
> 
> 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](https://en.wikipedia.org/wiki/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



Lelong number of curvature of Kawamata's hermitian metric
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>


