Holomorphic retraction $\implies$ holomorphic tubular neighbourhood? 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.
 A: 
Question. Is there a holomorphic
tubular neighbourhood of S in M?

I have a counterexample. Let $\phi:\; Z^2 \to Aut(C^n)$
be a group homomorphism, with $Aut(B)$ denote the group
of holomorphic automorphisms of $C^n$. We assume that
the image of $\phi$ preserves 0 and acts trivially
on $T_0 C^n$. To construct $\phi$ which has
infinite image take, for example,
$\phi(1,0)(x, y) = (x, y + x^2)$ on $C^2$.
The quotient $M:= C \times C^n/Z^2$,
with $Z^2$ acting by translations on $C$ and
as $\phi$ on $C^n$ is a manifold admitting a retraction
to the elliptic curve $E= C \times\{0\}/Z^2$,
Since the normal bundle of $E$ is trivial,
existence of "tubular neighbourhood" would
imply that $M = E \times B$ locally around $E$.
This would imply that there exists
a holomorphic map $p:\; C \to C^n$
such that for any $a\in Z^2\subset C$
one has $p(z+a) = \phi(a)(p(z))$.
For the example above, this would
imply that $p$ has polynomial growth,
hence it is algebraic, which is impossible.
This example is based on an idea of
Loray, Thom, Touzet, who proved that
for any elliptic curve with trivial tangent
bundle in a formal scheme there is a
canonical foliation for which
this curve is a closed fiber.
Frank Loray (IRMAR), Olivier Thom (IRMAR), Frédéric Touzet (IRMAR)
Two dimensional neighborhoods of elliptic curves:
formal classification and foliations.
https://arxiv.org/abs/1704.05214
(Theorem 3)
I have constructed a foliation with a unique
closed fiber, and checked that the
curve has no deformations. I guess I could have
just referred to this paper instead.
A: In fact the short exact sequence above $\textit{doesn't}$ splits even in the case of retraction and $k=1$. What I mean is the following.
Indeed, it does split as a sequence of $\mathcal{O}_S$-modules. Now take $S$ to be the diagonal in $M=S\times S$. Put $k=1$ and consider the sequence of corresponding $\mathcal{O}_M$-modules. This ext is so called Atiyah's class. I claim that a tubular neighborhood of $S$ defines a holomorphic connection on $T_S$.
In our case the normal bundle to $S$ is the tangent bundle $T_S$. The tubular neighborhood for each $x\in S$ defines identification $f_x$ of a neighborhood $0\in T_x$ with a neighborhood of $x\in S$. Hence for each nearby $x,y\in S$ there is well-defined ${f_x}^{-1}(y)\in T_x$. Using affine parallel transport $P_{x\to y}:T_x\to T_x$ at $T_x$ where $P_{x\to y}v=v+f^{-1}(y)$, you get a natural identification of $T_x$ with $T_y$ via $df|_{f^{-1}(x)}\circ P_{x\to y}$.
Our Atiyah's class $A\in Ext^1(\mathcal{O}_S,\mathcal{I}_S/\mathcal{I}_S^2)$  is the universal one: for a sheaf $E$ over $S$ you can consider $A_E={\pi_2}_*(\pi_1^*E\otimes A)\in Ext^1_S(E,\Omega^1_S\otimes E)$, where $\pi_i$ are projections $S\times S\to S$. It is well-known that $A_E$ is complete obstruction for existence of a holomorphic connection over $E$. This happens rarely, since Atiyah class $A_{E}$ can be used for expression of Chern classes of $E$ (see Markaryan's paper on the subject).
In particular, if Chern classes of $T_S$ are non zero then there is no holomorphic connection and, as a corollary, there is no tubular neighborhood.
PS From this point the existence of a tubular neighborhood of the diagonal may imply vanishing of a very rich structure provided by derived co-Lie algebra defined by $A_{\Omega}$. It would be nice to see if there are examples of proper diagonals admitting a tubular neighborhood other than abelian varieties.
