}}f
\cdot \eta^k$. The latter is the real jacobian of $f$, it vanishes on the critical set $\CC$ of $f$.\
The measure $\mu$ can also be defined *via* pluripotential theory: we have $\mu = T^k$, where $T$ is the Green current of $f$. The latter is a closed positive $(1,1)$ current on ${\mathbb {P}}^k$ with Hölder potentials. In particular, for any algebraic subset $A \subset {\mathbb {P}}^k$, there exist $c,{\gamma}> 0$ such that the $r$-neighbourhood of $A$ satisfies $\mu ( A[r] )
\leq c \, r^{\gamma}$ for any $r >
0$ (see [@DS4], Prop. 2.3.7). For any $\delta > 0$ and $\tilde c > 0$, we set $c_\delta:= (1-d^{-\delta})^{-1}$ and $\tilde c_\delta:= \tilde c (1-d^{-\delta})^{-1}$. In the sequel, $c > 0$ is a constant independent of $n$, it may differ from a line to another.

The class $\UU$ {#cccd}
---------------

Let us recall the definition of the class $\UU$ (see subsection \[byby\]).

\[unb\] Let $\UU$ be the set of functions $\psi : {\mathbb {P}}^k \to {\mathbb {R}}\cup \{ -
\infty \}$ satisfying:

- $e^\psi$ is $h$-Hölder on ${\mathbb {P}}^k$ for some $h >0$,

- $\NN_\psi:= \{ \psi = -\infty \}$ is a (possibly empty) proper algebraic set of ${\mathbb {P}}^k$,

