Mixed Hodge structures and Weierstrass $\sigma$-function

A $\sigma$-operator on a complexification $V_{\text{C}}$ of an $\mathbb{R}$-vector space $V_{\mathbb{R}}$ is an operator $A\in {\mathrm{E}\mathrm{n}\mathrm{d}}_{\text{C}}(V_{\text{C}})$ such that $\sigma (A) = 0$where$\sigma (z)$denotestheWeierstrass$\sigma$-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.