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

Abstract

We investigate slicing properties of $m$-subharmonic functions in product domains $\Omega = \Omega' \times \Omega'' \subset \mathbb{C}^n = \mathbb{C}^p \times \mathbb{C}^{n-p}$, where $p, m, n$ are integers satisfying $1 \leq p \leq m-1 < n-1$.\\ Given an $m$-subharmonic function $v$ on $\Omega$, we prove the existence of a pluripolar subset $E \subset \Omega'$ such that, for every $x' \in \Omega' \smallsetminus E$, the slice $v_{|\{x'\}\times \mathbb{C}^{n-p}}$ is well defined and $(m - q_{m,p})$-subharmonic on $\Omega''$, 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 $\Omega'$, the $m$-Lelong number of $v$ at $(x', x'')$ coincides, up to a multiplicative constant, with the $(m - q_{m,p})$-Lelong number of the slice $v_{|\{x'\}\times \Omega''}$ at $x''$.