Mixed Hodge structures and Weierstrass $\sigma$-function

TL;DR

This paper establishes a correspondence between real mixed Hodge structures and a special class of operators called strongly pseudo-real $\sigma$-operators, using properties of the Weierstrass $\sigma$-function.

Contribution

It introduces the concept of strongly pseudo-real $\sigma$-operators and proves their one-to-one correspondence with real mixed Hodge structures.

Findings

Defined strongly pseudo-real $\sigma$-operators.

Proved the bijective correspondence with real mixed Hodge structures.

Connected complex analysis with algebraic geometry concepts.

Abstract

A $\sigma$-operator on a complexification $V_{\C}$ of an $\R$-vector space $V_{\R}$ 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.