Jetmetriken und hyperbolische Geometrie. (On jet metrics and hyperbolic geometry) (Q1114823)

Let E be a unit complex disk and let Y be a complex manifold. Denote by J(m,Y) the jet-bundle over Y of order m. By definition a jetmetric of order m is a continuous non-negative real function on J(m,Y) such that \(f(ca)=| c| f(a)\) on the fibers of the bundle. Starting from f and some non-constant holomorphic map from E to Y it is possible to define some hermitian metric in E. We say that f is hyperbolic if for any holomorphic map from E to Y the induced metric has only isolated zeros on E and has the Gaussian curvature uniformly bounded by a negative constant. Let now Y be a complement of general union of at least three smooth quadrics in a projective plane. It is proved that Y has a hyperbolic metric of order 4.