Research on pluricanonical bundles and multiplier ideal sheaves for a degenerate family of complex manifolds

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For a smooth proper Kaehler morphism f : X→Y and a Nakano semi-positive vector bundle(E, h) on X, we showed that every direct image sheaf R^qf_* K_{ X/Y}(E) is locally free and that it is equipped with a natural Hermitian metric, called a Hodge metric, with Nakano semi-positive curvature. Moreover even when f : X→Y can be singular, we showed that the positivity of a direct image sheaf extends across the singular locus of f on Y.