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Sanjay Amrutiya

Publications and source records attributed to Sanjay Amrutiya.

13 recordsLinked to original sources
arXiv2026

Moduli of super-representations of quivers

In this note, we construct moduli spaces of super-representations of quivers by extending King's Geometric Invariant Theory (GIT) framework to the super setting. Using the even part of super general linear groups and a parity-shifting operator, we construct moduli spaces that topologically parameterize super-representations of a fixed super-dimension. We also outline the construction of super quiver varieties for super-representations of framed quivers (without doubling); and realise super-Grassmannians and super-flag varieties as geometric quotients within this setting.

math.RT↗
arXiv2026

Real Slices of Parabolic $\mathrm{SL}(r,\mathbb{C})$-Opers

Let $X$ be a Riemann surface equipped with an anti-holomorphic involution $σ_X$. We show that this induces a natural anti-holomorphic involution on the space of parabolic $\mathrm{SL}(r,\mathbb{C})$-opers. The fixed-point locus of this involution is defined as real slice. We further study the induced involutions on different descriptions of parabolic $\mathrm{SL}(r,\mathbb{C})$-opers, in particular differential operators, and prove that these involutions coincide.

math.DG↗
arXiv2026

Semi-finite vector bundles on complex tori

We study finite and semi-finite vector bundles on complex tori. We give an explicit decomposition of such bundles in terms of torsion and unipotent factors. As a consequence, we prove that the extended Nori fundamental group scheme of a complex torus decomposes as the product of its etale fundamental group scheme and its unipotent fundamental group scheme.

math.AG↗
arXiv2026

Smooth relative connections on quiver bundles

We develop a theory of smooth relative connections over the real path algebra $\mathbb{R}Q$ on smooth twisted quiver bundles. We give obstructions to the existence of a smooth relative connection on twisted quiver bundles. For tree-type quiver bundles, we establish a necessary and sufficient condition for the existence of a smooth relative connection. Additionally, we provide a framework for the representation theory of flat quiver bundles, relating the existence of flat relative connections to the underlying quiver representations.

math.DG↗
arXiv2025

On semi-finite vector bundles with connection over Kahler manifolds

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.

math.AG↗
arXiv2025

Notes On Fundamental Groupoid Schemes

In this article, we study the various fundamental groupoid schemes corresponding to Tannakian categories of certain types of vector bundles. We compute fundamental groupoid scheme of anisotropic conic, Klein bottle and abelian varieties. Additionally, we study the relation among various fundamental groupoid schemes by considering their representations.

math.AG↗
arXiv2024

Hodge decomposition theorem on compact $d$-Kähler manifolds

In this article, we will explore the fundamental concepts, including various basic concepts on $d$-complex manifolds, along with several differential operators and examine the relationships between them. A $d$-Kähler manifold is a $d$-complex manifold equipped with a metric that satisfies a specific condition. We prove the Hodge decomposition theorem on compact $d$-Kähler manifolds, which establishes a crucial relationship between certain de-Rham cohomology groups and Dolbeault cohomology groups on a compact $d$-Kähler manifold .

math.DG↗
arXiv2023

A gauge theoretic aspect of parabolic bundles over real curves

In this article, we study the gauge theoretic aspects of real and quaternionic parabolic bundles over a real curve $(X, σ_X)$, where X is a compact Riemann surface and σX is an anti-holomorphic involution. For a fixed real or quaternionic structure on a smooth parabolic bundle, we examine the orbits space of real or quaternionic connection under the appropriate gauge group. The corresponding gauge-theoretic quotients sit inside the real points of the moduli of holomorphic parabolic bundles having a fixed parabolic type on a compact Riemann surface $X$.

math.AG↗
arXiv2022

On d-Holomorphic Connections

We develop the theory of d-holomorphic connections on d-holomorphic vector bundles over a Klein surface by constructing the analogous Atiyah exact sequence for d-holomorphic bundles. We also give a criterion for the existence of d-holomorphic connection in d-holomorphic bundle over a Klein surface in the spirit of the Atiyah-Weil criterion for holomorphic connections.

math.AG↗
arXiv2021

Moduli of parabolic sheaves and filtered Kronecker modules

We give functorial moduli construction of pure parabolic sheaves, in the sense of Alvarez-Consul and A. King, using the moduli of filtered Kronecker modules we introduced in our earlier work. We also use a version of S. G. Langton's result due to K. Yokogawa to deduce the projectivity of moduli of pure parabolic sheaves of maximal dimension. As an application of functorial moduli construction, we can get the morphisms at the level of moduli stacks.

math.AG↗
arXiv2019

Moduli of filtered quiver representations

In this paper, we give a construction of the moduli space of filtered representations of a given quiver of fixed dimension vector with the appropriate notion of stability. The construction of the moduli of filtered representations uses the moduli of representations of ladder quiver. The ladder quiver is introduced using a given quiver and a linear type quiver. We also study determinantal theta functions on such moduli spaces.

math.AG↗
arXiv2018

A note on certain Tannakian group schemes

In this note, we prove that the F-fundamental group scheme is birational invariant for smooth projective varieties. We prove that the F-fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the F-fundamental group scheme and the Nori fundamental group scheme coincides. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles and prove that in this way, we do not get a non-trivial extension of the Nori fundamental group scheme for elliptic curves, unlike in characteristic zero.

math.AG↗