On the Stein property of the complement to an algebraic hypersurface in a toric variety
It was shown by \textit{C. O. Kiselman} [Acta Math. 117, 1-35 (1967; Zbl 0152.07602)] that the complement \({\mathbb P}^n \setminus V\) to an algebraic hypersurface \(V\) in a complex projective space \({\mathbb P}^n\) is a Stein manifold. In the article under consideration, the question is discussed about the Stein property of the complement \(M^n \setminus V\) to an algebraic surface \(V\) in an arbitrary toric variety \(M^n\). It is shown that, given a polynomial function \(F(z)\) of affine coordinates \(z \in {\mathbb C}^n\), it is possible to construct a canonical (native) toric variety \(M^n\), a compactification of the space \({\mathbb C}^n\), for which \(M^n \setminus V\) is a Stein manifold, where \(V\) is the closure of the hypersurface \(\{ z \in \mathbb C^n : F(z) = 0 \}\) in \(M^n\). This result is a generalization of the corresponding result of the author for the case of curves \(V\) in \({\overline{\mathbb C}}^2\).
scientific article; zbMATH DE number 1286462