On some cohomological properties of almost complex manifolds

TL;DR

This paper investigates special almost complex structures called pure and full on compact manifolds, exploring their existence, properties, and parametrizations, especially on nilmanifolds and solvmanifolds, and their relation to symplectic and Lefschetz conditions.

Contribution

It provides new conditions for the existence of pure and full almost complex structures on compact quotients of Lie groups and constructs explicit families on nilmanifolds and solvmanifolds.

Findings

Existence conditions for pure and full structures on certain manifolds

Construction of parametrized families of such structures

Relations to symplectic structures and Hard Lefschetz condition

Abstract

We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang, in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by discrete subgroups. We obtain families of pure and full almost complex structures on compact nilmanifolds and solvmanifolds. Some of these families are parametrized by real 2-forms which are anti-invariant with respect to the almost complex structures.