Extending holomorphic maps from Stein manifolds into affine toric varieties (Q2821724)

A complex space \(Y\) is said to satisfy the \textit{interpolation property} if every holomorphic map \(S\to Y\) from a subvariety \(S\) of a reduced Stein space \(X\) can be extended to a holomorphic map \(X\to Y\) as soon as it can be extended to a continuous map \(X\to Y\).NEWLINENEWLINEThe interpolation property has been classically studied for the case that \(Y\) is a complex manifold, where several equivalent characterizations have been proved. As an example, it is sufficient in this case to take \(S\) to be a contractible submanifold of \(X=\mathbb C^n\).NEWLINENEWLINEThe article under review undertakes the first steps towards developing a theory of the interpolation property for singular targets \(Y\). The authors consider the case where \(Y\) is an irreducible affine toric variety which is not assumed to be normal.NEWLINENEWLINEThey prove that without further assumptions on \(Y\), the interpolation property always holds true if \(S\) is a factorial subvariety of a reduced Stein space \(X\) such that \(H^p(X,\mathbb Z)\to H^p(S,\mathbb Z)\) is surjective for \(p\leq 2\).NEWLINENEWLINEThey also study the more general case of \textit{normal} subvarieties \(S\subset X\) and show that the interpolation property still holds in this case for nondegenerate maps \(S\to Y\) if the normalization of \(Y\) is \(\mathbb C^d\). In contrast, they prove that if the normalization of \(Y\) is not \(\mathbb C^d\), the interpolation property becomes false even in the simple case where \(S\) is the product of two annuli embedded in \(X=\mathbb C^4\).