Variations of Mixed Hodge Structure attached to the deformation theory of a Complex Variation of Hodge Structures

TL;DR

This paper constructs a Mixed Hodge Structure on the local deformation space of a variation of Hodge structures on a compact Kähler manifold, linking it to the holonomy of a VMHS and revisiting Goldman-Millson's deformation theory.

Contribution

It introduces a novel Mixed Hodge Structure on the representation scheme's local ring, connecting deformation theory with Hodge theory in a new way.

Findings

Established a Mixed Hodge Structure on the local complete ring of the representation scheme.

Proved the tautological representation corresponds to the holonomy of a VMHS.

Revisited and extended Goldman-Millson's deformation theory for Kähler groups.

Abstract

We construct a Mixed Hodge Structure on the local complete ring of the representation scheme at the holonomy of a VHS on a compact K\"ahler manifold and prove that the corresponding tautological representation is the holonomy of a VMHS. In order to carry out this construction we revisit the well-known work of Goldman and Millson on the deformation theory of representations of K\"ahler groups.