(Volume) density property of a family of complex manifolds including the Koras-Russell cubic threefold
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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.References
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Additional Information
- Matthias Leuenberger
- Affiliation: Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
- MR Author ID: 1103054
- Email: matthias.leuenberger@bluewin.ch
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3887-3902
- MSC (2010): Primary 32M05, 32M25, 14R10
- DOI: https://doi.org/10.1090/proc/13030
- MathSciNet review: 3513546