Global residue formula for logarithmic indices of one-dimensional foliations

We prove a global residual formula in terms of logarithmic indices for one-dimensional holomorphic foliations, with isolated singularities, and logarithmic along normal crossings divisors. We also give a formula for the total sum of the logarithmic indices if the singular set of the foliation is contained in the invariant divisor. As an application, we provide a formula for the number of singularities in the complement of the invariant divisor on complex projective spaces. Finally, we obtain a Poincaré-Hopf type formula for singular normal projective varieties.