Ramified coverings over a complete Kähler space

摘要

Let X and Y be reduced complex spaces with countable topology. Let π: X→Y be a locally semi-finite holomorphic map such that the analytic set π~{-1} (S(Y)) $ is nowhere dense in X. If Y is complete Kähler, then we prove that X is also complete Kähler. Especially if π: X →Y is a (not necessarily finitely sheeted) ramified covering over a complete Kähler space Y, then X is also complete Kähler.