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Non-algebraic examples of manifolds with the volume density property

Author: Alexandre Ramos-Peon
Journal: Proc. Amer. Math. Soc. 145 (2017), 3899-3914
MSC (2010): Primary 32M17, 32H02; Secondary 32M25, 14R10
Published electronically: March 23, 2017
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Abstract: Some Stein manifolds (with a volume form) have a large group of (volume-preserving) automorphisms: this is formalized by the (volume) density property, which has remarkable consequences. Until now all known manifolds with the volume density property are algebraic, and the tools used to establish this property are algebraic in nature. In this note we adapt a known criterion to the holomorphic case, and give the first examples of non-algebraic manifolds with the volume density property: they arise as suspensions or pseudo-affine modifications over Stein manifolds satisfying some technical properties. As an application we show that there are such manifolds that are potential counterexamples to the Zariski Cancellation Problem, a variant of the Tóth-Varolin conjecture, and the problem of linearization of $ \mathbb{C}^*$-actions on $ \mathbb{C}^3$.

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Additional Information

Alexandre Ramos-Peon
Affiliation: Mathematisches Institut, Universität Bern Sidlerstr. 5 3012 Bern, Switzerland

Received by editor(s): March 3, 2016
Received by editor(s) in revised form: August 4, 2016, and September 23, 2016
Published electronically: March 23, 2017
Additional Notes: The author was partially supported by Schweizerischer Nationalfonds Grant 153120
Communicated by: Franc Forstneric
Article copyright: © Copyright 2017 American Mathematical Society

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