Development mapping on CR manifolds and the manifolds with constant holomorphic sectional curvature.

Abstract

Let M be a connected real hypersurface in a complex manifold N of dimension n. M is called spherical if, for each point $p \in M,$ there is a local holomorphic coordinate system $(z\sb1,\...,z\sb{n})$ such that M is defined by $\vert z\sb1\vert\sp2 {+\...+} \vert z\sb{n}\vert\sp2 = 1.$. In this paper, we consider manifolds with pseudoconvex boundary and find necessary and sufficient conditions for the existence of the complete Kahler metric with constant holomorphic sectional curvature. In order to study this question, we compare the Chern numbers $c\sb2 c\sbsp{1}{n-2}$ and $({-}1)\sp{n}{n\over2(n+1)}c\sbsp{1}{n}$ of the complex space X whose boundary is smooth spherical CR-manifold M. We consider the case n = 2, and we assume that there is a unique strongly pseudoconvex Stein space X with $\partial$X = M, as Rossi showed for the case n $\ge$ 3. We first analyze the singularities of X, then we use the development map f of the CR-manifold M. Finally, we use the Fefferman's approximate solution of the Monge-Ampere equation to study the asymptotic behavior of the Kahler-Einstein metric near the boundary M. We have the following main results: (1) If we have only nice singularities such as quasihomogeneous singularities on X, then in the resolution of these singularities the degree d of the normal bundle of the exceptional curves C and the genus g of this curve satisfies d = 1 $-$ g if and only if the neighborhood bounded by the link has negative constant holomorphic curvature. (2) If the ramification divisor R$\sb{f}$ of the development map f is of normal crossing, then R$\sb{f}=\phi$ in order to have constant holomorphic sectional curvature on X. (3) Let M be a spherical CR-manifold and X be a strictly pseudoconvex smooth manifold of dimension n with $\partial$X = M. Then the canonical Kahler-Einstein metric on X satisfies$$\int\sb{X}(c\sb2 - {n\over 2(n+1)}c\sbsp{1}{2})c\sbsp{1}{n-2} < \infty.$$.