Singularities of logarithmic foliations

Abstract

A logarithmic 1-form on $\mathbb C\mathbb P^n$ can be written as

\omega=\biggl(\prod_0^m F_j\biggr)\sum_0^m \lambda_i\frac{dF_i}{F_i}=\lambda_0 \widehat F_0 \,dF_0+\cdots+\lambda_m \widehat F_m \,dF_m

with $\widehat F_i=(\prod_0^m F_j)/F_i$ for some homogeneous polynomials F_i of degree d_i and constants $\lambda_i\in{\mathbb C}^\star$ such that $\sum\lambda_id_i=0$. For general $F_i,\lambda_i$, the singularities of $\omega$ consist of a schematic union of the codimension 2 subvarieties F_i = F_j = 0 together with, possibly, finitely many isolated points. This is the case when all F_i are smooth and in general position. In this situation, we give a formula which prescribes the number of isolated singularities.