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In the smooth category, one has the following extension theorem.

Theorem. Let $i:S\hookrightarrow M$ and $j:S\hookrightarrow N$ be two closed embeddings of a smooth manifold $S$ into smooth manifolds $M$ and $N$. Suppose that there is a vector bundle isomorphism $F:TM|_S\to TN|_S$ which restricts to the identity on $TS$. Then, there are neighborhoods of $i(S)$ and $j(S)$ and a diffeomorphism $f$ between them such that $df_p=F_p$ for all $p\in S$.

However, the only proof that I know relies on choosing Riemannian metrics on $M$ and $N$ and constructing tubular neighborhoods using the exponential maps. See, e.g. [Guillemin-Sternberg, Semi-classical analysis, Proposition 9, Ch 2].

Is there an analogue of this theorem in the holomorphic category?

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  • $\begingroup$ I think this is a duplicate of mathoverflow.net/questions/114414/… $\endgroup$
    – Ben McKay
    Commented Oct 5, 2017 at 9:33
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    $\begingroup$ @BenMcKay The proof of the tubular neighborhood theorem implies the above theorem, but I don't see why they should be equivalent. $\endgroup$
    – user115379
    Commented Oct 5, 2017 at 9:41
  • $\begingroup$ But the answers to that question give counterexamples to the existence of your $f$: even if $M$ is projective space while $N$ is the total space of the normal bundle of the embedding $i \colon S \to M$, and $S$ is any nonlinear hypersurface. $\endgroup$
    – Ben McKay
    Commented Oct 5, 2017 at 9:45

1 Answer 1

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I believe that McKay is right, here is some additional information. Let $\mathcal{O}_M$ and $\mathcal{O}_N$ denote the sheaves of holomorphic functions on $M$ and $N$, and let $I_M$, $I_N$ denote the ideal sheaves of $S\subset M$ and $S\subset N$, respectively. What you are after is an isomorphism of ringed spaces $(S,\mathcal{O}_M|_S)\rightarrow (S,\mathcal{O}_N|_S)$, but such an isomorphism would induce isomorphisms $(S,\mathcal{O}_M/I_M^{n+1}|_S)\rightarrow (S,\mathcal{O}_N/I^{n+1}_N|_S)$ for all $n\geq 0$. Here $(S,\mathcal{O}_M/I_M^{n+1}|_S)$ encodes roughly the $n$-th jet of the embedding of $S$ into $M$, and similarly for $N$. As shown e.g. in "Griffiths, The extension problem in complex analysis II; embeddings with positive normal bundle", there are obstructions to the existence of these maps for each $n\geq 1$. Precisely, if $\nu_M$ denotes the normal bundle of $S$ in $M$, then the obstructions lie in $H^1(S;TN|_S\otimes (\nu_M^\ast)^n)$ (proposition 1.3 in Griffiths). Even if all those obstructions vanish you only get a formal map of neighborhoods (i.e. an isomorphism $(S,\lim_n \mathcal{O}_M/I_M^n|_S)\rightarrow (S,\lim_n \mathcal{O}_N/I_N^n|_S)$, and you need an additional argument to upgrade it to an actual map of neighborhoods.

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