A Kodaira type conjecture on almost complex 4 manifolds

Not long ago, Cirici and Wilson defined a Dolbeault cohomology on almost complex manifolds to answer Hirzebruch’s problem. In this paper, we define a refined Dolbeault cohomology on almost complex manifolds. We show that the condition ${\tilde{h}}^{1,0}={\tilde{h}}^{0,1}$ implies a symplectic structure on a compact almost complex $4$ manifold, where ${\tilde{h}}^{1,0}$ and ${\tilde{h}}^{0,1}$ are the dimensions of the refined Dolbeault cohomology groups with bi-degrees $(1,0)$ and $(0,1)$ respectively. Combining the partial answer to Donaldson’s tameness conjecture, we offer a sufficient condition for a compact almost complex $4$ manifold to become an almost Kähler one. Moreover, we prove that the condition ${\tilde{h}}^{1,0}={\tilde{h}}^{0,1}$ is equivalent to the generalized $\mathrm{\partial }\bar{\mathrm{\partial }}$-lemma. This can be regarded as an analogue of the Kodaira’s conjecture on almost complex $4$ manifolds. As an application, we show that the Kodaira-Thurston manifold satisfies the $\mathrm{\partial }\bar{\mathrm{\partial }}$-lemma. Meanwhile, we show that the Frölicher-type equality does not hold on a general almost complex $4$ manifold, which is different to the case of compact complex surfaces.