Slices and $m$-Lelong numbers of $m$-subharmonic functions

Abstract

We investigate slicing properties of $m$-subharmonic functions in product domains $Ω= Ω&#39; \times Ω&#39;&#39; \subset \mathbb{C}^n = \mathbb{C}^p \times \mathbb{C}^{n-p}$, where $p, m, n$ are integers satisfying $1 \leq p \leq m-1 < n-1$.\\ Given an $m$-subharmonic function $v$ on $Ω$, we prove the existence of a pluripolar subset $E \subset Ω&#39;$ such that, for every $x&#39; \in Ω&#39; \smallsetminus E$, the slice $v_{|\{x&#39;\}\times \mathbb{C}^{n-p}}$ is well defined and $(m - q_{m,p})$-subharmonic on $Ω&#39;&#39;$, where $q_{m,p}$ denotes the smallest integer greater than or equal to $\frac{mp}{n}$.\\ Moreover, we show that, outside a negligible subset of $Ω&#39;$, the $m$-Lelong number of $v$ at $(x&#39;, x&#39;&#39;)$ coincides, up to a multiplicative constant, with the $(m - q_{m,p})$-Lelong number of the slice $v_{|\{x&#39;\}\times Ω&#39;&#39;}$ at $x&#39;&#39;$.