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(Volume) density property of a family of complex manifolds including the Koras-Russell cubic threefold


Author: Matthias Leuenberger
Journal: Proc. Amer. Math. Soc. 144 (2016), 3887-3902
MSC (2010): Primary 32M05, 32M25, 14R10
DOI: https://doi.org/10.1090/proc/13030
Published electronically: March 17, 2016
MathSciNet review: 3513546
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Abstract: We present modified versions of existing criteria for the density property and the volume density property of complex manifolds. We apply these methods to show the (volume) density property for a family of manifolds given by $ x^2y=a(\bar z) + xb(\bar z)$ with $ \bar z =(z_0,\ldots ,z_n)\in \mathbb{C}^{n+1}$ and holomorphic volume form $ \mathrm {d} x/x^2\wedge \mathrm {d} z_0\wedge \ldots \wedge \mathrm {d} z_n$. The key step is to show that in certain cases transitivity of the action of (volume preserving) holomorphic automorphisms implies the (volume) density property, and then to give sufficient conditions for the transitivity of this action. In particular, we show that the Koras-Russell cubic threefold $ \lbrace x^2y + x + z_0^2 + z_1^3=0\rbrace $ has the density property and the volume density property.


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Additional Information

Matthias Leuenberger
Affiliation: Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
Email: matthias.leuenberger@bluewin.ch

DOI: https://doi.org/10.1090/proc/13030
Keywords: Density property, holomorphic automorphisms, Koras-Russell cubic threefold
Received by editor(s): July 14, 2015
Received by editor(s) in revised form: November 1, 2015, and November 5, 2015
Published electronically: March 17, 2016
Additional Notes: The author was partially supported by Schweizerischer Nationalfond Grant 200021-140235/1
Communicated by: Franc Forstneric
Article copyright: © Copyright 2016 American Mathematical Society