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 ${\mathbb{C}}^{\ast }$. Here $\mathbb{C}$ is the complex plane considered as a 2-dimensional real manifold, and ${\mathbb{C}}^{\ast }$ is the multiplicative group of complex numbers considered as a real group.