On the positivity of direct image bundles

In the present paper, we obtain an equivalent relation between the log-plurisubharmonicity of the relative Bergman kernel, the Griffiths and Nakano positivity for the direct image with the natural $L^2$ metric, by finding a converse of Berndtsson's theorem on the direct image. A converse of Berndtsson's generalization of Kiselman minimal principle is also obtained.