Mixed Hodge structures and Weierstrass \(\sigma\)-function (Q373676)

In a previous note [C. R., Math., Acad. Sci. Paris 350, No. 15--16, 777--780 (2012; Zbl 1254.14012)], the authors have observed a new characterisation of the existence of a real pure Hodge structure on a finite dimensional real vector space \(V\): it is equivalent to the existence of an endomorphism \(S\) on \(V\) which is annihilated by the Weierstrass \(\sigma\)-function associated with the lattice generated by \(\omega_1=1+i\) and \(\omega_2=1-i\). The key point is that \(S\) is diagonalisable, because \(\sigma\) has simple zeros.NEWLINENEWLINEIn the present note the preceding characterization is extended to the case of real mixed Hodge structures on \(V\). A definition of a strongly pseudo-real \(\sigma\)-operator on \(V\otimes \mathbb C\) is provided and the main theorem says that there in a \(1-1\) correspondence between MHS and strongly pseudo-real \(\sigma\)-operators on \(V\).