On Kodaira–Spencer’s problem on almost Hermitian 4-manifolds

In 1954, Hirzebruch reported a problem posed by Kodaira and Spencer: on compact almost complex manifolds, is the dimension [Formula: see text] 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 [Formula: see text]-manifolds.