Excess logarithmic residues for foliations by curves and applications

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 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.

• The paper introduces excess logarithmic residues that quantify the gap between logarithmic and ordinary normal sheaf invariants in one-dimensional foliations.
• It establishes global residue formulas linking explicit local Grothendieck residue expressions to Chern number integrals, reinforcing classical Baum–Bott results.
• Applications include effective Poincaré-type bounds for invariant hypersurfaces and a novel dynamical method for computing log discrepancies in singular surface resolutions.

Excess Logarithmic Residues for Foliations by Curves and Their Applications

Introduction and Motivation

The paper "Excess logarithmic residues for foliations by curves and applications" (2604.25395) develops a robust residue theory for one-dimensional holomorphic foliations logarithmic along a divisor. The principal innovation is the introduction of "excess logarithmic residues", defined through the comparison between the logarithmic normal sheaf and the ordinary normal sheaf for such foliations. These invariants provide a quantification of the deviation between the corresponding Baum--Bott residues—measuring the difference in the contributions from singularities when the foliation is analyzed in the logarithmic versus the non-logarithmic setup.

Global logarithmic residue formulas are proved, packaging Chern numbers as sums of explicit local residues. The new local invariants admit computable Grothendieck residue expressions, granting a powerful toolkit for both local and global study. The authors apply this theory to derive effective bounds—most notably, explicit Poincaré-type bounds for the degree of invariant hypersurfaces of logarithmic foliations on projective spaces based on non-negativity of the excess logarithmic residues. In a birational context, the framework gives a new dynamic interpretation and computation of log discrepancies in terms of componentwise logarithmic residues along the exceptional divisor in the minimal resolution of normal $\mathbb Q$-Gorenstein surfaces.

Logarithmic Foliations and Excess Residues

A holomorphic foliation $F$ by curves on a complex manifold $X$ tangent to a reduced divisor $D$ leads to two natural transverse structures: the ordinary normal sheaf $N_F = T_X/T_F$ and the logarithmic normal sheaf $N_{F/X}^{\log} = T_X(-\log D) / T_F$. There is a canonical sheaf morphism $N_{F/X}^{\log} \to N_F$, and the excess residue at a point $p \in D$ is defined to reflect the defect between the Chern--Weil representatives corresponding to these two sheaves.

Locally, the logarithmic residue admits a Grothendieck-type formula in adapted coordinates: $$\widehat{\operatorname{Res}^{\log}_{c_1^{n-i},\,D^i}}(F,D,p) = \operatorname{Res}_p \left[ \dfrac{ (\operatorname{tr} J_D(v))^{n-i} \left( v(f)/f|_D \right)^{i-1} \, dz_1 \wedge \cdots \wedge dz_{n-1} } {\bar a_1, \dots, \bar a_{n-1}} \right]_D,$$ where $J_D(v)$ is the Jacobian of the vector field restricted to $F$0, and the numerator encodes explicit terms reflecting the tangential and logarithmic behavior. The "variational residue" is precisely the difference of the ordinary and logarithmic local residues and is measured by a transgression form.

Main Theorems: Global Residue Formulas

The central result is the global residue theorem for logarithmic foliations with isolated singularities. For each integer $F$1 with $F$2, let $F$3 denote the finite set of isolated zeroes of the foliation induced on $F$4.

$F$5

with the analogous variational identity,

$F$6

For $F$7, this recovers the global logarithmic Baum--Bott theorem; for $F$8, these identities encapsulate how the geometry of $F$9 constrained by $X$0 influences topological invariants of the pair $X$1. The proof leverages Chern--Weil theory, the Bott partial connection, Stokes' theorem, and precise control of boundary contributions.

Applications to the Poincaré Problem

A significant application is a Poincaré-type bound for the degree of invariant hypersurfaces. Given a one-dimensional foliation $X$2 of degree $X$3 on $X$4, and an invariant hypersurface $X$5 of degree $X$6 logarithmic for $X$7, non-negativity of the total logarithmic residues yields: $X$8 This criterion is a strengthening of classical results by Soares, Carnicer, Campillo--Carnicer--de la Fuente Garcia, and Brunella--Mendes. In the case $X$9, the bound recovers and explains Carnicer's result for the degree of nondicritical separatrices via explicit residue computations. The local algebra finds connections with the work of Aleksandrov on logarithmic differentials and residues, extending the comparison of local indices (GSV, CS, classic and logarithmic) in the analysis of foliation singularities.

For example, in $D$0, if $D$1 is a nondicritical separatrix, then: $D$2 with equality if and only if all GSV indices vanish, i.e., the singularities along $D$3 are generalized curves.

Dynamical Computation of Log Discrepancies

A major theoretical implication is the dynamical computation of log discrepancies for surface singularities. Given a normal $D$4-Gorenstein surface $D$5, with a logarithmic resolution $D$6 extracting exceptional divisors $D$7, and a foliation $D$8 whose singular set is supported in $D$9, the componentwise logarithmic residues along the $N_F = T_X/T_F$0's of the lifted foliation $N_F = T_X/T_F$1 recover the log discrepancy vector: $N_F = T_X/T_F$2 where $N_F = T_X/T_F$3 is the intersection matrix $N_F = T_X/T_F$4 and $N_F = T_X/T_F$5 the vector of exceptional logarithmic residues. Log canonicity of $N_F = T_X/T_F$6 is thus determined by non-negativity of $N_F = T_X/T_F$7. The authors offer explicit computations for cyclic quotient singularities, confirming consistency with classical formulas for discrepancies.

Numerical Evidence and Explicit Computations

The paper contains explicit computations in projective and affine coordinates, demonstrating the practical calculability of the local and global residue invariants. For instance, examples on $N_F = T_X/T_F$8 and $N_F = T_X/T_F$9 are fully worked out, including the illustration of the residue calculation for cyclic quotient singularities, reconciling the residues with the expected values for log discrepancies.

Strong equalities between sums of residues and Chern-number integrals are established. For instance, for $N_{F/X}^{\log} = T_X(-\log D) / T_F$0 in $N_{F/X}^{\log} = T_X(-\log D) / T_F$1 logarithmic along $N_{F/X}^{\log} = T_X(-\log D) / T_F$2: $N_{F/X}^{\log} = T_X(-\log D) / T_F$3 matching the theoretical prediction.

Theoretical and Practical Implications

The introduction of excess logarithmic residues brings a new layer of refinement to the intersection theory and local analysis of foliations, offering sharper invariants for situations involving tangency with divisors. These tools provide dynamical and cohomological criteria for effectivity problems (such as the Poincaré problem), finiteness of invariant sets, and the birational geometry of log pairs.

In the context of algebraic geometry, the dynamical characterization of log canonicity has implications for the classification of singularities and may influence approaches to the Minimal Model Program in the logarithmic setting. The residue framework could potentially yield computational criteria for singularities in higher-dimensional spaces, possibly affecting the study of moduli of foliations and the birational geometry of varieties with foliated structures.

Conclusion

This work establishes a rigorous theory of excess logarithmic residues for one-dimensional holomorphic foliations logarithmic along divisors and demonstrates their substantial applicability in both local and global settings. The systematic comparison and explicit formulas for residues enrich the classical residue theory, contribute to effective bounds in the Poincaré problem, and encode dynamical information about singularities, including a novel approach to log discrepancies. The results open new perspectives for the investigation of foliated spaces with singularities and for the interplay between local dynamical invariants and global geometric structures.