Mixed Hodge structures and Weierstrass $\sigma$-function

Abstract

A $\sigma$-operator on a complexification $V_{\C}$ of an $\mathbb{R}$-vector space $V_{\mathbb{R}}$ is an operator $A \in \rm{End}_{\C} (V_{\C})$ such that $\sigma (A)=0$ where $\sigma (z)$ denotes the Weierstrass $\sigma$-function. In this paper we define the notion of the strongly pseudo-real $\sigma$-operator and prove that there is one to one correspondence between real mixed Hodge structures and strongly pseudo-real $\sigma$-operators.

5 pages