Algebraicity of real analytic hypersurfaces and blowing-down. (Q1608530)

The author proves the following theorem. Let \((M,0)\) be a germ of real analytic strongly pseudoconvex hypersurface in \({\mathbb{C}}^{n+1}\). Then the following two statements are equivalent. (i) \((M,0)\) is holomorphically equivalent to a germ of real algebraic hypersurface. (ii) \((M,0)\) is a blow-down to a real algebraic one, i.e. there is a real analytic CR-mapping \(G:(M,0)\to (\widetilde M,0)\) such that \(G\) extends to a holomorphic bimeromorphic map \(G:U\to \widetilde U\), where \(U, \widetilde U\) are neighborhoods of \(0\) in \({\mathbb{C}}^{n+1}\), and \(\widetilde M\) is a real algebraic (not necessary smooth) hypersurface in \({\mathbb{C}}^{n+1}\).