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

TL;DR

This paper constructs and analyzes real non-abelian mixed Hodge structures on the homotopy types of complex quasi-projective varieties, revealing new splitting properties and connections to spectral sequences and monodromy.

Contribution

It introduces a novel framework for mixed Hodge structures on schematic homotopy types, including their splitting and variation properties, advancing understanding of their algebraic and topological aspects.

Findings

Mixed Hodge structures are constructed on homotopy groups and fundamental groups.

Splitting occurs when tensoring with R[x] with a specific Hodge filtration.

Structures can be recovered from Gysin spectral sequences and monodromy actions.

Abstract

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 fundamental groups. We also show that these split on tensoring with the ring R[x] equipped with the Hodge filtration given by powers of (x-i), giving new results even for simply connected varieties. The mixed Hodge structures can thus be recovered from the Gysin spectral sequence of cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, these structures all become real variations of mixed Hodge structure.