Corollary 12.28.49FC7A89CD619

Proof. The restriction of a smooth strictly plurisubharmonic function on $U\setminus D$ to a complex leaf of $M_ r$ is strictly plurisubharmonic. At a maximum point on the compact set $M_ r$, this restriction has a local maximum on the leaf, which has positive dimension because $n\geq 2$; this contradicts the maximum principle.

In the finite-holonomy case, each fiber of the map of proposition 12.28.2 over a nonzero value is a connected compact complex hypersurface of positive dimension. By the maximum modulus principle, every holomorphic function on $S\setminus D$ is constant on it, so two distinct points of the fiber cannot be separated. In the infinite-holonomy case, theorem 12.28.3 makes every holomorphic function constant on the punctured tube, which again contradicts holomorphic separability. In both cases $S\setminus D$ is not Stein. Finally, theorem 12.27.3 gives the flat neighborhood for the nine-point blowups, whose normal bundle is non-torsion by assumption, and theorem 12.28.3, also with $K=\varnothing $, gives the last assertion. $\square$