On the connectedness of the singular set of holomorphic foliations

TL;DR

The paper proves a Bott-type connectedness result for the $(k-1)$-dimensional part of the singular set of a singular holomorphic foliation of dimension $k>1$ under the ampleness of the determinant of the normal bundle. Employing Baum–Bott residues, it localizes the top Chern class $c_1(N_{\mathscr F})^{\,n-k+1}$ to the $(k-1)$-dimensional singular components and deduces that all such components must lie in a single connected piece. This yields a topological obstruction to integrability for singular foliations, and in particular settles a question of Cerveau for codimension-one foliations on $\mathbb{P}^3$, relating the obstruction to the count of isolated singularities and to properties of the curve $C$ of 1-dimensional singularities. The paper also provides corollaries clarifying the relationship between singularity counts and cohomology in the $\mathbb{P}^3$ setting and presents examples showing the sharpness of the hypotheses, along with broader remarks on non-projective cases and tangent-sheaf phenomena.

Abstract

Let $\mathcal{F}$ be a singular holomorphic foliation of dimension $k>1$ on a projective $n$-manifold $X$. Assume that the determinant of the normal sheaf of $\mathcal{F}$ is ample (as is always the case when $X=\mathbb{P}^{n}$), and that the singular set $Sing(\mathcal{F})$ has dimension $\leq k-1$. We show that the union of those irreducible components of $Sing(\mathcal{F})$ 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 $\mathbb{P}^{3}$.

Theorems & Definitions (10)

• Theorem 1
• Corollary 1
• Corollary 2
• Example 4.1
• Example 4.2
• Example 4.3
• Remark 4.4
• Remark 4.5
• Remark 4.6
• Remark 4.7