On page 255 of Hartshorne's Algebraic Geometry, it is shown that "cohomology commutes with flat base extension":

Proposition III.9.3: Let $f : X \to Y$ be a separated morphism of finite type of noetherian schemes, and let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $u : Y' \to Y$ be a flat morphism of noetherian schemes. $$ \require{AMScd} \begin{CD} X' @>{v}>> X \\ @VV{g}V @VV{f}V \\ Y' @>{u}>> Y \end{CD} $$ Then for all $i \ge 0$ there are natural isomorphisms $$ u^*R^if_*(\mathcal{F}) \cong R^ig_*(v^*\mathcal{F}). $$

I wonder whether something similar holds in the category of complex analytic spaces, where we would perhaps require $\mathcal{F}$ to be coherent and $f$ proper instead. However, I could find no reference.

The closest result in the complex analytic setting that I could find is Theorem III.3.4 and its corollaries in the book Algebraic methods in the global theory of complex spaces by Bănică and Stănăşilă. It is however quite different: the (coherent) sheaf $\mathcal{F}$ is instead required to be flat with respect to the (proper) morphism $f$, while $u$ is any base change, not necessarily flat. Then the isomorphism holds if $\mathcal{F}$ satisfies some extra properties.

So my question is: does Proposition III.9.3 in Hartshorne hold in the complex analytic setting, with some obvious modifications? And if so, what would be the reference?

My main interest is the following special case. Let $M$ be a compact hyperkähler manifold and $X = \mathrm{Tw}(M)$ its twistor space, $Y = \mathbb{CP}^1$, $f$ the natural holomoprhic twistor projection from $\mathrm{Tw}(M)$ to $\mathbb{CP}^1$, and $u$ a branched cover of $\mathbb{CP}^1$ by a smooth curve. I would like to know that for $\mathcal{F} = E$ a vector bundle on $\mathrm{Tw}(M)$, the isomorphism holds for $i = 0$. Note that $\mathrm{Tw}(M)$ is a complex manifold which is non-algebraic (and non-Kähler), so the algebraic version of the result cannot be used as is.

  • $\begingroup$ I am not sure about the general case but the case of proper $f$ and (quasi)-finite $u$ follows from the theorem of formal functions (in the analytic context) and the case of $0$-dimensional spaces $Y$ and $Y'$ (in this situation the proof is identical to the algebraic one). It seems that this is the case you really care about. $\endgroup$ – gdb Jul 11 at 2:16

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