arXiv subjects

Soumya Ganguly

Publications and source records attributed to Soumya Ganguly.

9 recordsLinked to original sources

arXiv2026

On the Classification of Stein spaces with Bergman-Einstein metrics

For every $N\ge 2$, we prove that the Bergman metric on the regular locus of a finite ball quotient $\mathbb{B}^N/Γ$, where $Γ\subset \mathrm{U}(N)$ is finite and fixed-point-free, is Kähler-Einstein if and only if $Γ$ is trivial. Consequently, if $Ω$ is an $N$-dimensional normal Stein space with isolated singularities and compact, smooth, strongly pseudoconvex boundary admitting a real-algebraic CR realization, then the Bergman metric on $Ω_{\mathrm{reg}}$ is Kähler-Einstein if and only if $Ω$ is biholomorphic to $\mathbb{B}^N$. This proves an algebraic version of the Cheng-Huang-Xiao conjecture in every complex dimension $N\ge 2$.

math.CV↗

arXiv2026

Cohomology of complex supertori

We consider supertori which are quotients of affine superspace by translations by algebraically independent odd parameters. Specifically, we describe the ring structure of its space of global sections by generators and relations and completely determine the coherent cohomology groups of its structure sheaf by reducing it to a problem of Lie algebra cohomology. We also show that Poincaré duality on sheaf cohomology is compatible with that of the group cohomology of the translation group and give explicit tables of examples. Finally, we compute the Picard groups of these supertori.

math.AG↗

arXiv2026

Attention as In-Context Empirical Bayes: A Two-Stage View via Particle Dynamics

We study minimal attention-only transformers under all-token corruption and show they admit a two-stage empirical Bayes interpretation. A single attention step computes a kernel-weighted posterior mean with respect to the empirical distribution defined by the context. Depth refines this distribution through particle dynamics (Stage 1), while a long-range skip-connection carries the noisy input as a query for posterior inference (Stage 2), revealing distinct statistical roles for depth and attention residuals. The framework isolates a minimal setting in which the context itself induces a depth-dependent energy landscape governing in-context inference. We show that effective denoising can emerge without an explicit noise schedule: a fixed kernel bandwidth and finite integration horizon suffice, yielding a principled depth-noise relationship. We further establish a posterior-mean recovery guarantee for a class of well-behaved priors, where the empirical estimator converges to the Bayes-optimal predictor under asymptotic conditions. Connecting these dynamics to reverse-diffusion limits, our results provide a statistical interpretation of attention as in-context inference via sample-based posterior estimation, without explicit density modeling.

cs.LG↗

arXiv2026

Abundance of Bergman metrics with constant positive holomorphic sectional curvature

An outstanding open question, which has attracted renewed attention following the pioneering work of Huang--Li--Treuer, is whether, for a given positive integer $m$, there exists a complex manifold whose Bergman metric is locally isometric to $m$ times the Fubini--Study metric. Previously, this question had only been resolved in the case $m=1$. In this paper, we construct, for any pair of positive integers $(m,n)$ with $n \geq 2$, an $\mathbb{R}$-parameter (hence uncountable) family of Reinhardt domains in $\mathbb{C}^n$ whose Bergman metrics are all locally isometric to $m$ times the Fubini--Study metric. Moreover, we show that the domains in this family are mutually Bergman inequivalent. This not only answers the folklore question, but also suggests that a reasonable classification of the geometry of such complex manifolds is infeasible. We also note such examples cannot exist in dimension one. The results complete the remaining open case in the study of complex manifolds whose Bergman space separates points and whose Bergman metric has constant holomorphic sectional curvature. Our approach differs from existing methods in the literature. We reduce the construction to a mapping problem and apply a Brouwer fixed point argument to establish the existence of the desired domains.

math.CV↗

arXiv2025

Local algebraicity and localization of the Bergman kernel on Stein spaces with finite type boundaries

On a two dimensional Stein space with isolated, normal singularities, smooth finite type boundary, and locally algebraic Bergman kernel, we establish an estimate on the type of the boundary in terms of the local algebraic degree of the Bergman kernel. As an application, we characterize two dimensional ball quotients as the only Stein spaces with smooth finite type boundary and locally rational Bergman kernel. A key ingredient in the proof of the degree estimate is a new localization result for the Bergman kernel of a pseudoconvex, finite type domain in a complex manifold.

math.CV↗

arXiv2025

Rotational symmetries of domains and orthogonality relations

Let $Ω\subset \mathbb{C}^n$ be a domain whose Bergman space contains all holomorphic monomials. We derive sufficient conditions for $Ω$ to be Reinhardt, complete Reinhardt, circular or Hartogs in terms of the orthogonality relations of the monomials with respect to their $L^2$-inner products and their $L^2$-norms. More generally, we give sufficient conditions for $Ω$ to be invariant under a linear group action of an $r$-dimensional torus, where $r \in \{1,\ldots, n\}$.

math.CV↗

arXiv2024

On Laplacian and Distance Laplacian Spectra of Generalized Fan Graph & a New Graph Class

Given a graph $G$, the Laplacian matrix of $G$, $L(G)$ is the difference of the adjacency matrix $A(G)$ and $\text{Deg}(G)$, where $\text{Deg}(G)$ is the diagonal matrix of vertex degrees. The distance Laplacian matrix $D^L({G})$ is the difference of the transmission matrix of $G$ and the distance matrix of $G$. In the given paper, we first obtain the Laplacian and distance Laplacian spectrum of generalized fan graphs. We then introduce a new graph class which is denoted by $\mathcal{NC}(F_{m,n})$. Finally, we determine the Laplacian spectrum and the distance Laplacian spectrum of $\mathcal{NC}(F_{m,n})$.

math.CO↗

arXiv2024

Bergman-Einstein metrics on two-dimensional Stein spaces

We show that the Bergman metric of the ball quotients $\mathbb{B}^2/Γ$, where $Γ$ is a finite and fixed point free group, is Kähler-Einstein if and only if $Γ$ is trivial. As a consequence, we characterize the unit ball $\mathbb{B}^2$, among 2 dimensional Stein spaces with isolated normal singularities, proving an algebraic version of Cheng's conjecture for 2 dimensional Stein spaces.

math.CV↗

arXiv2024

Uniform $\mathcal{C}^k$ Approximation of $G$-Invariant and Antisymmetric Functions, Embedding Dimensions, and Polynomial Representations

For any subgroup $G$ of the symmetric group $\mathcal{S}_n$ on $n$ symbols, we present results for the uniform $\mathcal{C}^k$ approximation of $G$-invariant functions by $G$-invariant polynomials. For the case of totally symmetric functions ($G = \mathcal{S}_n$), we show that this gives rise to the sum-decomposition Deep Sets ansatz of Zaheer et al. (2018), where both the inner and outer functions can be chosen to be smooth, and moreover, the inner function can be chosen to be independent of the target function being approximated. In particular, we show that the embedding dimension required is independent of the regularity of the target function, the accuracy of the desired approximation, as well as $k$. Next, we show that a similar procedure allows us to obtain a uniform $\mathcal{C}^k$ approximation of antisymmetric functions as a sum of $K$ terms, where each term is a product of a smooth totally symmetric function and a smooth antisymmetric homogeneous polynomial of degree at most $\binom{n}{2}$. We also provide upper and lower bounds on $K$ and show that $K$ is independent of the regularity of the target function, the desired approximation accuracy, and $k$.

cs.LG↗