Analytic structure of q-pseudoconcave subsets of continuous graphs

2026/03/31 by Filippo Valnegri
Mathematics · #math.CV #msc:32F10 #msc:32U05 #msc:32V40

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arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

The aim of this article is to find under which conditions there exists a foliation by n-dimensional complex manifolds on a closed subset Z⊂ℂ^N, which is locally a continuous graph over a closed subset of ℂⁿ×ℝ. We prove that such foliation does exist if Z is q-pseudoconcave, for any q≥ n. We also prove that this bound on the index of pseudoconcavity is sharp. Namely, we construct many q-pseudoconcave subsets of ℂ^N, for q<n, which are smooth graphs over closed subsets of ℂⁿ×ℝ but with no analytic structure. As an application, we show that an n-pseudoconcave set sitting in a subset of ℂ^N, which is locally a differentiable graph over a domain of ℝ²ⁿ⁺¹, is foliated by n-dimensional complex manifolds.

Citations

• Compact pseudoconcave sets
• The core of the Levi distribution
• New inverse and implicit function theorems for differentiable maps with isolated critical points
• Foliations of continuous q-pseudoconcave graphs
• Plurisubharmonically separable complex manifolds
• Weakly complete complex surfaces