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There’s plenty about differential refinement of ordinary cohomology on nLab. Can one also have analytic or holomorphic refinement?
Much of sheaf cohomology in the context of algebraic/analytic geometry is by default (and sort of by definition of subject) cohomology with coefficients in sheaves of holomorphic/analytic functions. You might call this “holomorphic” or “analytic” cohomology for emphasis, though this is not common usage of terms (even though maybe one could argue that it should be).
It is noteworthy that deep at the historical roots, this is the very origin of the notion of sheaf and eventually of algebraic geometry in the first place: as opposed to smooth functions, there are very few holomorphic functions globally, and hence people who were interested in these much earlier had to pass from naive geometry to (what we now call) topos theory in order to glue local models to something interesting. This is at its heart the reason why today some people think that $\infty$-category theory is intrinsically a topic of algebraic geometry, while differential geometers are still managing to fight it by constructing ever more technology for generalized smooth manifolds.
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