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Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds


Author: Finnur Lárusson
Journal: Proc. Amer. Math. Soc. 143 (2015), 1159-1167
MSC (2010): Primary 32E10; Secondary 32H02, 32Q28, 54C35, 54C55, 55M15
DOI: https://doi.org/10.1090/S0002-9939-2014-12335-5
Published electronically: October 16, 2014
MathSciNet review: 3293731
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Abstract: The basic result of Oka theory, due to Gromov, states that every continuous map $ f$ from a Stein manifold $ S$ to an elliptic manifold $ X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $ f$ at once, in a way that depends continuously on $ f$ and leaves $ f$ fixed if it is holomorphic to begin with. In other words, is $ \mathcal {O}(S,X)$ a deformation retract of $ \mathcal {C}(S,X)$? We prove that it is if $ S$ has a strictly plurisubharmonic Morse exhaustion with finitely many critical points, in particular, if $ S$ is affine algebraic. The only property of $ X$ used in the proof is the parametric Oka property with approximation with respect to finite polyhedra, so our theorem holds under the weaker assumption that $ X$ is an Oka manifold. Our main tool, apart from Oka theory itself, is the theory of absolute neighbourhood retracts. We also make use of the mixed model structure on the category of topological spaces.


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Additional Information

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/S0002-9939-2014-12335-5
Keywords: Stein manifold, Oka manifold, parametric Oka property, deformation retract, absolute neighbourhood retract, mixed model structure.
Received by editor(s): June 23, 2013
Published electronically: October 16, 2014
Additional Notes: The author was supported by Australian Research Council grant DP120104110.
The author is grateful to Jaka Smrekar for helpful discussions.
Communicated by: Franc Forstneric
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.