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Let $X$ be an open complex manifold, e.g., the complement of a simple normal crossing divisor $D$ in a (smooth) projective manifold $M$. Let $T^{1,0}X$ be the holomorphic tangent bundle of $X$. Let $K \subset X$ be a compact set with $U \subset X$ a sufficiently small open neighbourhood of $K$. Let $V$ be a holomorphic vector field on $K$, i.e., a holomorphic section $V\in H^0(K,T^{1,0}X)$. Does there exist an extension $\widetilde{V} \in H^0(U, T^{1,0}X)$ such that $\widetilde{V} = V$ on $K$?

Note that I'm not asking whether the sheaf of holomorphic vector fields is a fine sheaf, I'm only asking whether holomorphic vector fields defined on compact sets admit local extensions to open neighbourhoods. I'm also aware of the failure of the tubular neighbourhood theorem to exist in the holomorphic category (in complete generality), but the literature on such a theorem appears quite vast and formidable.

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By Theorem II.9.5 in Bredon's Sheaf Theory, for any closed subset $K$ of a paracompact space $X$ and for any sheaf of abelian groups $F$ on $X$, the canonical map $$\mathop{\rm colim} F(U) \to F(K)$$ is an isomorphism, where $U$ runs over all open neighborhoods of $K$.

In particular, every element of $F(K)$ is the restriction of some element of $F(U)$, where $U$ is an open neighborhood of $K$.

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  • $\begingroup$ Thank you, this is exactly what I was after. Thanks for the reference, too! $\endgroup$
    – user105074
    Jul 4 '21 at 1:51
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    $\begingroup$ The question fixes $U$ before fixing the vector field, so each vector field having an extension to some $U$ is not enough. $\endgroup$
    – Ben McKay
    Jul 4 '21 at 6:39
  • $\begingroup$ @BenMcKay: No, this question does not fix $U$ before fixing the vector field. It says $U$ is a sufficiently small open neighborhood $U$ of $K$, not $U$ is an open neighborhood of $K$, and the adjective “sufficiently small” means $U$ is chosen after $V$. $\endgroup$ Jul 4 '21 at 15:05
  • $\begingroup$ @BenMcKay I apologise for the possibly misleading question, Dmitri was correct in his reading, however. $\endgroup$
    – user105074
    Jul 4 '21 at 19:22
  • $\begingroup$ @DmitriPavlov Thank you again! $\endgroup$
    – user105074
    Jul 4 '21 at 19:22

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