Weil domain

A special case of an analytic polyhedron. A bounded domain $ D $ in $ n $- dimensional space $ \mathbf C ^ {n} $ is said to be a Weil domain if there exist $ N \geq n $ functions $ f _ {i} ( z) $, $ i= 1 \dots N $, holomorphic in a fixed neighbourhood $ U ( \overline{D}\; ) $ of the closure $ \overline{D}\; $, such that

1) $ D= \{ {z } : {| f _ {i} ( z) | < 1, i = 1 \dots N, z \in U ( \overline{D}\; ) } \} $;

2) the faces of the Weil domain $ D $, i.e. the sets

$$ \sigma _ {i} = \{ {z \in D } : { | f _ {i} ( z) | = 1 ,\ | f _ {j} ( z) | \leq 1 ,\ j \neq i } \} , $$

have dimension $ 2n - 1 $;

3) the edges of the Weil domain $ D $, i.e. the intersections of any $ k $( $ 2 \leq k \leq n $) different faces, have dimension $ \leq 2n - k $.

The totality of all $ n $- dimensional edges of a Weil domain is called the skeleton of the domain. The Bergman–Weil representation applies to Weil domains. These domains are named for A. Weil [1], who obtained the first important results for these domains.

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