Real 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 $\mathbb C$, equivariant with respect to $\mathbb C^*$. Here $\mathbb C$ is the complex plane considered as a 2-dimensional real manifold, and $\mathbb C^*$ is the multiplicative group of complex numbers considered as a real group.