(Volume) Density Property of a family of complex manifolds including the Koras-Russell Cubic

TL;DR

The paper develops modified criteria for the density and volume density properties of complex manifolds and applies them to a family including the Koras-Russell Cubic, establishing these properties for specific examples.

Contribution

It introduces new criteria for density properties and proves these properties for a family of complex manifolds, including the Koras-Russell Cubic Threefold.

Findings

The family of manifolds given by x^2 y = a(z) + x b(z) has the volume density property.

The Koras-Russell Cubic Threefold has both the density and volume density properties.

Transitivity of volume-preserving automorphisms implies density properties under certain conditions.

Abstract

We present modified versions of existing criteria for the density property and the volume density property of complex manifolds. We apply this 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 volume form $\mathrm{d} x/x^2\wedge \mathrm{d} z_0\wedge\ldots\wedge\mathrm{d} z_n$. The key step is showing that in certain cases transitivity of the action of (volume preserving) holomorphic automorphisms implies the (volume) density property, and then giving 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.