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Let $M$ be a complex manifold and $X \subset M$ a complex submanifold. We may assume that $X$ is compact, if that's helpful.

Can we always find a neighbourhood $U$ of $X$ in $M$ together with a holomorphic map $r : U \to X$ which restricts to the identity map on $X$?

In the $C^\infty$-case, any tubular neighborhood gives such retractions. Of course, the tubular neighbourhood theorem may fail in the holomorphic case, but the existence of a retraction is a priori weaker.

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  • $\begingroup$ This is essentially a dupe of this question as it was observed in inkspot's answer there that there is no holomorphic retraction either. $\endgroup$ Jan 9, 2021 at 15:44

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No, this is actually very rare. Indeed the existence of such a retraction implies that the exact sequence $$0\rightarrow T_X\rightarrow T_{M|X}\rightarrow N_{X/M}\rightarrow 0$$ splits. In particular, the coboundary map $H^0(N_{X/M})\rightarrow H^1(X,T_X)$ is zero, which means that first order deformations of $X$ in $M$ are trivial as deformations of $X$. This is false in most examples, for instance hypersurfaces in $\mathbb{P}^n$, and many others.

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