arXiv subjects
Publications and source records attributed to Anoop Singh.
Take a holomorphic Lie algebroid $(V,\phi)$ over a rationally connected smooth complex projective variety $X$. We show that, under certain conditions, a vector bundle $E$ over $X$ admits a $(V,\phi)$-connection if and only if $E$ is trivial. Moreover, we prove that under the same conditions, any $(V,\phi)$-connection over $X$ is flat.
Given a holomorphic Lie algebroid $(V, \phi)$ on a compact connected Riemann surface $X$, we give a necessary and sufficient condition for a parabolic vector bundle on $X$, with parabolic structure over a nonzero reduced effective divisor, to admit a parabolic Lie algebroid connection for the Lie algebroid $(V, \phi)$.
Let $(V, \phi)$ be a holomorphic Lie algebroid over an irreducible smooth complex projective variety $X$ of dimension at least three, and let $E$ be a holomorphic vector bundle on $X$. We establish a necessary and sufficient condition for the existence of a holomorphic $(V, \phi)$--connection on $E$.
We consider the moduli space of parabolic connections with rational generic weights over a compact Riemann surface of genus $g \geq 3$. We determine the Chow group of the moduli space of parabolic connections such that the underlying parabolic bundle is stable. We also discuss the rationality and rationally connectedness of the moduli space of parabolic connections.
Given a holomorphic Lie algebroid $(V, \phi)$ on a compact connected Riemann surface $X$, we give a necessary and sufficient condition for a holomorphic vector bundle $E$ on $X$ to admit a holomorphic Lie algebroid connection. If $(V, \phi)$ is nonsplit, then every holomorphic vector bundle on $X$ admits a holomorphic Lie algebroid connection for $(V, \phi)$. If $(V, \phi)$ is split, then a holomorphic vector bundle $E$ on $X$ admits a holomorphic Lie algebroid connection if and only if the degree of each indecomposable component of $E$ is zero.
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.
Let $X$ be a compact connected Riemann surface and $(V, \phi)$ a holomorphic Lie algebroid on $X$ such that the holomorphic vector bundle $V$ is stable. We give a necessary and sufficient condition on holomorphic vector bundles $E$ on $X$ to admit a Lie algebroid connection.
Let $X$ be a compact connected Riemann surface of genus $g \geq 2$ and $G$ a connected reductive affine algebraic group over $\mathbb{C}$. We prove the semiprojectivity of the moduli spaces of semistable $G$-Higgs bundles and $G$-bundles with $\lambda$-connections of fixed topological type $d\in \pi_1(G)$. As an application, in the smooth case we describe the resulting Bialynicki - Birula decomposition and derive cohomological and motivic consequences.
We investigate relative holomorphic connections on a principal bundle over a family of compact complex manifolds. A sufficient condition is given for the existence of a relative holomorphic connection on a holomorphic principal bundle over a complex analytic family. We also introduce the notion of relative equivariant bundles and establish its relation with relative holomorphic connections on principal bundles.
We explore algebro-geometric properties of the moduli space of holomorphic Lie algebroid ($ \mathcal{L} $) connections on a compact Riemann surface $X$ of genus $g \,\geq\, 3$. A smooth compactification of the moduli space of $\mathcal{L}$-connections, such that underlying vector bundle is stable, is constructed; the complement of the moduli space inside the compactification is a divisor. A criterion for the numerical effectiveness of the boundary divisor is given. We compute the Picard group of the moduli space, and analyze Lie algebroid Atiyah bundles associated with an ample line bundle. This enables us to conclude that regular functions on the space of certain Lie algebroid connections are constants. Moreover, under some condition, it is shown that the moduli space of $\mathcal{L}$-connections does not admit non-constant algebraic functions. Rationally connectedness of the moduli spaces is explored.
Let $X$ be a compact Riemann surface of genus $g \geq 3$ and $S$ a finite subset of $X$. Let $\xi$ be fixed a holomorphic line bundle over $X$ of degree $d$. Let $\mathcal{M}_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}_{pc}(r, \alpha, \xi)$ ) denote the moduli space of parabolic connections of rank $r$, degree $d$ and full flag rational generic weight system $\alpha$, (respectively, with the fixed determinant $\xi$) singular over the parabolic points $S \subset X$. Let $\mathcal{M}'_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}'_{pc}(r, \alpha, \xi)$) be the Zariski dense open subset of $\mathcal{M}_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}_{pc}(r, \alpha, \xi)$ )parametrizing all parabolic connections such that the underlying parabolic bundle is stable. We show that there is a natural compactification of the moduli spaces $\mathcal{M}'_{pc}(r, d, \alpha)$, and $\mathcal{M}'_{pc}(r, \alpha, \xi)$ by smooth divisors. We describe the numerically effectiveness of these divisors at infinity. We determine the Picard group of the moduli spaces $\mathcal{M}_{pc}(r, d, \alpha)$, and $\mathcal{M}_{pc}(r, \alpha, \xi)$. Let $\mathcal{C}(L)$ denote the space of holomorphic connections on an ample line bundle $L$ over the moduli space $\mathcal{M}(r, d, \alpha)$ of parabolic bundles. We show that $\mathcal{C}(L)$ does not admit any non-constant algebraic function.
In this article, we compute the pseudo-effective cones of various projective bundles $\mathbb{P}_X(E)$ over higher dimensional varieties $X$ under some assumptions on $X$ as well as on the vector bundle $E$. We also compute the volume function on fibre product $\mathbb{P}(E)\times_C\mathbb{P}(F)$ of two projective bundles over a smooth irreducible complex projective curve $C$. In particular, we show that the volume function on the fiber product of two ruled surfaces is of polynomial type.
We investigate the relative opers over the complex analytic family of compact complex manifolds of relative dimension one. We introduce the notion of relative opers arising from the second fundamental form associated with a relative holomorphic connection. We also investigate the relative differential operators over the complex analytic family of compact complex manifolds whose symbol is the identity automorphism. We show that the set of equivalent relative opers arising from the second fundamental form is in bijective correspondence with the set of equivalent relative differential operators whose symbol is the identity automorphism.
Let $(X,\,D)$ be an $m$-pointed compact Riemann surface of genus at least $2$. For each $x \,\in\, D$, fix full flag and concentrated weight system $\alpha$. Let $P \mathcal{M}_{\xi}$ denote the moduli space of semi-stable parabolic vector bundles of rank $r$ and determinant $\xi$ over $X$ with weight system $\alpha$, where $r$ is a prime number and $\xi$ is a holomorphic line bundle over $X$ of degree $d$ which is not a multiple of $r$. We compute the Chen-Ruan cohomology of the orbifold for the action on $P \mathcal{M}_{\xi}$ of the group of $r$-torsion points in ${\rm Pic}^0(X)$.
We investigate the relative logarithmic connections on a holomorphic vector bundle over a complex analytic family. We give a sufficient condition for the existence of a relative logarithmic connection on a holomorphic vector bundle singular over a relative simple normal crossing divisor. We define the relative residue of relative logarithmic connection and express relative Chern classes of a holomorphic vector bundle in terms of relative residues.
Let $X$ be a compact Riemann surface of genus $g \geq 3$. We consider the moduli space of holomorphic connections over $X$ and the moduli space of logarithmic connections singular over a finite subset of $X$ with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces, and show that they are constant under certain condition. We show the Torelli type theorem for the moduli space of logarithmic connections. We also describe the rational connectedness of these moduli spaces.
Let $k$ be a perfect field, and $X$ an irreducible smooth projective curve over $k$. We give a criterion for a vector bundle over $X$ to admit a logarithmic connection singular over a finite subset of $X$ with given residues, where residues are assumed to be rigid.
We introduce the notion of Hitchin variety over $\C$. Let $L$ be a holomorphic line bundle over a Hitchin variety $X$. We investigate the space of all global sections of sheaf of differential operators $\cat{D}^k (L)$ and symmetric powers of sheaf of first order differential operators $\cat{S}^k(\cat{D}^1 (L))$ over $X$ and show that for a projective Hithcin variety both the spaces are one dimensional. As an application, we show that the space $\cat{C}(L)$ of holomorphic connections on $L$ does not admit any non-constant regular function.