Mixed twistor structures

TL;DR

This paper introduces mixed twistor structures, extending mixed Hodge structures, to develop a weight theory for representations of fundamental groups of smooth projective varieties, with applications to hyperk"ahler manifolds.

Contribution

It generalizes classical mixed Hodge theory to the twistor setting, enabling new insights into geometric structures and their associated representations.

Findings

Mixed twistor structures can be applied to hyperk"ahler manifolds.

The theory supports replacing 'Hodge' with 'twistor' in classical results.

Jet spaces of hypercomplex manifolds naturally carry mixed twistor structures.

Abstract

The purpose of this paper is to introduce the notion of mixed twistor structure, a generalization of the notion of mixed Hodge structure. The utility of this notion is to make possible a theory of weights for various things surrounding arbitrary representations of the fundamental group of a smooth projective variety. We give some examples of generalizations of classical results for variations of mixed Hodge structure, to the twistor setting. This supports a ``meta-theorem'' (which we state but don't prove) that one can everywhere replace the word ``Hodge'' by the word ``twistor''. We show that the jet spaces of hyperk\"ahler or more generally hypercomplex manifolds have natural mixed twistor structures which determine the hypercomplex structure in a formal neighborhood of a point.