On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents
If \(M \subset \mathbb{C}^2\) is a real analytic surface with a nondegenerate complex tangent at a point (= the origin), being thus locally given by \(z_2 = z_1 \overline {z}_1 + \gamma z^2_1 + \gamma \overline {z}^2_1 + H(z_1, \overline {z}_1)\) (\(H\) is a convergent power series in \(z_1\) and \(\overline {z}_1\) starting with third order terms and \(\gamma\) is the Bishop invariant for the hyperbolic case, \({1\over 2} < \gamma < \infty\)) which is formally equivalent to a quadric \(Q_\gamma : z_2 = z_1 \overline {z}_1 + \gamma z^2_1 + \gamma \overline {z}^2_1\), the author shows that \(M\) is actually equivalent to \(Q\) through biholomorphic mappings provided that \(\gamma \lambda^2 - \lambda + \gamma = 0\) for \(\lambda\) either a root of unity or satisfying a condition called the diophantine condition. Also the author deals with holomorphic flatness of a real analytic surface in \(\mathbb{C}^2\) obtaining a result which indicates that the holomorphic flatness (transforming by holomorphic mappings a manifold into a real hyperplane) is not equivalent to the convergence of normalizations for surfaces near hyperbolic complex tangents.
real analytic surface in \(\mathbb{C}^ 2\)
scientific article; zbMATH DE number 745324