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Anoop Singh

Publications and source records attributed to Anoop Singh.

At least 19 recordsLinked to original sources
arXiv2026

A remark on the Moduli of parabolic connections

We consider the moduli space of stable parabolic connections of full flag type with generic rational weights on a compact Riemann surface $X$ of genus $g \ge 3$, with singularities along the parabolic points. We compute the Chow groups of the fixed determinant moduli space consisting of parabolic connections whose underlying parabolic bundle is stable. We show that the moduli space of stable parabolic connections is not rationally connected, whereas the fixed determinant moduli space is rationally connected and in fact rational. We also prove that the determinant map to the Picard variety is universal among morphisms to abelian varieties. Finally, we show that the algebraic group of rank one holomorphic connections acts trivially on the cohomology of the moduli space of stable parabolic connections.

math.AG↗
arXiv2026

Early Signatures of Memorization in Diffusion Models via Basin Geometry and Cyclic Denoising

Diffusion models generalize early in training and later reproduce individual training samples. Standard tests detect memorization only once one-shot generation produces near-copies, leaving a released model unaudited until its outputs fail. We show that memorization is encoded in the geometry of the learned energy landscape before it appears in generated samples, a state we call latent memorization. Using score divergence and basin volume, we find that localized basins form around training samples and separate them from held-out samples before the first memorized sample appears, with an onset that follows the same $O(n)$ scaling as the memorization time. We probe these basins with cyclic denoising, which repeatedly applies partial noising and denoising. Under the exact empirical score, we prove that cycling started near an isolated training sample recovers it and returns to it over any finite number of cycles with high probability. In trained models, cycling recovers training images from CelebA and CIFAR-10 checkpoints whose one-shot samples contain no copies, and at a CelebA checkpoint with 0.1% one-shot copies, 500 cycles raise the memorized fraction above 30%. Cycling also reveals degenerate attractors that match no single training image and fade as training proceeds, so residence in a basin does not by itself imply memorization. These findings hold on a Gaussian mixture, CelebA, and CIFAR-10 across optimizers, architectures, noise schedules, and training-set sizes, and extend to off-the-shelf Stable Diffusion v1.4, where the cycled conditional-unconditional divergence gap separates memorized from non-memorized prompts with an AUC of 0.944 and a TPR of 0.866 at 1% FPR. More broadly, what a diffusion model has memorized is a property of the geometry and stability of its learned distribution, and assessing it requires examining this structure rather than generated outputs alone.

cs.LG↗
arXiv2026

Holomorphic Lie algebroid connections over rationally connected varieties

Take a holomorphic Lie algebroid $(V,ϕ)$ over a rationally connected smooth complex projective variety $X$. We show that, under certain conditions, a vector bundle $E$ over $X$ admits a $(V,ϕ)$-connection if and only if $E$ is trivial. Moreover, we prove that under the same conditions, any $(V,ϕ)$-connection over $X$ is flat.

math.AG↗
arXiv2026

Existence of holomorphic Lie algebroid connections in higher dimensions

Let $(V, ϕ)$ 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 $(V, ϕ)$--connection on $E$.

math.AG↗
arXiv2026

Semiprojectivity of the moduli of principal $G$-bundles with $λ$-connections

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 $λ$-connections of fixed topological type $d\in π_1(G)$. As an application, in the smooth case we describe the resulting Bialynicki - Birula decomposition and derive cohomological and motivic consequences.

math.AG↗
arXiv2025

Parabolic vector bundles and Lie algebroid connections

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.

math.AG↗
arXiv2025

A criterion for holomorphic Lie algebroid connections

Given a holomorphic Lie algebroid $(V, ϕ)$ 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 $(V, ϕ)$ is nonsplit, then every holomorphic vector bundle on $X$ admits a holomorphic Lie algebroid connection for $(V, ϕ)$. If $(V, ϕ)$ 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.

math.AG↗
arXiv2024

On pseudoeffective cones of projective bundles and volume 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.

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↗
arXiv2023

Relative connections on Principal bundles and relative equivariant structures

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.

math.AG↗
arXiv2022

A note on the moduli spaces of holomorphic and logarithmic connections over a compact Riemann surface

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.

math.AG↗
arXiv2022

Line bundles on the moduli space of parabolic connections over a compact Riemann surface

Let $X$ be a compact Riemann surface of genus $g \geq 3$ and $S$ a finite subset of $X$. Let $ξ$ be fixed a holomorphic line bundle over $X$ of degree $d$. Let $\mathcal{M}_{pc}(r, d, α)$ (respectively, $\mathcal{M}_{pc}(r, α, ξ)$ ) denote the moduli space of parabolic connections of rank $r$, degree $d$ and full flag rational generic weight system $α$, (respectively, with the fixed determinant $ξ$) singular over the parabolic points $S \subset X$. Let $\mathcal{M}'_{pc}(r, d, α)$ (respectively, $\mathcal{M}'_{pc}(r, α, ξ)$) be the Zariski dense open subset of $\mathcal{M}_{pc}(r, d, α)$ (respectively, $\mathcal{M}_{pc}(r, α, ξ)$ )parametrizing all parabolic connections such that the underlying parabolic bundle is stable. We show that there is a natural compactification of the moduli spaces $\mathcal{M}'_{pc}(r, d, α)$, and $\mathcal{M}'_{pc}(r, α, ξ)$ by smooth divisors. We describe the numerically effectiveness of these divisors at infinity. We determine the Picard group of the moduli spaces $\mathcal{M}_{pc}(r, d, α)$, and $\mathcal{M}_{pc}(r, α, ξ)$. Let $\mathcal{C}(L)$ denote the space of holomorphic connections on an ample line bundle $L$ over the moduli space $\mathcal{M}(r, d, α)$ of parabolic bundles. We show that $\mathcal{C}(L)$ does not admit any non-constant algebraic function.

math.AG↗
arXiv2022

On the relative opers in dimension one

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.

math.DG↗
arXiv2021

Chen-Ruan cohomology and moduli spaces of parabolic bundles over a Riemann surface

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 $α$. Let $P \mathcal{M}_ξ$ denote the moduli space of semi-stable parabolic vector bundles of rank $r$ and determinant $ξ$ over $X$ with weight system $α$, where $r$ is a prime number and $ξ$ 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 $P \mathcal{M}_ξ$ of the group of $r$-torsion points in ${\rm Pic}^0(X)$.

math.AG↗
arXiv2021

On the relative logarithmic connections and relative residue formula

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.

math.AG↗