Real mixed Hodge structures

TL;DR

This paper characterizes the category of real mixed Hodge structures as vector bundles with connections on the complex plane, invariant under the action of the multiplicative group, linking algebraic and geometric perspectives.

Contribution

It establishes an equivalence between real mixed Hodge structures and certain equivariant vector bundles with connections, providing a geometric interpretation.

Findings

Category of real mixed Hodge structures is equivalent to equivariant vector bundles with connections.

Connections are not necessarily flat, broadening the class of geometric objects considered.

Provides a new geometric framework for understanding mixed Hodge structures.

Abstract

We identify the category of real mixed Hodge structures with the category of vector bundles with connections (not necessarily flat) on C, equivariant with respect to C^*. Here C is the complex plane considered as a 2-dimensional real manifold, and C^* is the multiplicative group of complex numbers considered as a real group.