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

Leuenberger, Matthias (2016). (Volume) density property of a family of complex manifolds including the Koras-Russell cubic threefold. Proceedings of the American Mathematical Society, 144(9), pp. 3887-3902. American Mathematical Society 10.1090/proc/13030

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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 x²y=a(z) + xb(z) with z =(z₀,...,zn) ⋲ Cn+¹ and holomorphic volume form dx/x²ʌdz₀ ʌ...ʌdzn. 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 {x²y + x + z2/0 + z3/1 =0} has the density property and the volume density property.

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