On the hyperconvexity of holomorphically convex domains in the space \({\mathbb{C}}^ n\) (Q1071136)

This paper presents a partial answer to J.-L. Stehle's conjecture that every holomorphically convex domain D in \({\mathbb{C}}^ n\) with \(D=(\bar D\overset\circ)\) is hyperconvex, meaning there exists a plurisubharmonic function p on a neighborhood of \(\bar D,\) which is negative on D, such that \(\{z\in D| p(z)\leq c\}\) is relatively compact in D for any \(c<0\). The author introduces a class \({\mathcal F}_ D\) of holomorphic functions on D and proves the conjecture for \({\mathcal F}_ D\)-convex domains D in \({\mathbb{C}}^ n\) with \(D=(\bar D\overset\circ)\). \({\mathcal F}_ D\) consists of those holomorphic functions f on D to which a \(c\in {\mathbb{C}}\) exists such that f(0)\(\neq c\) and the current associated to \(V:=\{z\in D| f(z)=c\}\) has finite order. The proof uses the result of Lelong and Skoda on the existence of an entire function \(F: {\mathbb{C}}^ n\to {\mathbb{C}}\) with \(V=F^{-1}(0)\).