Number of singularities of a foliation on ${\mathbb P}^n$

Abstract

Let D be a one dimensional foliation on a projective space, that is, an invertible subsheaf of the sheaf of sections of the tangent bundle. If the singularities of D are isolated, Baum-Bott formula states how many singularities, counted with multiplicity, appear. The isolated condition is removed here. Let m be the dimension of the singular locus of D. We give an upper bound of the number of singularities of dimension m, counted with multiplicity and degree, that D may have, in terms of the degree of the foliation. We give some examples where this bound is reached. We then generalize this result for a higher dimensional foliation on an arbitrary smooth and projective variety.