Lelong number of curvature of Kawamata's hermitian metric

Question

Let $X,Y$ are two projective varieties and $f:X\to Y$ is an Iitaka fibration. Consider the following singular hermitian metric $$h(\sigma,\sigma)=\left(\int_{X_y}|\sigma|^{\frac{2}{m!}}\right)^{m!}$$ where $y\in Y$ and $\sigma$ is a section of $$\frac{1}{m!}f_*\mathcal O_X(m!K_{X/Y})|_{X_y}$$

then, the curvature of hermitian metric $h$, i.e., $\sqrt{-1}\Theta_h$ has zero Lelong number? In other words, the Weil-Petersson current on moduli space of Calabi-Yau varieties has zero Lelong number?

Note that direct image of relative line bundle is nef

and if the line bundle $L\to X$ be a nef and big then there exists a hermitian metric on $L$ with vanishing Lelong number. But this question is specific

Lelong number gives a lot of information to us. For example, Let $X$ be a projective manifold and $(L, h)$ a positive , singular hermitian line bundle , whose Lelong numbers vanish everywhere. Then $L$ is nef

Answer 1

By theorem 2 and Poroposition 6, of the paper, Georg Schumacher and Hajime Tsuji, Quasi-projectivity of moduli spaces of polarized varieties, Annals of Mathematics,159(2004), 597–639

the Weil-Petersson metric has zero Lelong number.

http://annals.math.princeton.edu/wp-content/uploads/annals-v159-n2-p03.pdf

So, this tells us that the direct image of relative line bundle is nef

Or Theorem 3.4 http://www.mathematik.uni-marburg.de/~schumac/doubar.pdf