Related papers: $\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k, \ma…

When a non-singular complex projective surface $X$ satisfies that $K_X\sim 0$, we shall show that there are only finitely many isomorphic classes as abstract schemes in the set of moduli scheme of $H$-semistable sheaves with fixed Chern…

We show that for any constructible sheaf F on a smooth algebraic variety X over a field of arbitrary characteristic its singular support SS(F) is equidimensional of dimension dim X. Here SS(F) is the minimal closed subset of the cotangent…

Let $K$ be a field of characteristic zero complete with respect to a non-trivial, non-Archimedean valuation. We relate the sheaf $\widehat{\mathcal{D}}$ of infinite order differential operators on smooth rigid $K$-analytic spaces to the…

In 2011, Guralnick and Tiep proved that if $G$ was a Chevalley group with Borel subgroup $B$ and $V$ an irreducible $G$-module in cross characteristic with $V^B = 0$, then the the dimension of $H^1(G,V)$ is determined by the structure of…

In this article we construct a categorical resolution of singularities of an excellent reduced curve $X$, introducing a certain sheaf of orders on $X$. This categorical resolution is shown to be a recollement of the derived category of…

We propose a new point of view on quantum cohomology, strongly motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is the D-module which "quantizes" a…

We show that the arithmetic D-module associated to an overconvergent F-isocrystal over a smooth curve is holonomic. We first prove that unipotent F-isocrystals are holonomic D-module by using the fact that such F-isocrystals come from…

In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian…

We study the growth of representations of the Lie algebra of vector fields on the affine space that admit a compatible action of the polynomial algebra. We establish the Bernstein inequality for these representations, enabling us to focus…

A function that is analytic on a domain of $\mathbb{C}^n$ is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. We define and study the Bernstein-Sato polynomial…

Let $k$ be a field with separable closure $\bar{k}\supset k$, and let $X$ be a qcqs $k$-scheme. We use the theory of profinite Galois categories developed by Barwick-Glasman-Haine to provide a quick conceptual proof that the sequences…

In this paper we prove the constructibility on the subanalytic sites of the sheaves of tempered holomorphic solutions of holonomic D-modules on complex analytic manifolds. Such a result solves a conjecture of M. Kashiwara and P. Schapira…

Consider a field $k$ of characteristic $0$, not necessarily algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame polynomial $f\in k[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic…

Given a field $K$ equipped with a set of discrete valuations $V$, we develop a general theory to relate reduction properties of skew-hermitian forms over a quaternion $K$-algebra $Q$ to quadratic forms over the function field $K(Q)$…

Consider a field k of characteristic p > 0, G_r the r-th Frobenius kernel of a smooth algebraic group G, DG_r the Drinfeld double of G_r, and M a finite dimensional DG_r-module. We prove that the cohomology algebra H*(DG_r,k) is finitely…

We investigate infinite dimensional modules for a linear algebraic group $\mathbb G$ over a field of positive characteristic $p$. For any subcoalgebra $C \subset \mathcal O(\mathbb G)$ of the coordinate algebra of $\mathbb G$, we consider…

Labourie and the author independently showed that a convex real projective structure on an oriented surface of genus at least 2 is equivalent to a conformal structure plus a holomorphic cubic differential U. We analyze the behavior of the…

We examine the quantum symmetric and exterior algebras of finite-dimensional \uqg-modules first systematically studied by Berenstein and Zwicknagl, and resolve some questions that they raised. We show that the difference (in the…

Let $n \geq 1$ be an integer, let $V=(\mathbb{Z}/2\mathbb{Z})^{n}$ and let $X$ be a $V$-CW-complex. If $X$ is a finite $CW$-complexe, the equivariant modulo $2$ cohomology of the $V$-CW-complexe $X$, denoted by $H_{V}^{*}(X,…

Suppose $X$ is a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Let $G$ be a finite group acting faithfully on $X$ over $k$ such that $G$ has non-trivial, cyclic Sylow…