A remark on pseudoconvex domains with analytic complements in compact Kähler manifolds

Abstract

For an effective divisor $A$ with support $B$ in a compact Kähler manifold $M$ of dimension $\geq 3$, the following are antinomic. a) $M\backslash B$ has a $C^{\infty}$ plurisubharmonic exhaustion function whose Levi form has pointwise at least 3 positive eigenvalues outside a compact subset of $M\backslash B$. b) $[A]|B$, the normal bundle of $A$, is topologically trivial.