariant holomorphic Lie algebroid connection. If $(V,\,ϕ)$ is split, then it is proved that the following four statements are equivalent: An equivariant principal $G$--bundle $E_G$ admits an equivariant holomorphic Lie algebroid connection. The equivariant principal $G$--bundle $E_G$ admits an equivariant holomorphic connection. The principal $G$--bundle $E_G$ admits a holomorphic connection. For every triple $(P,\, L(P),\, χ)$, where $L(P)$ is a Levi subgroup of a parabolic subgroup $P\, \subset\, G$ and $χ$ is a holomorphic character of $L(P)$, and every $Γ$--equivariant holomorphic reduction of structure group $E_{L(P)}$ of $E_G$ to $L(P)$, the degree of the line bundle over $X$ associated to $E_{L(P)}$ for $χ$ is zero. The correspondence between $Γ$--equivariant principal $G$--bundles over $X$ and parabolic $G$--bundles on $X/Γ$ translates the above result to the context of parabolic $G$--bundles.
We show that a Frobenius reciprocity map on character varieties of surfaces is a Poisson embedding.
Given a symplectic (respectively, orthogonal) parabolic vector bundle over a compact Riemann surface, we prove that its pullback and direct image through a map between compact Riemann surfaces inherit a natural symplectic (respectively, orthogonal) structure. If the parabolic bundle is endowed with a parabolic Higgs field or a parabolic connection which are compatible with the symplectic (respectively, orthogonal) structure, then its pullback and direct image are also compatible with the resulting symplectic (respectively, orthogonal) structure. We also show that these constructions are preserved through the Nonabelian Hodge Correspondence.
We define a class of nonsingular holomorphic foliations on compact complex tori which generalizes (in higher codimension) the turbulent foliations of codimension one constructed by Ghys. For those smooth turbulent foliations we prove that all transversely holomorphic Cartan geometries are flat. We also establish a uniqueness result for the transversely holomorphic Cartan geometries.
Let $G$ be a connected simply connected semisimple complex algebraic group and $P\, \subset\, G$ a parabolic subgroup. We give a necessary and sufficient condition for a line bundle -- on the blow-up of the generalized flag variety $G/P$ along a smooth Schubert variety -- to be ample (respectively, nef). Furthermore, it is shown that every such nef line bundle is actually globally generated. As a consequence, we are able to describe when such a blow-up is (weak) Fano.
The notion of virtual global generation (VGG) for a vector bundle has multiple possible generalization from the case of curves to higher dimensional normal projective varieties. We study relationship between these notions. All these notions agree for curves but in higher dimension we show that this is not the case.
Let $X_0$ be an irreducible smooth projective curve defined over $\overline{\mathbb Q}$ and $\mathbb E$ a vector bundle on $X_0$. We give a criterion for connections on the base change ${\mathbb E}\otimes_{\overline{\mathbb Q}}{\mathbb C} \rightarrow X_0\times_{{\rm Spec}\,\overline{\mathbb Q}} {\rm Spec}\,\mathbb{C} $ to $\mathbb C$ to be the base change of some connection on $\mathbb E$. A similar criterion is given for Higgs fields on ${\mathbb E}\otimes_{\overline{\mathbb Q}}{\mathbb C}$.
We prove that a vector bundle $E$ over a smooth complex projective variety $M$ is étale trivial if and only if $E$ is semiample and $c_1(E) \in H^2(M, {\mathbb Q})$ vanishes. Also, a vector bundle $E$ over a smooth complex projective curve is semiample if and only if it is virtually globally generated.
In the paper "The second cohomology of nilpotent orbits in classical Lie algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P. Chatterjee, and C. Maity, explicit descriptions of the second and first real de Rham cohomology groups of a general homogeneous space of a Lie group are given, extending an earlier result in "On the exactness of Kostant-Kirillov form and the second cohomology of nilpotent orbits, Internat. J. Math. 23 (2012), no. 8, 1250086" by I. Biswas and P. Chatterjee. From the computational viewpoint, they turned out to be new and very useful, and in fact played a crucial role in determining the second cohomology of nilpotent orbits as done in the above two papers. In this paper, we give computable and explicit descriptions of the third and fourth real de Rham cohomologies of a general homogeneous space, in terms of the associated Lie-theoretic data, along the lines mentioned above. We also draw numerous corollaries of our main results in important special settings. Moreover, as a consequence, we obtain a new and interesting invariant by showing that for a large class of homogeneous spaces, the difference between the third and fourth Betti numbers coincides with the difference between the numbers of simple factors of the ambient group and the associated closed subgroup.
Take an irreducible smooth complex projective curve $X$ of genus $g$, with $g\,\geq\, 3$. Let $r$ be an even positive integer. We prove that the Brauer group of the moduli stack of stable parabolic $\textnormal{PSp}(r,\mathbb{C})$--bundles on $X$, of full-flag parabolic data along a set of marked points on $X$, coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of stable parabolic $\textnormal{PSp}(r,\mathbb{C})$--bundles. Under certain conditions on the parabolic types, we also compute the Brauer group of the smooth locus of this coarse moduli space. Similar computations are also done for the case of partial flags.
Let $X$ be a compact connected Kähler manifold. We consider the category $\mathcal{C}^\mathrm{EC}(X)$ of flat holomorphic connections $(E,\, \nabla^E)$ over $X$ satisfying the condition that the underlying holomorphic vector bundle $E$ admits a filtration of holomorphic subbundles preserved by the connection $\nabla^E$ such that the monodromy of the induced connection on each successive quotient has finite image. The category $\mathcal{C}^\mathrm{EC}(X)$, equipped with the neutral fiber functor that sends any object $(E,\, \nabla^E)$ to the fiber $E_{x_0}$, where $x_0\, \in\, X$ is a fixed point, defines a neutral Tannakian category over $\mathbb{C}$. Let $\varpi^{\mathrm{EC}}(X,\, x_0)$ denote the affine group scheme corresponding to this neutral Tannakian category $\mathcal{C}^\mathrm{EC}(X)$. Let $π^{\mathrm{EN}}(X,\, x_0)$ be an extension of the Nori fundamental group scheme over $\mathbb{C}$. We show that $π^{\mathrm{EN}}(X,\, x_0)$ is a closed subgroup scheme of $\varpi^{\mathrm{EC}}(X,\, x_0)$. Finally, we discuss an example illustrating that if $X$ is not Kähler, then the natural homomorphism $π^{\mathrm{EN}}(X,\, x_0)\, \longrightarrow\, \varpi^{\mathrm{EC}}(X,\, x_0)$ might fail to be an embedding.
Quantum information theory reveals a clear distinction between local and nonlocal correlations through the entanglement across spatially separated subsystems. The orthogonal complement of an unextendible biseparable basis (UBB) consists entirely of genuine multipartite entangled states, representing the most robust form of such nonlocal correlations. In this letter, we provide a sufficient condition for any subspace to be genuinely entangled, enabling the systematic construction of high-dimensional genuinely entangled subspaces (GESs) from UBBs. Our construction yields the largest known GES ever obtained from a UBB. In fact, every state in this subspace is 1-distillable across every bipartition which is one of the crucial result we obtained. Furthermore, we prove that every UBB is indistinguishable under LOCC protocols, underscoring a distinct manifestation of quantum nonlocality. The UBBs we construct exhibit strong nonlocality in this scenario, making cryptographic protocols secure not only against LOCC-based attacks but also against coordinated group attacks. We introduce a no-go condition that certifies such an extreme form of nonlocality. All previously known UBBs satisfy this condition, which may lead to the misconception that strong nonlocality is an inherent property of every UBB. However, we construct a UBB that violates the no-go condition and exhibits locality across certain bipartitions, challenging conventional notions of unextendibility and nonlocality in multipartite quantum systems.
Let G be a complex reductive group and D a finite subset of a compact Riemann surface X. It was shown in [BJ] that the moduli space of G-characters of the complement of D in X has a natural Poisson structure. We show that the moduli space of logarithmic G-connections on X singular over D has a Poisson structure. It is proved that the monodromy map from the moduli space of logarithmic G-connections to the moduli space of G-characters is Poisson structure preserving.
Let $G$ be a semisimple, simply connected, affine algebraic group defined over $\mathbb C$. Consider the Liouville symplectic structure on the total space $T^*G((t))$ of the cotangent bundle of the loop group $G((t))$, where $t$ is a formal parameter. We show that the Liouville symplectic structure on $T^*G((t))$ induces the symplectic structures on the moduli stack of framed principal Higgs $G$-bundles on a compact connected Riemann surface $X$ and also on the moduli spaces of framed $G$-connections on $X$. Similar symplectic structures -- on the moduli stack of framed principal Higgs $G$-bundles, with finite order framing, and also framed connections on $X$, with finite order framing -- were constructed earlier by various authors. Our results show that they all have a common origin.
Let ${\mathcal M}$ be a moduli space of stable vector bundles of rank $r$ and determinant $ξ$ on a compact Riemann surface $X$. Fix a semistable holomorphic vector bundle $F$ on $X$ such that $χ(E\otimes F)= 0$ for $E \in \mathcal M$. Then any $E\in \mathcal M$ with $H^0(X, E\otimes F) = 0 = H^1(X, E\otimes F)$ has a natural holomorphic projective connection. The moduli space of pairs $(E,\, \nabla)$, where $E\, \in\, \mathcal M$ and $\nabla$ is a holomorphic projective connection on $E$, is an algebraic $T^*{\mathcal M}$--torsor on $\mathcal M$. We identify this $T^*{\mathcal M}$--torsor on $\mathcal M$ with the $T^*{\mathcal M}$--torsor given by the sheaf of connections on an ample line bundle over $\mathcal M$.