On the connectedness of the singular set of holomorphic foliations

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 $\le k-1$. 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 ${\mathbb{P}}^3$.

Improved exposition. Dedicated to Professor Nigel Hitchin on the occasion of his 80th Birthday