## 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 $$f(\partial \Delta) \subset X$$.

### MSC:

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

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