Dual Nakano positivity and singular Nakano positivity of direct image sheaves

Let $f:X\to Y$ 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 $K_{X\mathrm{/}Y}\otimes L\otimes \mathcal{I}(h)$, 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.