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 $x''$.
| Comments: | 17 pages, 2 figures, 1 table |
| Subjects: | Complex Variables (math.CV) |
| MSC classes: | 32U05, 31C10, 32U25, 32C30, 32U40 |
| Cite as: | arXiv:2605.25027 [math.CV] |
| (or arXiv:2605.25027v1 [math.CV] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25027 arXiv-issued DOI via DataCite (pending registration) |
From: Noureddine Ghiloufi [view email]
[v1]
Sun, 24 May 2026 12:06:57 UTC (115 KB)
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