Approximation and extension theorems for normal affine toric varieties

Abstract

We study approximation and extension problems for holomorphic maps from Stein spaces to normal affine toric varieties, continuing the work of Lärkäng and Lárusson. Normality allows us to obtain twisted factorizations for arbitrary holomorphic maps from factorial complex spaces, including degenerate maps. This yields approximation and extension theorems under weaker cohomological assumptions. We also prove a simultaneous approximation and extension theorem under an additional codimension condition on the inverse image of the singular locus and a cohomological condition on the union of the interpolation submanifold and the approximation set. Finally, an explicit example with target $\{w^2=uv\}\cong\mathbb{C}^2/\{\pm1\}$ shows that simultaneous approximation and extension can fail even when the relevant cohomological obstructions vanish. Thus normality alone does not recover the full approximation and interpolation property of Oka manifolds.