Density property for hypersurfaces $$UV = P({\bar X})$$

Abstract

We study hypersurfaces of $${\mathbb{C}}^{n+2}_{{\bar x},u,v}$$ given by equations of form $$UV = P({\bar X})$$ where the zero locus of a polynomial p is smooth reduced. The main result says that the Lie algebra generated by algebraic completely integrable vector fields on such a hypersurface coincides with the Lie algebra of all algebraic vector fields. Consequences of this result for some conjectures of affine algebraic geometry and for the Oka-Grauert-Gromov principle are discussed.