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 {w2=uv}≅ℂ2/{±1} 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.
Related
- Suslin's cancellation conjecture on smooth real affine varieties with few real points
- On the first-order deformations of seminormal affine toric varieties
- Extension Theorems in Measure and Integration Theory as Completeness Axioms
- Rational Points on Affine Varieties over Finite Fields and Polygonal Number Sequences
- Neutral components of automorphism groups of (semi)rigid affine varieties
BibTeX
@misc{uchimoto2026approximation,
title = {{Approximation and extension theorems for normal affine toric varieties}},
author = {Ryo Uchimoto},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22926},
eprint = {2610.22926},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22926},
}
RIS
TY - UNPB
TI - Approximation and extension theorems for normal affine toric varieties
AU - Ryo Uchimoto
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22926
UR - https://arxiv.org/abs/2610.22926
AB - 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}≅ℂ^2/{±1} 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.
ER -