# The Hodge theoretic fundamental group and its cohomology

@article{Arapura2009TheHT, title={The Hodge theoretic fundamental group and its cohomology}, author={Donu Arapura}, journal={arXiv: Algebraic Geometry}, year={2009} }

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 variety to be a Hodge theoretic K(pi,1), which roughly means that the cohomology of variations of mixed Hodge structure can be determined from the group.

#### 12 Citations

Formality and splitting of real non-abelian mixed Hodge structures

- Mathematics
- 2009

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… Expand

Real non-abelian mixed Hodge structures for quasi-projective varieties: formality and splitting

- Mathematics
- 2011

We define and construct mixed Hodge structures on real schematic homotopy types of complex quasi-projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic… Expand

Morgan’s mixed Hodge structures and nonabelian Hodge structures

- Mathematics
- 2021

Abstract We refine the Morgan’s work on mixed Hodge structures on Sullivan’s 1-minimal models by using non-abelian Hodge theory. As an application, we give explicit representatives of real unipotent… Expand

An Abelian Category of Motivic Sheaves

- Mathematics
- 2008

The goal of this paper is to construct a category of motivic "sheaves" on an algebraic variety defined over a subfield of C, using Nori's method. This categoryis abelian and it possesses faithful… Expand

Linear Shafarevich conjecture

- Mathematics
- 2009

In this paper we settle armatively Shafarevich’s uniformization conjecture for varieties with linear fundamental groups. We prove the strongest to date uniformization result | the universal covering… Expand

Contemporary Mathematics Homomorphisms between Kähler groups Donu Arapura To

- 2013

This paper is an expanded version of my talk at the Jaca conference; as such it is somewhere in between a survey and a research article. Algebraic geometry and topology have, of course, been… Expand

Tannaka duality for enhanced triangulated categories

- Mathematics
- 2013

We develop Tannaka duality theory for dg categories. To any dg functor from a dg category A to finite-dimensional complexes, we associate a dg coalgebra C via a Hochschild homology construction. When… Expand

DGA-Models of variations of mixed Hodge structures

- Mathematics
- 2018

We define objects over Morgan's mixed Hodge diagrams which will be algebraic models of unipotent variations of mixed hodge structures over Kahler manifolds. As an analogue of Hain-Zucker's… Expand

Tannakian formalism for fiber functors over tensor categories

- Mathematics, Computer Science
- Period. Math. Hung.
- 2018

It is shown that (under some technical conditions) if the fiber functor has a section, then the source category is equivalent to the category of comodules over a Hopf algebra in the target category. Expand

A ug 2 00 9 HOMOMORPHISMS BETWEEN KÄHLER GROUPS DONU ARAPURA To

- 2009

To Anatoly Libgober This paper is an expanded version of my talk at the Jaca conference; as such it is somewhere in between a survey and a research article. Algebraic geometry and topology have, of… Expand

#### References

SHOWING 1-10 OF 35 REFERENCES

Formality and splitting of real non-abelian mixed Hodge structures

- Mathematics
- 2009

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… Expand

An Abelian Category of Motivic Sheaves

- Mathematics
- 2008

The goal of this paper is to construct a category of motivic "sheaves" on an algebraic variety defined over a subfield of C, using Nori's method. This categoryis abelian and it possesses faithful… Expand

A Category of Motivic Sheaves

- Mathematics
- 2008

The goal of this paper is to construct a category of motivic "sheaves" on an algebraic variety defined over a subfield of C, using Nori's method. This categoryis abelian and it possesses faithful… Expand

Representations of algebraic groups.

- Mathematics
- 1963

Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the… Expand

Variation of mixed Hodge structure. I

- Mathematics
- 1985

Around 1970, Griffiths introduced the notion of a variation of Hodge structure on a complex manifold S (see [17, w 2]). It constitutes the axiomatization of the features possessed by the local… Expand

Shafarevich Maps and Automorphic Forms

- Mathematics
- 1995

This text studies various geometric properties and algebraic invariants of smooth projective varieties with infinite fundamental groups. This approach allows for much interplay between methods of… Expand

The hodge de rham theory of relative malcev completion

- Mathematics
- 1996

Abstract Suppose that X is a smooth manifold and ρ : π1(X,x) → S is a representation of the fundamental group of X into a real reductive group with Zariski dense image. To such data one can associate… Expand

Cosimplicial Objects in Algebraic Geometry

- Mathematics
- 1993

Let X be an arc-connected and locally arc-connected topological space and let I be the unit interval. Applying the connected component functor to each fibre of the fibration of the total space map(I,… Expand

Mixed hodge modules

- Mathematics
- 1990

Introduction 221 § 1. Relative Monodromy Filtration 227 §2. Mixed Hodge Modules on Complex Spaces (2. a) Vanishing Cycle Functors and Specializations (Divisor Case) 236 (2.b) Extensions over Locally… Expand

Rational homotopy theory

- 2011

1 The Sullivan model 1.1 Rational homotopy theory of spaces We will restrict our attention to simply-connected spaces. Much of this goes through with nilpotent spaces, but this will keep things… Expand