Almost-complex invariants of families of six-dimensional solvmanifolds

TL;DR

This paper computes various almost-complex and almost-Hermitian invariants for families of six-dimensional solvmanifolds, providing new insights into their geometric structures and obstructions to symplectic forms.

Contribution

It introduces explicit calculations of invariants on solvmanifolds and establishes an obstruction criterion for symplectic structures in almost-complex manifolds.

Findings

Computed invariants $h^{p,0}_{ar ext{Dol}}$, $h^{p,0}_{ar ext{d}}$ for families of solvmanifolds.

Identified an obstruction to the existence of symplectic structures on certain almost-complex manifolds.

Extended understanding of invariants in higher-dimensional almost-Hermitian geometry.

Abstract

We compute almost-complex invariants $h^{p,0}_{\overline\partial}$, $h^{p,0}_{\text{Dol}}$ and almost-Hermitian invariants $h^{p,0}_{\bar\delta}$ on families of almost-K\"ahler and almost-Hermitian $6$-dimensional solvmanifolds. Finally, as a consequence of almost-K\"ahler identities we provide an obstruction to the existence of a symplectic structure on a given compact almost-complex manifold. Notice that, when $(X,J,g,\omega)$ is a compact almost Hermitian manifold of real dimension greater than four, not much is known concerning the numbers $h^{p,q}_{\overline\partial}$.