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Alexander Polishchuk

Publications and source records attributed to Alexander Polishchuk.

At least 19 recordsLinked to original sources
arXiv2026

$L^2$-property for algebraic stacks over local non-archimedean fields

We introduce an $L^2$-norm on the space of Schwartz half-densities over algebraic stacks over local non-archimedean fields. We show that these $L^2$-norms are finite for the stacks of $PGL_2$-bundles on $\mathbb{P}^1$ with parabolic structures at $\ge 3$ points. The latter property was conjectured in the context of the analytic Langlands correspondence of arXiv:2103.01509.

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arXiv2025

Analog of theta-lifting for a curve over dual numbers over a finite field

We continue the study of automorphic functions associated with a curve $C$ over the ring $k[\epsilon]/(\epsilon^2)$, where $k$ is a finite field, begun in arXiv:2303.16259. Namely, we study an example of theta-lifting in this framework and show that it can be understood in terms of the orbit decomposition of the space of automorphic functions $\mathcal{S}(\rm{SL}_2(F)\backslash \rm{SL}_2(\mathbb{A}_C))$ introduced in loc.cit. We prove that all strongly cuspidal functions in $\mathcal{S}(\rm{SL}_2(F)\backslash \rm{SL}_2(\mathbb{A}_C))$ can be constructed using theta-lifting for an appropriate double covering $\tilde{C}\to C$.

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arXiv2025

NC-smooth thickenings and Jacobians

We prove that a bounded complex of coherent sheaves on an abelian variety A, whose Fourier-Mukai transform has support of dimension at most 1, extends to a perfect complex on the standard NC-smooth thickening of A and on the deformation quantization of any constant Poisson bracket on A. We discuss a related conjectural characterization of Jacobians in terms of NC-smooth thickenings.

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arXiv2024

Schwartz $κ$-densities for the moduli stack of rank $2$ bundles on a curve over a local field

Let $\rm{Bun}$ be the moduli stack of rank $2$ bundles with fixed determinant on a smooth proper curve $C$ over a local field $F$. We show how to associate with a Schwartz $κ$-density, for $\rm{Re}(κ)\ge 1/2$, a smooth function on the corresponding coarse moduli space of very stable bundles. In the non-archimedean case we also prove that the stack $\rm{Bun}$ is $κ$-bounded in the sense of Definition 2.10 of [arXiv:2112.08139] for any $κ\in\mathbb{C}$.

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arXiv2024

Schwartz $\kappa$-densities on the moduli stack of rank $2$ bundles near stable bundles

Let $C$ be a curve over a non-archimedean local field of characteristic zero. We formulate algebro-geometric statements that imply boundedness of functions on the moduli space of stable bundles of rank $2$ and fixed odd degree determinant over $C$, coming from the Schwartz space of $\kappa$-densities on the corresponding stack of bundles (earlier we proved that these functions are locally constant on the locus of very stable bundles). We prove the relevant algebro-geometric statements for curves of genus $2$ and for non-hyperelliptic curves of genus $3$.

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arXiv2024

Superperiods and superstring measure near the boundary of the moduli space of supercurves

We study the behavior of the superperiod map near the boundary of the moduli space of stable supercurves and prove that it is similar to the behavior of periods of classical curves. We consider two applications to the geometry of this moduli space in genus $2$, denoted as $\bar{\mathcal S}_2$. First, we characterize the canonical projection of $\bar{\mathcal S}_2$ in terms of its behavior near the boundary, proving in particular that $\bar{\mathcal S}_2$ is not projected. Secondly, we combine the information on superperiods with the explicit calculation of genus $2$ Mumford isomorphism, due to Witten, to study the expansion of the superstring measure for genus $2$ near the boundary. We also present the proof, due to Deligne, of regularity of the superstring measure on $\bar{\mathcal S}_g$ for any genus.

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arXiv2024

Exceptional pairs on del Pezzo surfaces and spaces of compatible Feigin-Odesskii brackets

We prove that for every relatively prime pair of integers $(d,r)$ with $r>0$, there exists an exceptional pair $({\mathcal O},V)$ on any del Pezzo surface of degree 4, such that $V$ is a bundle of rank $r$ and degree $d$. As an application, we prove that every Feigin-Odesskii Poisson bracket on a projective space can be included into a 5-dimensional linear space of compatible Poisson brackets. We also construct new examples of linear spaces of compatible Feigin-Odesskii Poisson brackets of dimension $>5$, coming from del Pezzo surfaces of degree $>4$.

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arXiv2024

Algebra of global sections of $ψ$-bundles on $\bar{M}_{0,n}$

We consider the ${\mathbb Z}^n$-graded algebra of global sections of line bundles generated by the standard line bundles $L_1,\ldots,L_n$ on $\bar{M}_{0,n}$. We find a simple presentation of this algebra by generators and quadratic relations. As an application we prove that the moduli space $\bar{M}_{0,n}[ψ]$ of $ψ$-stable curves of genus $0$ is Cohen-Macaulay and normal, and the natural map $\bar{M}_{0,n}\to \bar{M}_{0,n}[ψ]$ is a rational resolution.

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arXiv2024

Compatible Poisson Brackets Associated with Elliptic Curves in $G(2,5)$

We prove that a pair of Feigin-Odesskii Poisson brackets on ${\mathbb P}^4$ associated with elliptic curves given as linear sections of the Grassmannian $G(2,5)$ are compatible if and only if this pair of elliptic curves is contained in a del Pezzo surface obtained as a linear section of $G(2,5)$.

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arXiv2024

Feigin-Odesskii brackets associated with Kodaira cycles and positroid varieties

We establish a link between open positroid varieties in the Grassmannians $G(k,n)$ and certain moduli spaces of complexes of vector bundles over Kodaira cycle $C^n$, using the shifted Poisson structure on the latter moduli spaces and relating them to a certain twist of the standard Poisson structure on $G(k,n)$. %by a bivector field on its maximal torus. This link allows us to solve a classification problem for extensions of vector bundles over $C^n$. Based on this solution we further classify the symplectic leaves of all positroid varieties in $G(k,n)$ with respect to the twisted standard Poisson structure. Moreover, we get an explicit description of the moduli stack of symplectic leaves of $G(k,n)$ with the twisted standard Poisson structure as an open substack of the stack of vector bundles on $C^n$.

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arXiv2024

On supercurves of genus 1 with an underlying odd spin structure

We study the standard family of supercurves of genus 1 with an underlying odd spin structures. We give a simple algebraic description of this family and of the compactified family of stable supercurves with one Neveu-Schwarz puncture. We also describe the Gauss-Manin connection on the first de Rham cohomology of this family and compute the superperiods of global differentials.

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arXiv2023

Elliptic bihamiltonian structures from relative shifted Poisson structures

In this paper, generalizing the construction of \cite{HP1}, we equip the relative moduli stack of complexes over a Calabi-Yau fibration (possibly with singular fibers) with a shifted Poisson structure. Applying this construction to the anticanonical linear systems on surfaces, we get examples of compatible Poisson brackets on projective spaces extending Feigin-Odesskii Poisson brackets. Computing explicitly the corresponding compatible brackets coming from Hirzebruch surfaces, we recover the brackets defined by Odesskii-Wolf in \cite{OW}.

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arXiv2023

Bosonization of Feigin-Odesskii Poisson varieties

The derived moduli stack of complexes of vector bundles on a Gorenstein Calabi-Yau curve admits a 0-shifted Poisson structure. Feigin-Odesskii Poisson varieties are examples of such moduli spaces over complex elliptic curves. Using moduli stack of chains we construct an auxiliary Poisson varieties with a Poisson morphism from it to a Feigin-Odesskii variety. We call it the \emph{bosonization} of Feigin-Odesskii variety. As an application, we show that the Feigin-Odesskii Poisson brackets on projective spaces (associated with stable bundles of arbitrary rank on elliptic curves) admit no infinitesimal symmetries.

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arXiv2023

Hecke operators for curves over non-archimedean local fields and related finite rings

We study Hecke operators associated with curves over a non-archimedean local field $K$ and over the rings $O/{\mathfrak m}^N$, where $O\subset K$ is the ring of integers. Our main result is commutativity of a certain "small" local Hecke algebra over $O/{\mathfrak m}^N$, associated with a connected split reductive group $G$ such that $[G,G]$ is simple and simpy connected. The proof uses a Hecke algebra associated with $G(K(\!(t)\!))$ and a global argument involving $G$-bundles on curves.

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