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Indranil Biswas

Publications and source records attributed to Indranil Biswas.

At least 109 records · Page 6Linked to original sources
arXiv2024

Chen--Ruan cohomology and orbifold Euler characteristic of moduli spaces of parabolic bundles

We consider the moduli space of stable parabolic Higgs bundles of rank $r$ and fixed determinant, and having full flag quasi-parabolic structures over an arbitrary parabolic divisor on a smooth complex projective curve $X$ of genus $g$, with $g\,\geq\, 2$. The group $Γ$ of $r$-torsion points of the Jacobian of $X$ acts on this moduli space. We describe the connected components of the various fixed point loci of this moduli under non-trivial elements from $Γ$. When the Higgs field is zero, or in other words when we restrict ourselves to the moduli of stable parabolic bundles, we also compute the orbifold Euler characteristic of the corresponding global quotient orbifold. We also describe the Chen--Ruan cohomology groups of this orbifold under certain conditions on the rank and degree, and describe the Chen--Ruan product structure in special cases.

math.AG↗
arXiv2024

On the fundamental group schemes of certain quotient varieties

In \cite{armstrong}, M. Armstrong proved a beautiful result describing fundamental groups of quotient spaces. In this paper we prove an analogue of Armstrong's theorem in the setting of $F$-divided \cite{dS07} and essentially finite \cite{Nori76} fundamental group schemes.

math.AG↗
arXiv2024

Bertini type results and their applications

We prove Bertini type theorems and give some applications of them. The applications are in the context of Lefschetz theorem for Nori fundamental group for normal varieties as well as for geometric formal orbifolds. In another application, it is shown that certain class of a smooth quasi-projective variety contains a smooth curve such that irreducible lisse \ell-adic sheaves on the variety with "ramification bounded by a branch data" remains irreducible when restricted to the curve.

math.AG↗
arXiv2024

Torsors on moduli spaces of principal $G$-bundles

Let $G$ be a semisimple complex algebraic group with a simple Lie algebra $\mathfrak{g}$, and let $\mathcal{M}^0_{G}$ denote the moduli stack of topologically trivial stable $G$-bundles on a smooth projective curve $C$. Fix a theta characteristic $κ$ on $C$ which is even in case $\dim{\mathfrak{g}}$ is odd. We show that there is a nonempty Zariski open substack ${\mathcal U}_κ$ of $\mathcal{M}^0_{G}$ such that $H^i(C,\, \text{ad}(E_G)\otimesκ) \,=\, 0$, $i\,=\, 1,\, 2$, for all $E_G\,\in\, {\mathcal U}_κ$. It is shown that any such $E_G$ has a canonical connection. It is also shown that the tangent bundle $T{U}_κ$ has a natural splitting, where $U_κ$ is the restriction of $\mathcal{U}_κ$ to the semi-stable locus. We also produce an isomorphism between two naturally occurring $Ω^1_{{M}^{rs}_{G}}$--torsors on the moduli space of regularly stable ${M}^{rs}_{G}$.

math.AG↗
arXiv2024

Line bundles on the moduli space of Lie algebroid connections over a curve

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.

math.AG↗
arXiv2024

Strong quantum nonlocality: Unextendible biseparability beyond unextendible product basis

An unextendible biseparable basis (UBB) is a set of orthogonal pure biseparable states which span a subspace of a given Hilbert space while the complementary subspace contains only genuinely entangled states. These biseparable bases are useful to produce genuinely entangled subspace in multipartite system. Such a subspace could be more beneficial for information theoretic applications if we are able to extract distillable entanglement across every bipartition from each state of this subspace. In this manuscript, we have derived a rule for constructing such a class of UBB which exhibits the phenomenon of strong quantum nonlocality. This result positively answers the open problem raised by Agrawal et al. [Phys. Rev. A 99, 032335 (2019)]; that there exists a UBB which can demonstrate the phenomenon of strong quantum nonlocality in the perspective of local irreducibility paradigm.

quant-ph↗
arXiv2024

Higher genus maxfaces with Enneper end

We have proven the existence of new higher-genus maxfaces with Enneper end. These maxfaces are not the companions of any existing minimal surfaces, and furthermore, the singularity set is located away from the ends. The nature of the singularities is systematically investigated.

math.DG↗
arXiv2024

Characterization of genuine ramification using formal orbifolds

We give a characterization of genuinely ramified maps of formal orbifolds in the Tannakian framework. In particular we show that a morphism is genuinely ramified if and only if the pullback of every stable bundle remains stable in the orbifold category. We also give some other characterizations of genuine ramification. This generalizes the results of [BKP1] and [BP1]. In fact, it is a positive characteristic analogue of results in [BKP2].

math.AG↗
arXiv2024

Pushforward of structure sheaf and virtual global generation

Let $f:X\rightarrow Y$ be a generically smooth morphism between irreducible smooth projective curves over an algebraically closed field of arbitrary characteristic. We prove that the vector bundle $((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is virtually globally generated. Moreover, $((f_*{\mathcal O}_X)/{\mathcal O}_Y)^*$ is ample if and only if $f$ is genuinely ramified.

math.AG↗
arXiv2024

Infinitesimal deformations of some Quot schemes

Let $E$ be a vector bundle on a smooth complex projective curve $C$ of genus at least two. Let $\mathcal{Q}(E,d)$ be the Quot scheme parameterizing the torsion quotients of $E$ of degree $d$. We compute the cohomologies of the tangent bundle $T_{\mathcal{Q}(E,d)}$. In particular, the space of infinitesimal deformations of $\mathcal{Q}(E,d)$ is computed. Kempf and Fantechi computed the space of infinitesimal deformations of $\mathcal{Q}(\mathcal{O}_C,d)\,=\, C^{(d)}$. We also explicitly describe the infinitesimal deformations of $\mathcal{Q}(E,d)$.

math.AG↗
arXiv2024

Genuinely ramified maps and monodromy

For any genuinely ramified morphism $f\, :\, Y\, \longrightarrow\, X$ between irreducible smooth projective curves we prove that $\overline{(Y\times_X Y) \setminus Δ}$ is connected, where $Δ\, \subset\, Y\times_X Y$ is the diagonal. Using this result the following are proved: If $f$ is further Morse then the Galois closure is the symmetric group $S_d$, where $d\,=\, \text{degree}(f)$. The Galois group of the general projection, to a line, of any smooth curve $X\,\subset\, \PP^n$ of degree $d$, which is not contained in a hyperplane and contains a non-flex point, is $S_d$.

math.AG↗
arXiv2024

Effective cone of a Grassmann bundle over a curve defined over $\overline{\mathbb F}_p$

Let $X$ be an irreducible smooth projective curve defined over $\overline{\mathbb F}_p$ and $E$ a vector bundle on $X$ of rank at least two. For any $1\, \leq\, r\, <\, {\rm rank}(E)$, let ${\rm Gr}_r(E)$ be the Grassmann bundle over $X$ parametrizing all the $r$ dimensional quotients of the fibers of $E$. We prove that the effective cone in ${\rm NS}({\rm Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$ coincides with the pseudo-effective cone in ${\rm NS}({\rm Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$. When $r\,=\,1$ or ${\rm rank}(E)-1$, this was proved by A. Moriwaki.

math.AG↗
arXiv2024

Ramified covering maps of singular curves and stability of pulled back bundles

Let $f : X \rightarrow Y$ be a generically smooth nonconstant morphism between irreducible projective curves, defined over an algebraically closed field, which is étale on an open subset of $Y$ that contains both the singular locus of $Y$ and the image, in $Y$, of the singular locus of $X$. We prove that the following statements are equivalent: \begin{enumerate} \item The homomorphism of étale fundamental groups $$f_* : π_1^{\rm et}(X) \rightarrowπ_1^{\rm et}(Y)$$ induced by $f$ is surjective. \item There is no nontrivial étale covering $ϕ: Y' \rightarrow Y$ admitting a morphism $q: X \rightarrow Y'$ such that $ϕ\circ q = f$. \item The fiber product $X\times_Y X$ is connected. \item $\dim H^0(X, f^*f_* {\mathcal O}_X)= 1$. \item ${\mathcal O}_Y \subset f_*{\mathcal O}_X$ is the maximal semistable subsheaf. \item The pullback $f^*E$ of every stable sheaf $E$ on $Y$ is also stable. \end{enumerate}

math.AG↗
arXiv2023

Characteristic forms for holomorphic families of local systems

Let $f:\mathcal{X}\to S$ be a proper holomorphic submersion of complex manifolds and $G$ a complex reductive linear algebraic group with Lie algebra $\mathfrak{g}$. Assume also given a holomorphic principal $G$-bundle $\mathcal{P}$ over $\mathcal{X}$ which is endowed with a holomorphic connection $\nabla$ relative to $f$ that is flat (this to be thought of as a holomorphic family of compact complex manifolds endowed with a holomorphic principal $G$-bundle with flat connection). We show that a refinement of the Chern-Weil homomorphism yields a graded algebra homomorphism $\mathbb{C}[\mathfrak{g}]^G\to \bigoplus_{n\ge 0} H^0(S,\,Ω^n_{S,cl}\otimes R^nf_*\mathbb{C})$, where $Ω^n_{S,cl}$ stands for the sheaf of closed holomorphic $n$-forms on $S$. If the fibers of $f$ are compact Riemann surfaces and we take as our invariant the Killing form, then we recover Goldman's closed holomorphic $2$-form on the base $S$.

math.AG↗