ON LOCALLY HYPERCONVEX MORPHISMS

We consider locally hyperconvex morphisms of complex spaces. The main result says that if pi : Omega --> X is a locally hyperconvex morphism of complex spaces, then pi is globally hyperconvex. In particular, if X is hyperconvex, then Omega is also hyperconvex. As application, if X is a complex space and D subset of subset of X is an open set which is locally hyperconvex, then D is hyperconvex provided that there exis ts a strongly plurisubharmonic function defined on a neighborhood of ( D) over bar. This happens, for instance, if X is K-complete in the sen se of Grauert. Also, we show that hyperconvex analytic always admit a fundamental systems of hyperconvex sets.