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

2015/07/14 by Matthias Leuenberger, Leuenberger, Matthias
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.AG #math.CV #msc:14R10 #msc:32M05 #msc:32M25

paper · pdf · doi:10.48550/arxiv.1507.03842

15p

arxiv created 2015/07/14 · arxiv updated 2015/07/15

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 x²y=a( z) + xb( z) with z =(z₀,…,z_n)∈ℂⁿ⁺¹ and volume form d x/x²\wedge d z₀\wedge…\wedged 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 x²y + x + z₀² + z₁³=0\rbrace has the density property and the volume density property.

Related

• Non-algebraic Examples of Manifolds with the Volume Density Property
• A Criterion for the Density Property of Stein Manifolds
• New examples of Stein manifolds with volume density property
• Chaotic holomorphic automorphisms of Stein manifolds with the volume density property
• The Density Property for Vector Bundles