Searcharxiv⌕ Search

arXiv subjects

D. Arinkin

Publications and source records attributed to D. Arinkin.

15 recordsLinked to original sources
arXiv2024

Proof of the geometric Langlands conjecture IV: ambidexterity

This paper performs the following steps toward the proof of GLC in the de Rham setting: (i) We deduce GLC for G=GL_n; (ii) We prove that the Langlands functor L_G constructed in [GLC1], when restricted to the cuspidal category, is ambidextrous; (iii) We reduce GLC to the study of a certain classical vector bundle with connection on the stack of irreducible local systems; (iv) We prove that GLC is equivalent to the contractibility of the space of generic oper structures on irreducible local systems; (v) Using [BKS], we deduce GLC for classical groups.

math.AG↗
arXiv2024

Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE

This paper is the second in a series of five that together prove the geometric Langlands conjecture. Our goals are two-fold: (1) Formulate and prove the Fundamental Local Equivalence (FLE) at the critical level; (2) Study the interaction between Kac-Moody localization and the global geometric Langlands functor of ref. [GLC1]. This paper contains an extensive Appendix, whose primary goals are: (a) Development the theory of ind-coherent sheaves in infinite type; (b)Development of the formalism of factorization categories.

math.AG↗
arXiv2022

Automorphic functions as the trace of Frobenius

We prove that the trace of the Frobenius endofunctor of the category of automorphic sheaves with nilpotent singular support maps isomorphically to the space of unramified automorphic functions, settling a conjecture from [AGKRRV1]. More generally, we show that traces of Frobenius-Hecke functors produce shtuka cohomologies.

math.AG↗
arXiv2022

The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support

We define a new geometric object--the stack of local systems with restricted variation. We formulate a version of the categorical geometric Langlands conjecture that makes sense for any constructible sheaf theory (such as l-adic sheaves). We formulate a conjecture that makes precise the connection between the category of automorphic sheaves and the space of automorphic functions.

math.AG↗
arXiv2017

The category of singularities as a crystal and global Springer fibers

We prove the "Gluing Conjecture" on the spectral side of the categorical geometric Langlands correspondence. The key tool is the structure of crystal on the category of singularities, which allows to reduce the conjecture to the question of homological triviality of certain homotopy types. These homotopy types are obtained by gluing from a global version of Springer fibers.

math.AG↗
arXiv2013

*-Quantizations of Fourier-Mukai transforms

We study deformations of Fourier-Mukai transforms in general complex analytic settings. We start with two complex manifolds X and Y together with a coherent Fourier-Mukai kernel P on their product. Suppose that P implements an equivalence between the coherent derived categories of X and Y. Given an arbitrary formal quantization of X we construct a unique quantization of Y such that the Fourier-Mukai transform deforms to an equivalence of the derived categories of the quantizations. Here quantizations are understood in the framework of stacks of algebroids.

math.AG↗
arXiv2010

Autoduality of compactified Jacobians for curves with plane singularities

Let C be an integral projective curve with planar singularities. Consider its Jacobian J and the compactified Jacobian J'. We construct a flat family P of Cohen-Macaulay sheaves on J' parametrized by J'; over J, the family P is the Poincare line bundle. We prove that the Fourier-Mukai transform given by P is an auto-equivalence of the derived category of J'.

math.AG↗
arXiv2010

Cohomology of line bundles on compactified Jacobians

Let C be an integral projective curve with surficial singularities. We prove that topologically trivial line bundles on the compactified Jacobian of C are in one-to-one correspondence with line bundles on C (the autoduality conjecture), and compute the cohomology of the line bundles. We also show that the natural Fourier-Mukai functor between the derived categories of quasi-coherent sheaves on the Jacobian and on the compactified Jacobian is fully faithful.

math.AG↗
arXiv2010

Rigid irregular connections on P^1

N.Katz's middle convolution algorithm provides a description of rigid connections on the projective line with regular singularities. We extend the algorithm by adding the Fourier transform to it. The extended algorithm provides a description of rigid connections with arbitrary singularities.

math.AG↗
arXiv2008

Fourier transform and middle convolution for irregular D-modules

S.Block and H.Esnault constructed the local Fourier transform for D-modules. We present a different approach to the local Fourier transform, which makes its properties almost tautological. We apply the local Fourier transform to compute the local version of Katz's middle convolution.

math.AG↗
arXiv2007

Moduli spaces of d-connections and difference Painleve equations

We show that difference Painleve equations can be interpreted as isomorphisms of moduli spaces of d-connections on the projective line with given singularity structure. We also derive a new difference equation. It is the most general difference Painleve equation known so far, and it degenerates to both difference Painleve V and classical (differential) Painleve VI equations.

math.AG↗
arXiv2007

Tau-function of discrete isomonodromy transformations and probability

We introduce the tau-function of a rational d-connection and its isomonodromy transformations. We show that in a continuous limit our tau-function agrees with the Jimbo-Miwa-Ueno tau-function, compute the tau-function for the isomonodromy transformations leading to difference Painleve V and difference Painleve VI equations, and prove that the gap probability for a wide class of discrete random matrix type models can be viewed as the tau-function for an associated d-connection.

math.AG↗
arXiv2004

Moduli of connections with a small parameter on a curve

We study GL(2)-bundles with connections with a small parameter on a smooth projective curve. We describe an open subset in the moduli space of such bundles. The description degenerates into the Hitchin fibration as the parameter tends to zero.

math.AG↗