Nonabelian mixed Hodge structures

TL;DR

This paper introduces the concept of nonabelian mixed Hodge structures, defines their properties, and explores their applications to nonabelian cohomology, including an explicit example involving the complexified 2-sphere.

Contribution

It provides the first formal definition of nonabelian mixed Hodge structures and constructs a framework for their associated nonabelian cohomology with concrete examples.

Findings

Defined nonabelian mixed Hodge structures (namhs).

Constructed nonabelian cohomology $H=Hom(X_M, V)$ for smooth projective varieties.

Computed an example with the complexified 2-sphere, showing $Hom(X_M,V)$ is a namhs.

Abstract

We propose a definition of ``nonabelian mixed Hodge structure'' together with a construction associating to a smooth projective variety $X$ and to a nonabelian mixed Hodge structure $V$, the ``nonabelian cohomology of $X$ with coefficients in $V$'' which is a (pre-)nonabelian mixed Hodge structure denoted $H=Hom(X_M, V)$. We describe the basic definitions and then give some conjectures saying what is supposed to happen. At the end we compute an example: the case where $V$ has underlying homotopy type the complexified 2-sphere, and mixed Hodge structure coming from its identification with $\pp ^1$. For this example we show that $Hom (X_M,V)$ is a namhs for any smooth projective variety $X$.