Non-algebraic examples of manifolds with the volume density property

Ramos Peon, Alexandre (2017). Non-algebraic examples of manifolds with the volume density property. Proceedings of the American Mathematical Society, 145(9), pp. 3899-3914. American Mathematical Society 10.1090/proc/13565

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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 C⁺-actions on C³.

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