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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) densityproperty, which has remarkable consequences. Until now all known manifolds withthe volume density property are algebraic, and the tools used to establish thisproperty are algebraic in nature. In this note we adapt a known criterion tothe holomorphic case, and give the first known examples of non-algebraicmanifolds with the volume density property: they arise as suspensions orpseudo-affine modifications over Stein manifolds satisfying some technicalproperties. As an application we show that there are such manifolds that arepotential counterexamples to the Zariski Cancellation Problem, a variant of theToth-Varolin conjecture, and the problem of linearization of C*-actions on C^3.

Non-algebraic Examples of Manifolds with the Volume Density Property