Homogeneous real hypersurfaces (Q1902224)

Let \(M\) be an analytic real hypersurface through the origin in \(\mathbb{C}^n\). The hypersurface \(M\) is called weighted homogeneous if it is locally equivalent, via a biholomorphic map which preserves the origin, to a hypersurface given by an equation of the form \(P(z, \overline z) = 0\), where \(P\) is a polynomial which is homogeneous with respect to a nonisotropic group of dilations. The main result is Theorem 4.1. Let \(M\) be an analytic real hypersurface through the origin in \(\mathbb{C}^n\) and suppose there is an approximate infinitesimal dilation \(Y \in \text{hol} (M)\). Then \(M\) is weighted homogeneous. Theorem 4.1 does not require that \(M\) be rigid and there is no nondegeneracy hypothesis or finite type hypothesis on \(M\). However, if \(M\) is weighted homogeneous, then it is rigid.