  For proper surjective holomorphic maps from K"ahler manifolds to analytic
spaces, we give a decomposition theorem for the cohomology groups of the
canonical bundle twisted by Nakano semi-positive vector bundles by means of the
higher direct image sheaves, by using the theory of harmonic integrals
developed by Takegoshi. As an application, we prove a vanishing theorem of
Koll'ar-Ohsawa type by combining the L^2-method for the dbar-equation.
