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Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.

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Cohomology of Homogeneous Complex Manifolds

In general, the answer to your question is "no". For example, let $\mathbb{C}$ act on the standard square two-torus by translations. Clearly, this action preserves the complex structure. All the cohom …
Ken Richardson's user avatar