What is relation between Weil-Petersson metric on holomorphic fibre space $f:X\to Y$ of compact complex manifolds $X,Y$ . (let fibres are Calabi-Yau manifolds)

And Ricci curvature of Narasimhan-Simha Hermitian metric on direct image of relative line bundle $f_*(K_{X/Y})$.

See : Shigeharu TAKAYAMA Singularities of Narasimhan-Simha type metrics on direct images of relative pluricanonical bundles Tome 66, n o 2 (2016), p. 753-783.