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

Abstract

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

17 pages, 2 figures, 1 table