Related papers: The Hodge filtration and parametrically prime divi…

One of the most mysterious aspects of Saito's theory of Hodge modules are the Hodge and weight filtrations that accompany the pushforward of a Hodge module under an open embedding. In this paper we consider the open embedding in a product…

Let X be a smooth, projective variety over the field of complex numbers. On the space H of its rational cohomology of degree i we have the arithmetic filtration F^p. On the other hand, on the space of cohomology of degree i of X with…

In this paper, we improve the classical multiplicative ergodic theorem.

Let $(\Omega,\mathscr F,\mathbb P) $ be a probability space and let $(\mathscr F_n)$ be a binary filtration, i.e. exactly one atom of $\mathscr F_{n-1}$ is divided into two atoms of $\mathscr F_n$ without any restriction on their respective…

Let p be a prime and let F_pbar be the algebraic closure of the finite field of p elements. Let f(x) be any one variable rational function over F_pbar with n poles of orders d_1, ...,d_n. Suppose p is coprime to d_i for every i. We prove…

Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a relative ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*\Omega^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. Using the…

We define a natural filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ of a complex analytic variety $(V,0)$ related with the geometry of arcs on the variety and describe some properties of it.

Fedorov and Sabbah--Yu calculated the (irregular) Hodge numbers of hypergeometric connections. In this paper, we study the irregular Hodge filtrations on hypergeometric connections defined by rational parameters, and provide a new proof of…

The Function Field Sieve algorithm is dedicated to computing discrete logarithms in a finite field GF(q^n), where q is small an prime power. The scope of this article is to select good polynomials for this algorithm by defining and…

For $i : Z \to X$ a closed immersion of smooth varieties, we study how the $V$-filtration along $Z$ and the Hodge filtration on a mixed Hodge module $M$ on $X$ interact with each other. We also give a formula for the functors $i^*$, $i^!$…

We introduce and study the filtration on the space of automorphic functions (in the everywhere unramified situation for the function field case) obtained by transferring the filtration on the spectral side of the classical Langlands…

For a regular function f on a smooth complex quasi-projective variety, J.-D. Yu introduced in arXiv:1203.2338 a filtration (the irregular Hodge filtration) on the de Rham complex with twisted differential d+df, extending a definition of…

We study a notion of derived foliations on schemes and derived schemes of arbitrary characteristics. We introduce the Hodge filtration associated to a derived foliation, which functorialy filters derived de Rham cohomology. We use this…

Earlier the authors considered and, in some cases, computed Poincare series of two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular on the complex plane). A…

In this article there are two main results. The first result gives a formula, in terms of a log resolution, for the graded pieces of the Hodge filtration on the cohomology of a unitary local system of rank one on the complement of an…

We investigate the relation between the Hodge theory of a smooth subcanonical $n$-dimensional projective variety $X$ and the deformation theory of the affine cone $A_X$ over $X$. We start by identifying $H^{n-1,1}_{\mathrm{prim}}(X)$ as a…

Let $X$ be a Hirzebruch surface, and let $H$ be any ample divisor. In this paper, we algorithmically determine when the moduli space of semistable sheaves $M_{X,H}(r,c_1,c_2)$ is nonempty. Our algorithm relies on certain stacks of…

Let G=SL(E) be the special linear algebraic group on E where E is a finite dimensional vector space over a field K of characteristic zero. In this paper we study the canonical filtration of the dual G-module of global sections of a…

We present an algorithm to compute the Hodge ideals of $\mathbb{Q}$-divisors associated to any reduced effective divisor $D$. The computation of the Hodge ideals is based on an algorithm to compute parts of the $V$-filtration of Malgrange…

We study the relation between the Hodge filtration of the de Rham cohomology of a proper smooth supervariety $X$ and the usual Hodge filtration of the corresponding reduced variety $X_0$.