Some pseudo-K\"ahler Einstein $4$-symmetric spaces with a "twin" special almost complex structure

TL;DR

This paper explores specific 4-symmetric symplectic spaces with pairs of invariant almost complex structures, one integrable and Einstein, the other maximally non-integrable and Ricci Hermitian, revealing new geometric properties.

Contribution

It identifies and analyzes pairs of invariant almost complex structures on 4-symmetric symplectic spaces, highlighting a novel 'twin' structure with one integrable Einstein metric and one non-integrable Ricci Hermitian structure.

Findings

Existence of pairs of invariant almost complex structures on 4-symmetric spaces.

One structure yields a pseudo-Kähler Einstein metric.

The other is maximally non-integrable and Ricci Hermitian.

Abstract

On $4$-symmetric symplectic spaces, invariant almost complex structures -- up to sign -- arise in pairs. We exhibit some $4$-symmetric symplectic spaces, with a pair of "natural" compatible (usually not positive) invariant almost complex structures, one of them being integrable and the other one being maximally non integrable (i.e. the image of its Nijenhuis tensor at any point is the whole tangent space at that point). The integrable one defines a pseudo-K\"ahler Einstein metric on the manifold, and the non integrable one is Ricci Hermitian (in the sense that the almost complex structure preserves the Ricci tensor of the associated Levi Civita connection) and special in the sense that the associated Chern Ricci form is proportional to the symplectic form.