Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration. (English) Zbl 1039.32014

Let \(\Delta\) denote the closed unit disk in \(\mathbb C\), let \(B\) denote the open unit ball in \(\mathbb C^m\) and let \(X\) be a compact subset of \((\partial B) \times \mathbb C^n\). Put \(X_z=\{w \in \mathbb C^n; (z,w) \in X \}\).
The author proves several results concerning the structure of the polynomial convex hull \(\widehat X\) of \(X\). One example: suppose that \(X_z\) is the closure of a completely circled pseudoconvex domain, \(X\) admits a nonnegative continuous defining function \(\rho\), \(X=\{(z,w) \in (\partial B) \times \mathbb C^n; \rho(z,w) \leq 1 \}\), \(\rho\) is homogeneous plurisubharmonic in \(w\) for any fixed \(z\), and \(\rho(z,w) \neq 0\) if \(w \neq 0\). Then, through any interior point \((z_0,w_0)\) of \(\widehat X\) passes an analytic disk parametrized by a holomorphic mapping \(f:\Delta \rightarrow {\overline B} \times \mathbb C^n\) which is smooth up to the boundary of \(\Delta\) and \(f(\partial \Delta) \subset X\).


32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
32W20 Complex Monge-Ampère operators
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