# Special homogeneous almost complex structures on symplectic manifolds

## TL;DR

This paper investigates special homogeneous almost complex structures on symplectic manifolds, particularly those with Chern-Ricci forms proportional to the symplectic form, revealing their geometric and topological properties.

## Contribution

It characterizes non Chern-Ricci flat structures as co-adjoint orbits and identifies conditions under which compact co-adjoint orbits admit such structures, with examples.

## Key findings

- Non Chern-Ricci flat structures are covered by co-adjoint orbits.
- Compact co-adjoint orbits admit special structures under topological conditions.
- Examples include twistor spaces and Griffiths period domains.

## Abstract

Homogeneous compatible almost complex structures on symplectic manifolds are studied, focusing on those which are special, meaning that their Chern-Ricci form is a multiple of the symplectic form. Non Chern-Ricci flat ones are proven to be covered by co-adjoint orbits. Conversely, compact isotropy co-adjoint orbits of semi-simple Lie groups are shown to admit special compatible almost complex structures whenever they satisfy a necessary topological condition. Some classes of examples including twistor spaces of hyperbolic manifolds and discrete quotients of Griffiths period domains of weight two are discussed.