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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\mathbb {C}^4$.

- David A. Cox, John Little, and Donal O’Shea,
*Using algebraic geometry*, 2nd ed., Graduate Texts in Mathematics, vol. 185, Springer, New York, 2005. MR**2122859** - David A. Cox, John B. Little, and Henry K. Schenck,
*Toric varieties*, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR**2810322**, DOI 10.1090/gsm/124 - Franc Forstnerič,
*Oka manifolds*, C. R. Math. Acad. Sci. Paris**347**(2009), no. 17-18, 1017–1020 (English, with English and French summaries). MR**2554568**, DOI 10.1016/j.crma.2009.07.005 - Franc Forstnerič,
*Stein manifolds and holomorphic mappings*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 56, Springer, Heidelberg, 2011. The homotopy principle in complex analysis. MR**2975791**, DOI 10.1007/978-3-642-22250-4 - Franc Forstnerič,
*Oka manifolds: from Oka to Stein and back*, Ann. Fac. Sci. Toulouse Math. (6)**22**(2013), no. 4, 747–809 (English, with English and French summaries). With an appendix by Finnur Lárusson. MR**3137250**, DOI 10.5802/afst.1388 - Franc Forstnerič and Finnur Lárusson,
*Survey of Oka theory*, New York J. Math.**17A**(2011), 11–38. MR**2782726** - M. Gromov,
*Oka’s principle for holomorphic sections of elliptic bundles*, J. Amer. Math. Soc.**2**(1989), no. 4, 851–897. MR**1001851**, DOI 10.1090/S0894-0347-1989-1001851-9 - Anargyros Katsabekis and Apostolos Thoma,
*Toric sets and orbits on toric varieties*, J. Pure Appl. Algebra**181**(2003), no. 1, 75–83. MR**1971806**, DOI 10.1016/S0022-4049(02)00305-5 - Finnur Lárusson,
*Mapping cylinders and the Oka principle*, Indiana Univ. Math. J.**54**(2005), no. 4, 1145–1159. MR**2164421**, DOI 10.1512/iumj.2005.54.2731 - Finnur Lárusson,
*Smooth toric varieties are Oka.*arXiv:1107.3604 - Henry B. Laufer,
*Imbedding annuli in $\textbf {C}^{2}$*, J. Analyse Math.**26**(1973), 187–215. MR**346189**, DOI 10.1007/BF02790429 - Ezra Miller and Bernd Sturmfels,
*Combinatorial commutative algebra*, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR**2110098** - Mircea Mustata,
*Semigroups and affine toric varieties,*Chapter 1 of lecture notes on toric varieties. Accessed at http://www.math.lsa.umich.edu/$\sim$mmustata/toric_var.html on 28 October 2014. - R. Michael Range,
*Holomorphic functions and integral representations in several complex variables*, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR**847923**, DOI 10.1007/978-1-4757-1918-5 - Alexander Schrijver,
*Theory of linear and integer programming*, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Ltd., Chichester, 1986. A Wiley-Interscience Publication. MR**874114** - Bernd Sturmfels,
*Algorithms in invariant theory*, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993. MR**1255980**, DOI 10.1007/978-3-7091-4368-1

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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

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