Corollary 11.17.4B53EB4EFC135

Proof. The chart of $Z$ where the $\mathcal O_ X$-coordinate is nonzero is identified with the total space of $E$, and $Y$ is its zero section. Suppose that $Z\setminus Y$ is $r$-convex. Since $\dim Z=n+r$, a smooth exhaustion function $\psi $ then has at least $n+1$ positive Levi eigenvalues outside a compact set $K$. Choose a relatively compact disk-bundle neighborhood $D\Subset E$ of $Y$. The set $Z\setminus D$ is compact and disjoint from $Y$. Choose a regular value $c$ of $\psi $ greater than its maximum on $Z\setminus D$ and on $K$.

Since $\{ \psi \leq c\} $ is compact in $Z\setminus Y$, the set $\Omega =\{ \psi >c\} \cup Y$ is an open neighborhood of $Y$. Its closure is a compact subset of $D$, and its boundary is the regular level set $\{ \psi =c\} $. The defining function $f=c-\psi $ has at least $n+1$ negative Levi eigenvalues there. Restriction to the complex tangent hyperplane removes at most one negative direction, leaving at least $n$. Since $E$ is ample, theorem 11.17.2 produces a smooth complex norm on $E^*$ with plurisubharmonic square, a contradiction. $\square$