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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Extending holomorphic maps from Stein manifolds into affine toric varieties
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by Richard Lärkäng and Finnur Lárusson PDF
Proc. Amer. Math. Soc. 144 (2016), 4613-4626 Request permission

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 $\mathbb {C}^4$.

References
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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
  • MR Author ID: 347171
  • Email: finnur.larusson@adelaide.edu.au
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 144 (2016), 4613-4626
  • MSC (2010): Primary 14M25; Secondary 32E10, 32Q28
  • DOI: https://doi.org/10.1090/proc/13108
  • MathSciNet review: 3544514