**Motivation**: Currents with mild singularities seems to have vanishing Lelong number, so the following question seems to be natural. 


Definition(**Minimal singularities**):Let $T$ be a closed positive $(1,1)$ current on projective variety $X$. $T$ is said to be of
minimal singularities, if for every closed positive $(1,1)$-current $T\_0$
with $[T\_0]=[T]$, there exists a $L^1$-function $φ$ such that


$$T'=T+\sqrt{-1}\partial\bar\partial\phi$$
and is bounded from above


Definition:(**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


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



> 
> I am wondering to know minimal singularities have vanishing Lelong
>  number?
> 
> 
> 



A question about minimal singularities
No, in any $(1,1)$ cohomology class which is big but not nef, any closed positive $(1,1)$ current with minimal singularities has nontrivial Lelong number somewhere, by Boucksom (Propositions 3.1 and 3.8 [here](https://arxiv.org/abs/math/0204336)).


For an explicit example, take $X$ a projective algebraic surface, $L$ an ample line bundle on $X$, let $\mu:\tilde{X}\to X$ be the blowup at a point, with exceptional divisor $E$, and take your class to be the first Chern class of $\mu^\*L\otimes\mathcal{O}\_{\tilde{X}}(E)$. This class is big but not nef, since its intersection number with $E$ is $-1$, and so $E$ is contained in the non-nef locus of this class (Theorem 6.2 [here](https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/coneduality.pdf)). By the article of Boucksom this shows that every closed positive current in this class must have positive Lelong number at all points of $E$.


