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

Abstract

We study hypersurfaces of C n+2 x,u,v given by equations of form uv = p(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.

Structure of the group of automorphisms of the spectral $$2$$ -ball

“…Anyway little is known about their holomorphic automorphisms. It was lastly shown (see [5,7]) that the group generated by shears and overshears (for the definition see e.g. [8]) is dense in the group of holomorphic automorphisms.…”

Algebraic density property of homogeneous spaces

“…In [15] this class was enlarged by hypersurfaces of form uv = p(x) and in [16] by connected complex algebraic groups except for C + , C * (for which the density property is not true) and the higher dimensional tori (for which the validity of this property is still unknown). Furthermore, it was shown in [15], [16] that these varieties have the algebraic density property.…”

“…Hence SL 2 /C * is isomorphic to a hypersurface S in C 3 u,v,z given by the equation uv = z 2 −1/4. In particular, it has the algebraic density property by [15].…”

Analytical Derivation of the Fuss Relations for Bicentric Hendecagon and Dodecagon

“…In actual derivation of the Fuss relations we use only upper half-plane (y ≥ 0) because such polygons are symmetric with respect to x-axes. Slightly different derivation is for odd and for even sided polygons [6], as is also described later in this article.…”