Invariants of almost complex and almost K\"ahler manifolds

TL;DR

This paper investigates invariants of almost complex and almost Kähler manifolds, exploring their relationships and invariance properties, especially in four dimensions and for special classes like solvable Lie group quotients.

Contribution

It establishes new relationships between cohomological invariants and proves invariance properties in specific geometric contexts.

Findings

$h^{n,0}_J=0$ if $J$ is non integrable

$h^{p,0}_d$ is almost Kähler invariant

Information on $h^{1,1}_d$ for solvable Lie group quotients

Abstract

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 $\left\{\left[\alpha\right]\in H^{p+q}_{dR}(M^{2n};\mathbb{C}) \,\vert\,\alpha\in A^{p,q}(M^{2n}),\, d\alpha = 0 \right\}$. When $2n=4$, it has many interesting properties. Endow $(M^{2n},J)$ with an almost 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 almost K\"ahler invariant when $2n=4$. In this paper we study the relationship between $h^{p,q}_J$ and $h^{p,q}_d$ in dimension $2n\ge4$. We prove $h^{n,0}_J=0$ if $J$ is non integrable and show that $h^{p,0}_d$ is almost K\"ahler invariant. If $M^{2n}$ is a compact quotient of a completely solvable Lie group and $(J,g,\omega)$ is left invariant, we find information also on $h^{1,1}_d$. Finally we study the $\mathcal{C}^\infty$-pure and $\mathcal{C}^\infty$-full properties of $J$ on $n$-forms for the special dimension $2n=4m$.