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Consider $V:= \mathbb{C}^n$ such that $n \ge 3$ and take $p \in V$. Let $\nu \colon \tilde{V} \rightarrow V$ be a composite of blow-ups along smooth centres such that $\nu$ is an isomorphism outside $p$ and $F:= \nu^{-1}(p)$ is a SNC divisor. Let $F' \subset F$ be a divisor which is an union of several components of $F$. We can see that the vanishing of the local cohomology group supported on $F$

$ H^2_F(\tilde{V}, \mathcal{O}_{\tilde{V}})=0 $

by the local duality and the Grauert-Riemenschneider vanishing theorem.

Question Do we also have $H^2_{F'}(\tilde{V}, \mathcal{O}_{\tilde{V}})=0$ for the above $F'$?

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