New criterion for the algebraic volume density property
Presented by Prof. Frank KUTZSCHEBAUCH
Type: Oral presentation
Track: Several Complex Variables
This is a report on a joint work with Shulim Kaliman. We start by reminding definitions and importance of the notions of density property and volume density property introduced by Varolin. To prove that a manifold has one of these properties can be a cumbersome calculation with Lie brackets of vector fields. We present a criterion, proved using the theory of coherent sheafs, which can reduce the calculations to a minimum. In particular it can be applied to give a very short proof of our main result in Kaliman, S., Kutzschebauch, F.: The algebraic volume density property for affine algebraic manifolds. Invent. Math. 181 (2010), 605--647, which says that linear algebraic groups have the volume density property with respect to the left invariant (Haar) form. As another application on can prove that all homogenous spaces $G/H$ (G linear algebraic, H reductive) admitting a left invariant form have the volume density property.