(Volume) density property of a family of complex manifolds including the Koras-Russell cubic threefold (Q2814408)

The density property has been introduced by \textit{D. Varolin} [J. Geom. Anal. 11, No. 1, 135--160 (2001; Zbl 0994.32019)] to describe precisely when an automorphism group of a complex manifolds is ``large''. The Andersen-Lempert Theorem [\textit{E. Andersén} and \textit{L. Lempert}, Invent. Math. 110, No. 2, 371--388 (1992; Zbl 0770.32015); \textit{F. Forstnerič} and \textit{J.-P. Rosay}, Invent. Math. 112, No. 2, 323--349 (1993; Zbl 0792.32011)], which is essentially a Runge-type approximation theorem for holomorphic automorphisms, can then be generalized from the complex-Euclidean space to Stein manifolds with the density property.NEWLINENEWLINEA complex manifold \(X\) enjoys the density property if the Lie algebra generated by its complete holomorphic vector fields is dense in the Lie algebra of all of its holomorphic vector fields. Moreover, if there exists a nowhere vanishing section \(\omega\) of the canonical bundle of \(X\), then one can define also the following property: \((X, \omega)\) enjoys the volume density property if the Lie algebra generated by its complete \(\omega\)-preserving holomorphic vector fields is dense in the Lie algebra of all of its \(\omega\)-preserving holomorphic vector fields.NEWLINENEWLINEIn this paper, the author establishes the density property for a family of Stein manifolds given by \(X := \{x^2 y = a + x b\} \subset \mathbb{C}_x \times \mathbb{C}_y \times \mathbb{C}_z^{n+1}\) with \(a\) and \(b\) being certain polynomials in \(z\), as well as the volume density property for these manifolds w.r.t. the holomorphic volume form \(dx/x^2 \wedge dz\). As the most interesting example, it includes the well-known Koras-Russell cubic \(x^2 y = -(z_0^2 + z_1^3) - x\) which is known to be diffeomorphic to \(\mathbb R^6\) but not algebraically isomorphic to \(\mathbb C^3\).