Excess logarithmic residues for foliations by curves and applications

We introduce excess logarithmic residues for one-dimensional holomorphic foliations tangent to a divisor. They arise from the comparison between the logarithmic normal sheaf and the ordinary normal sheaf of the foliation, and measure the local variation between the logarithmic and classical Baum–Bott contributions. We prove a global residue formula expressing the corresponding Chern numbers as sums of local residues. We then derive a Poincaré-type bound for invariant hypersurfaces from the non-negativity of the relevant logarithmic residues. Finally, for a normal $\mathbb{Q}$-Gorenstein surface $Y$, we show that the componentwise logarithmic residues of a lifted foliation along the exceptional divisor of a functorial resolution recover the log discrepancies of the singularities of $Y$, giving a dynamical and foliated test for log canonicity of these singularities.