On the connectedness of the singular set of holomorphic foliations

2025/06/10 by Omegar Calvo-Andrade, Calvo-Andrade, Omegar, Maurício Corrêa +5
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2506.08942

openalex publication_date 2025/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a singular holomorphic foliation of dimension k>1 on a projective n-manifold X. Assume that the determinant of the normal sheaf of F is ample (as is always the case when X=ℙⁿ), and that the singular set Sing(F) has dimension ≤ k-1. We show that the union of those irreducible components of Sing(F) of dimension exactly k-1 is necessarily connected. Consequently, we obtain a Bott-type topological obstruction to the integrability of singular holomorphic distributions, echoing Bott's vanishing theorem, and we answer a question of Cerveau for codimension-one foliations on ℙ³.

Citations

• On functoriality of Baum-Bott residues
• Classification of the invariants of foliations by curves of low degree on the three-dimensional projective space
• Codimension one distributions and stable rank 2 reflexive sheaves on threefolds

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