This paper extends Dolbeault cohomology and its surrounding theory to arbitrary almost complex manifolds. We define a spectral sequence converging to ordinary cohomology, whose first page is the Dolbeault cohomology, and develop a harmonic theory which injects into Dolbeault cohomology. Lie-theoretic analogues of the theory are developed which yield important calculational tools for Lie groups and nilmanifolds. Finally, we develop applications to maximally non-integrable manifolds, including nearly Kähler 6-manifolds, and show Dolbeault cohomology can be used to prohibit the existence of nearly Kähler metrics.

中文翻译：

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几乎复杂流形的 Dolbeault 上同调

本文将 Dolbeault 上同调及其周围理论扩展到任意的几乎复流形。我们定义了一个收敛到普通上同调的谱序列，其第一页是 Dolbeault 上同调，并发展了一个注入 Dolbeault 上同调的调和理论。该理论的李理论类似物被开发出来，为李群和尼尔流形提供了重要的计算工具。最后，我们开发了最大不可积流形的应用程序，包括近 Kähler 6 流形，并表明 Dolbeault 上同调可用于禁止近 Kähler 度量的存在。