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](https://mathoverflow.net/questions/114414/is-there-any-holomorphic-version-of-the-tubular-neighborhood-theorem?rq=1), the tubular neighbourhood theorem may fail in the holomorphic case, but the existence of a retraction is *a priori* weaker.



Existence of holomorphic retraction
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.


