# Narasimhan-Simha Hermitian metric vs Weil-Petersson metric

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.

• I can't see the PDF, but the WP metric can be defined as the curvature form of the natural metric on that pushforward, so its likely those are the same thing, at least when the family is polarized. (See Tian's article on the smoothness of Kuranishi space for Calabi-Yaus.) – Gunnar Þór Magnússon Mar 21 '16 at 12:45
• There must be some connection between Narasimhan-Simha Hermitian metric and semi-Ricci flat metric. Tsuji posted some conjectures about it. – user21574 Mar 21 '16 at 13:03