Almost-complex invariants of families of six-dimensional solvmanifolds

Abstract

We compute almost-complex invariants $h_{\overline{\partial }}^{p,0}$, $h_{\text{Dol}}^{p,0}$ and almost-Hermitian invariants $h_{\bar{\delta }}^{p,0}$ on families of almost-Kähler and almost-Hermitian $6$-dimensional solvmanifolds. Finally, as a consequence of almost-Kähler 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 $h_{\overline{\partial }}^{p,q}$.