We define, for a compact subset K of complex Euclidean space containing the origin, the so-called Borel map (at the origin). We discuss its properties in detail and state, in the case when K is subanalytic, two conjectures relating the injectivity and surjectivity of the Borel map with properties of the polynomial hull of K. We give strong evidence for the validity of the conjectures (e.g. the open mapping property) and show that they are true when K is convex.

中文翻译：

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$$\mathbb {C}^m$$ 的紧致亚解析子集的 Borel 映射

我们为包含原点的复欧几里得空间的紧致子集K定义所谓的 Borel 映射（在原点）。我们详细讨论了它的性质，并说明了在K是亚解析的情况下，将 Borel 映射的单射性和满射性与K的多项式包的性质联系起来的两个猜想。我们为猜想的有效性（例如开放映射性质）提供了有力的证据，并证明当K是凸的时它们是正确的。