A Kodaira type conjecture on almost complex 4 manifolds

2023/07/27 by Dexie Lin, Lin, Dexie
Mathematics · #53A45 #53C23 #53D35 #57R30 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2307.14690

openalex publication_date 2023/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 h^1,0= h^0,1 implies a symplectic structure on a compact almost complex 4 manifold, where h^1,0 and 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 h^1,0= h^0,1 is equivalent to the generalized ∂∂-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 ∂∂-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.

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