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 h(partial derivative)(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 Kahler 4-manifolds.