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

In this work we introduce a GSV type index for varieties invariant by holomorphic Pfaff systems (possibly non locally decomposables) on projective manifolds. We prove a non-negativity property for the index. As an application, we prove that the nonnegativity of the GSV-index gives us an obstruction to the solution of the Poincaré problem for Pfaff systems on projectives spaces.

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