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Let $f:X\to Y$ be a surjective projective map and $L$ be a holomorphic linebundle on $X$ equipped with a (singular) semi-positive Hermitian metric $h$. Inthis article, by studying the canonical metric on the direct image sheaf of thetwisted relative canonical bundles $K_{X/Y}\otimes L\otimes\mathscr{I}(h)$, weobtain that this metric has dual Nakano semi-positivity when $h$ is smooth andthere is no deformation by $f$ and that this metric has locally Nakanosemi-positivity in the singular sense when $h$ is singular.

Dual Nakano positivity and singular Nakano positivity of direct image sheaves