Let $M$ be a complex manifold and $S \subset M$ a compact complex submanifold together with a holomorphic retraction $$r : M \to S,$$ i.e. a holomorphic map which restricts to the identity on $S$.

**Question.** *Is there a holomorphic tubular neighbourhood of $S$ in $M$?*

(That is, a neighbourhood $U$ of the normal bundle of $S$ in $M$ together with a biholomorphism from $U$ to a neighbourhood of $S$ in $M$ which restricts to the identity on $S$.)

**Remarks.**

(1) If a tubular neighbourhood exists, then the bundle map gives such a retraction (after replacing $M$ by a neighbourhood of $S$ in $M$).

(2) There are well-known obstructions to holomorphic tubular neighbourhoods. For instance [1], if a tubular neighbourhood exists then the short exact sequences $$0 \to \mathcal{I}_S / \mathcal{I}_S^{k + 1} \to \mathcal{O}_M/\mathcal{I}_S^{k+1} \to \mathcal{O}_S \to 0$$ split for all $k \ge 1$. But, of course, in our case all these sequences do split via $r^* : \mathcal{O}_S \to \mathcal{O}_M$. There is a further condition in [1] called $k$-comfortably embedded, but I'm not sure how it relates to a retraction.

**References.**

[1] Abate, M.; Bracci, F.; Tovena, F. Embeddings of submanifolds and normal bundles. Adv. Math. 220 (2009), no. 2, 620–656.