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### Siu decomposition

Let $T$ be a closed positive current of bidegree $(p,p)$ on a complex $n$-fold $X$. We denote by $\nu(T,x)$ its Lelong number at a point $x\in X$. The Lelong super-level sets are defined by $E_c(T):=\{x\in
X,\nu(T,x)\geq c\}$ for $c>0$, and a well known result of Y.T.Siu \[Siu74\] asserts that $E_c(T)$ is an analytic subset of $X$, of codimension at least $p$. As a consequence, for any analytic subset $Y$ of $X$, the generic Lelong number of $T$ along $Y$, defined by $$\nu(T,Y):=\inf\{\nu(T,x),x\in Y\},$$ is also equal to $\nu(T,x)$ for a very general $x\in Y$. It is also true that, for any irreducible analytic subset $Y$ of codimension $p$ in $X$, the current\
$T-\nu(T,Y)[Y]$ is positive. The symbol $[Y]$ denotes the integration current on $Y$, which is defined by integrating test forms on the smooth locus of $Y$. Since $E_+(T):=\cup_{c>0}E_c(T)$ is a countable union of $p$-codimensional analytic subsets, it contains an at most countable family $Y_k$ of $p$-codimensional irreducible analytic subsets. By what we have said, $T-\nu(T,Y_1)[Y_1]-...-\nu(T,Y_k)[Y_k]$ is a positive current for all $k$, thus the series $\sum_{k\geq 0}\nu(T,Y_k)[Y_k]$ converges, and we have $$T=R+\sum_k\nu(T,Y_k)[Y_k]$$ for some closed positive $(p,p)$-current $R$ such that each $E_c(R)$ has codimension $>p$. 