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Extending holomorphic maps from Stein manifolds into affine toric varieties


Authors: Richard Lärkäng and Finnur Lárusson
Journal: Proc. Amer. Math. Soc. 144 (2016), 4613-4626
MSC (2010): Primary 14M25; Secondary 32E10, 32Q28
DOI: https://doi.org/10.1090/proc/13108
Published electronically: April 19, 2016
MathSciNet review: 3544514
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Abstract: 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 Forstnerič, 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$.


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Additional Information

Richard Lärkäng
Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia
Address at time of publication: Department of Mathematics, University of Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany, and Department of Mathematics, Chalmers University of Technology and the University of Gothenburg, 412 96 Gothenburg, Sweden
Email: larkang@chalmers.se

Finnur Lárusson
Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia
Email: finnur.larusson@adelaide.edu.au

DOI: https://doi.org/10.1090/proc/13108
Keywords: Stein manifold, Stein space, affine toric variety, holomorphic map, extension
Received by editor(s): January 11, 2016
Published electronically: April 19, 2016
Additional Notes: The authors were supported by Australian Research Council grant DP120104110.
Communicated by: Franc Forstneric
Article copyright: © Copyright 2016 American Mathematical Society