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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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p-adic Poincaré Lemma

On Q3, maybe an additional geometric remark. First, to take the non-torsion field $k$ as coefficients, you should use the pro-étale topology, not the étale one, as Scholze does (see Scholze and Bhatt' …
Frédéric Paugam's user avatar