Let $X$ be a complex manifold, let $S \subset X$ be a complex submanifold, and let $N_S := TX|_S / TS$ be the normal bundle.
Definition. A tubular neighborhood of $S$ in $X$ is a neighborhood $U$ of the zero section of $N_S$ together with a biholomorphism from $U$ to a neighborhood of $S$ in $X$ which is the identity on the zero section.
In contrast to the smooth category, there is a well-known obstruction to the existence of tubular neighborhoods:
Proposition (See e.g. [1, Proposition 2.5]). If $S$ has a tubular neighborhood in $X$, then the short exact sequence $$ 0 \to TS \to TX|_S \to N_S \to 0\tag{1} $$ splits.
As indicated in this mathoverflow answer, this can be turned into a cohomological obstruction. There are many examples where this sequence does not split, such as any non-linear submanifold of $\mathbb{P}^n$ [1].
On the other hand, I am currently looking at a large family of pairs $(X, S)$ where (1) does split. I suspect that most of them still have no tubular neighborhood, but I am not sure how to proceed.
Question. What are other obstructions to the existence of tubular neighborhoods beside the non-splitness of (1)?
References.
[1] Morrow, J.; Rossi, H. Submanifolds of $\mathbb{P}^n$ with splitting normal bundle sequence are linear. Math. Ann. 234 (1978), no. 3, 253–261.
