Excess logarithmic residues for foliations by curves and applications

TL;DR

The paper introduces excess logarithmic residues for holomorphic foliations, linking local residue calculations to global geometric invariants and singularity measures.

Contribution

It defines excess logarithmic residues, establishes a global residue formula, and connects these residues to log discrepancies and singularity invariants.

Findings

Derived a global residue formula for Chern numbers.

Established a Poincaré-type bound for invariant hypersurfaces.

Connected logarithmic residues to log discrepancies of singularities.

Abstract

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 Poincar\'e-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.