Invariants of almost complex and almost Kähler manifolds

Tom Holt, Riccardo Piovani, Adriano Tomassini

Published 2022-09-15Version 1

Let $(M^{2n},J)$ be a compact almost complex manifold. The almost complex invariant $h^{p,q}_J$ is defined as the complex dimension of the cohomology space $\{[\alpha]\in H^{p+q}_{dR}(M;\mathbb{C}) \,\vert\,\alpha\in A^{p,q}_J(M) \}$. When $2n=4$, it has many interesting properties. Endow $(M^{2n},J)$ with a Hermitian metric $g$. The number $h^{p,q}_d$, i.e., the complex dimension of the space of Hodge-de Rham harmonic $(p,q)$-forms, is an almost K\"ahler invariant when $2n=4$. In this paper we show that $h^{1,0}_d$ is not an almost complex invariant and study $h^{p,q}_J$ and $h^{p,q}_d$ in dimension $2n\ge4$. We prove that $h^{n,0}_J=0$ if $J$ is non integrable and that $h^{p,0}_d$ is an almost K\"ahler invariant. We also focus on the special dimension $2n=4m$.