Related papers: The Mixed Hodge Structure on the Fundamental Group…

Let $X$ be a compact Riemann surface of genus $g$ and let $x \in X$. We derive the classical presentation of $\pi_1(X,x)$ (i.e the one given by $2g$ generators $a_1,b_1, \dots, a_g,b_g$ and the relation $\prod_{i=1}^g[a_i,b_i] = 1$) from…

Collino proved that the fundamental group of a certain Zariski open set of the symmetric square of a hyperelliptic curve is isomorphic to the integral Heisenberg group. We compute the mixed Hodge structure on this fundamental group, and…

Given a complex variety $X$, a linear algebraic group $G$ and a representation $\rho$ of the fundamental group $\pi\_1(X,x)$ into $G$, we develop a framework for constructing a functorial mixed Hodge structure on the formal local ring of…

Let $f:\mathbb{C}^{n+1} \to \mathbb{C}$ be a germ of hypersurface with isolated singularity. One can associate to $f$ a polarized variation of mixed Hodge structure $\mathcal{H}$ over the punctured disc, where the Hodge filtration is the…

We define and construct mixed Hodge structures on real schematic homotopy types of complex projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. We also show that these split on…

Let R be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the…

In this paper, we explore a notion of nonabelian Hodge structure on the fundamental group of an algebraic variety. This is approach is compared to some alternative approaches due to Morgan, Hain and others. We also give criteria for a…

We describe locally the representation varieties of fundamental groups for smooth complex varieties at representations coming from the monodromy of a variation of mixed Hodge structure. Given such a manifold $X$ and such a linear…

This paper was motivated by the following question: Recall that for a smooth projective variety X whose polarized Hodge structure on H^n(X,Q)_{prim} leads to a period point ...

On a smooth algebraic curve X with genus greater than 1 we consider a flat principal bundle with a reductive structure group S and a vector bundle associated with it. To this set of information we put in correspondence a pro-algebraic group…

The results of this paper can be divided into two parts, geometric and arithmetic. Let $X$ be a smooth projective curve over $\mathbb{C}$, and $e,\infty\in X(\mathbb{C})$ be distinct points. Let $L_n$ be the mixed Hodge structure of…

We study the unipotent completion $\Pi^{DR}_{un}(x_0, x_1, X_K)$ of the de Rham fundamental groupoid [De] of a smooth algebraic variety over a local non-archimedean field K of characteristic 0. We show that the vector space…

We construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology $H_{orb}^k(X)$ for projective $SL$-orbifolds $X$ satisfying a ``Hard Lefschetz Condition''. Furthermore, the total cohomology…

Let $\Sigma_{g,r}$ denote the $r$-punctured closed Riemann surface of genus $g$. For every $g\geq 0$, we determine the four-variable generating function for the mixed Hodge numbers of the unordered configuration spaces of $\Sigma_{g,1}$.…

We give a constructive proof of the Hodge conjecture for complex $K3$ surfaces that does not rely on Torelli-type results. Starting with an arbitrary rational $(1,1)$-class $\alpha\in H^{1,1}(X,\mathbb{Q})$, we algorithmically build a…

In this article, we prove a rigidity criterion for period maps of admissible variations of graded-polarizable mixed Hodge structure, and establish rigidity in a number of cases, including families of quasi-projective curves, projective…

We prove that a variation of graded-polarizable mixed Hodge structure over a punctured disk with unipotent monodromy, has a limiting mixed Hodge structure at the puncture (i.e., it is admissible in the sense of [SZ]) which splits over $\R$,…

Let $X$ be a compact Riemann surface, $\Sigma$ a finite set of points and $M = X\setminus \Sigma$. We study the $L^2$ cohomology of a polarized complex variation of Hodge structure on a Galois covering of the Riemann surface of finite type…

A bidouble cover is a flat $G:=\left(\mathbb{Z}/2\mathbb{Z}\right)^2$-Galois cover $X \rightarrow Y$. In this situation there exist three intermediate quotients $Y_1,Y_2$ and $Y_3$ which correspond to the three subgroups…

We discuss the variations of mixed Hodge structure for cohomology with compact support of quasi-projective simple normal crossing pairs. We show that they are graded polarizable admissible variations of mixed Hodge structure. Then we prove…