Pseudoconvex Domains in ℙⁿ: A Question on the 1-Convex Boundary Points

Abstract

Let ℙⁿ be n-dimensional complex projective space, let Ω ⊊ ℙⁿ be a pseudoconvex domain, and let δ(z) be the distance from z ∈ Ω to the boundary of Ω with respect to the Fubini-Study metric. According to a fundamental theorem of A. Takeuchi [T], the function — log δ is plurisubharmonic and enjoys an estimate

$$ \sqrt { - 1\partial \overline {\partial (} } - \log \delta ) \geqslant \frac{1} {3}\omega _{FS,} $$

where ω_FS denotes the Kähler form of the Fubini-Study metric, and the left hand side of the inequality is defined as a current. From Takeuchi’s theorem it follows that Ω admits a strictly plurisubharmonic exhaustion function, so that Ω is a Stein manifold. This means in particular that the boundary of Ω is connected if n ≥ 2 (cf. [G-R]).