A Kodaira type conjecture on almost complex 4 manifolds

Dexie Lin
Mathematics, Differential Geometry, Differential Geometry (math.DG), Geometric Topology (math.GT), Symplectic Geometry (math.SG)
2023-07-26 16:00:00
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 Hodge number of the refined Dolbeault cohomology groups with bi-degrees $(1,0)$ and $(0,1)$ respectively. Moreover, we prove that the condition $\tilde{h}^{1,0}=\tilde h^{0,1}$ is equivalent to the $\partial\bar\partial$-lemma. This can be regarded as an analogue of the Kodaira's conjecture on almost complex $4$ manifolds. Meanwhile, we show that the equality $b_1=\tilde h^{0,1}+\tilde h^{1,0}$ does not hold on a general almost complex $4$ manifold, which is non-similar to the case of compact complex surfaces.
PDF: A Kodaira type conjecture on almost complex 4 manifolds.pdf
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