The dimensions of the graded quotients of the cohomology of a plane curve complement U = P \ C with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on H(U,C).

In this work we propose a generalization of the Heat Kernel Signature (HKS). The HKS is a point signature derived from the heat kernel of the Laplace-Beltrami operator of a surface. In the theory of exterior calculus on a Riemannian manifold, the Laplace-Beltrami operator of a surface is a special case of the Hodge Laplacian which acts on r-forms, i. e. the Hodge Laplacian on 0-forms (functions...

In this article we introduce the mixed Hodge structure of the Brieskorn module of a polynomial f in C, where f satisfies a certain regularity condition at infinity (and hence has isolated singularities). We give an algorithm which produces a basis of a localization of the Brieskorn module which is compatible with its mixed Hodge structure. As an application we show that the notion of a Hodge cy...

We study certain complexes of differential forms, including reverse de Rham complexes, on (real or complex) Poisson manifolds, especially holomorphic log-symplectic ones. relate these to the degeneracy divisor and rank loci bivector. In some good cases we compute local cohomology complexes. Kahlerian case, deduce a relation between multiplicity Hodge numbers manifold. also show that vanishing o...

Contents Introduction 1 Chapter 1. Introduction to the Weil conjectures 3 1. A first look 3 2. Formal statement of the conjectures 8 3. Zeta functions 11 4. A plan to prove the conjectures 14 5. Some history of the proofs of the conjectures 18 A. Computer calculations 20 B. Computations for diagonal hypersurfaces 25 Chapter 2. Topological interlude: the cohomology of algebraic varieties 35 1. L...

Let p be a prime number. The subject of p-adic Hodge theory concerns the interplay between different objects arising from the cohomology of algebraic varieties over p-adic fields. A good introduction to the subject can be found in the notes of Brinon and Conrad [3]; here, we limit discussion to two proofs which we feel are simpler than their counterparts in the literature. Our original motivati...

In his lectures in [G1], M. Green gives a lucid explanation how fruitful the infinitesimal method in Hodge theory is in various aspects of algebraic geometry. A significant idea is to use Koszul cohomology for Hodge-theoretic computations. The idea originates from Griffiths work [Gri] where the Poincaré residue representation of the cohomology of a hypersurface played a crucial role in proving ...