Dual Nakano positivity and singular Nakano positivity of direct image sheaves

Abstract

Let f : X → Y be a surjective projective map and L be a holomorphic line bundle on X equipped with a (singular) semi-positive Hermitian metric h. In this article, by studying the canonical metric on the direct image sheaf of the twisted relative canonical bundles K X/Y ⊗ L ⊗ I (h), we obtain that this metric has dual Nakano semi-positivity when h is smooth and there is no deformation by f and that this metric has locally Nakano semi-positivity in the singular sense when h is singular.