(Volume) density property of a family of complex manifolds including the Koras-Russell cubic threefold

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 2 y = a ( z ¯ ) + x b ( z ¯ ) x^2y=a(\bar z) + xb(\bar z) with z ¯ = ( z 0 , … , z n ) ∈ C n + 1 \bar z =(z_0,\ldots ,z_n)\in \mathbb {C}^{n+1} and holomorphic volume form d x / x 2 ∧ d z 0 ∧ … ∧ d z n \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 { x 2 y + x + z 0 2 + z 1 3 = 0 } \lbrace x^2y + x + z_0^2 + z_1^3=0\rbrace has the density property and the volume density property.