Extending holomorphic maps from Stein manifolds into affine toric varieties
| Autor: | Finnur Larusson, Richard Lärkäng |
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| Rok vydání: | 2016 |
| Předmět: |
Pure mathematics
Subvariety Mathematics - Complex Variables Mathematics::Complex Variables Applied Mathematics General Mathematics 14M25 (Primary) 32E10 32Q28 (Secondary) 010102 general mathematics Holomorphic function Toric variety Submanifold 01 natural sciences Contractible space 0103 physical sciences FOS: Mathematics 010307 mathematical physics Affine transformation Complex Variables (math.CV) 0101 mathematics Complex manifold Mathematics::Symplectic Geometry Interpolation Mathematics |
| Zdroj: | Proceedings of the American Mathematical Society. 144:4613-4626 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/proc/13108 |
| Popis: | A complex manifold $Y$ is said to have the interpolation property if a holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be a contractible submanifold of $X=\mathbb{C}^n$ gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstneri\v{c}, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular targets. The targets that we study are affine toric varieties, not necessarily normal. We prove that every affine toric variety satisfies a weakening of the interpolation property that is much stronger than the convex interpolation property, but the full interpolation property fails for most affine toric varieties, even for a source as simple as the product of two annuli embedded in $\mathbb{C}^4$. Comment: 14 pages, v2 and v3: minor corrections and clarifications. To appear in Proceedings of the AMS |
| Databáze: | OpenAIRE |
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