Let $M$ be a compact Kahler Calabi-Yau variety which admit Ricci flat metric $\tilde\omega$, $Ric(\tilde \omega)=0$, then the Lelong number $\tilde \omega$ is zero?


In general if $\omega$ satisfies in the Kahler-Einstein equation $$Ric(\omega)=\lambda\omega$$ then is the Lelong number of $\omega$ vanishes?



Lelong number of Ricci flat metric
***Motto***: Canonical metric has vanishing Lelong number or (some times called Lelong-Demailly number)


Let me start with the definition of Lelong number.


Let
$W\subset \mathbb C^n$
be a domain, and $\Theta$ a positive current of degree $(q,q)$ on
$W$. For a point $p\in W$
one defines
$$\mathfrak v(\Theta,p,r)=\frac{1}{r^{2(n-q)}}\int\_{|z-p|<r}\Theta(z)\wedge (dd^c|z|^2)^{n-q}$$
The
Lelong number
of $\Theta$ at
$p$
is defined as


$$\mathfrak v(\Theta,p)=\lim\_{r \to 0}\mathfrak v(\Theta,p,r)$$


Let $\Theta$ be the curvature of singular hermitian metric $h=e^{-u}$, one has


$$\mathfrak v(\Theta,p)=\sup\{\lambda\geq 0: u\leq \lambda\log(|z-p|^2)+O(1)\}$$


By using Skoda-Lelong theory ,


If $X$ be a smooth projective variety and let $D$
be a nef and big $\mathbb R$-divisor on $X$.
Then the first Chern class $c\_1(D)$ can be represented
by a closed positive (1,1)-current
$T$
with
$\mathfrak v(T)=0$ 


If there exists a modification
$f:Y\to X$ such that there exists a Zariski decomposition
$f^\*L=P+N$
of
$f^\*L$
on
$Y$.
Then
there
exists
a
closed
positive
$(1,
1)$
current
$S$
such
that
$c\_1(P)=[S]$
and
$\mathfrak v(S)=0$


Since $c\_1(X)=[Ric(\omega)]$ and $Ric(\omega)=\lambda\omega$ hence canonical metric has vanishing Lelong number


Moreover fibrewise Ricci flat metric and Weil-Petersson metric have $\omega\_{SF}$ and $\omega\_{WP}$ have zero Lelong number and hence when Kodaira dimension is positive then along $\pi:X\to X\_{can}=\text{Proj }R(X,K\_X)$
the canonical metric $\omega\_{can}\in [-K\_{X\_{can}}+\pi\_\*(K\_{X/X\_{can}})]$ which satisfies in $$Ric(\omega\_{can})=-\omega+\omega\_{WP}$$ has vanishing Lelong number(here we used of this fact that direct image of relative line bundle plus anti-canonical bundle of base is nef and big)


