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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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when is a map of analytic Brauer groups induced by inclusion injective?

This got a bit long for a comment, so here's an answer. The answer is it is almost never injective. In general, for compact $X$, $Br(X)$ is infinite (at least if $H^2(X,O_X)\neq 0$), while $Br(U)$ is …
Benjamin Antieau's user avatar