On the Kodaira-Spencer's problem on almost Hermitian $4$-manifolds

Abstract

In 1954, Hirzebruch reported a problem posed by Kodaira and Spencer: on compact almost complex manifolds, is the dimension $h_{\bar{\partial }}^{p,q}$ of the kernel of the Dolbeault Laplacian independent of the choice of almost Hermitian metric? In this paper, we review recent progresses on the original problem and we introduce a similar one: on compact almost complex manifolds, find a generalization of Bott-Chern and Aeppli numbers which is metric-independent. We find a solution to our problem valid on almost Kähler $4$-manifolds.

arXiv comment16 pages. No figures. Comments are welcome