Excess logarithmic residues for foliations by curves and applications

1. Problem Statement

The paper addresses the study of one-dimensional holomorphic foliations ($\mathcal{F}$) on a complex manifold $X$ that are tangent to a reduced divisor $D$.

• Context: Classical Baum–Bott theory relates the global Chern classes of the normal bundle of a foliation to local residues at its singularities.
• Gap: When a foliation is tangent to a divisor $D$, the standard normal sheaf $N_{\mathcal{F}} = T_X / T_{\mathcal{F}}$ does not fully capture the geometry relative to $D$. Instead, one must consider the logarithmic normal sheaf $N^{\log}_{\mathcal{F}/X} = T_X(-\log D) / T_{\mathcal{F}}$, where $T_X(-\log D)$ is the sheaf of vector fields tangent to $D$.
• Objective: The authors aim to quantify the "excess" or difference between the classical (ordinary) geometric invariants and the logarithmic ones. They seek to define a residue theory that measures the local variation between these two transverse geometries and apply this to global problems like the Poincaré problem and the classification of surface singularities.

2. Methodology

The authors develop a theory based on Chern–Weil theory adapted to logarithmic structures and Grothendieck residues.

• Sheaf Comparison: They analyze the canonical morphism induced by the inclusion $T_X(-\log D) \hookrightarrow T_X$:
  $$N^{\log}_{\mathcal{F}/X} \longrightarrow N_{\mathcal{F}}$$
  This morphism defines the "excess" term.
• Residue Definitions:
  • Ordinary Residue ($\text{Res}$): Derived from the Chern class of the ordinary normal sheaf $N_{\mathcal{F}}$.
  • Logarithmic Residue ($\widehat{\text{Res}}^{\log}$): Derived from the Chern class of the logarithmic normal sheaf $N^{\log}_{\mathcal{F}/X}$.
  • Variational (Excess) Residue ($\text{Var}$): Defined as the difference:
    $$\text{Var} = \text{Res} - \widehat{\text{Res}}^{\log}$$
    This measures the local defect caused by replacing the ambient tangent bundle with the logarithmic tangent bundle.
• Local Formulas: Using adapted coordinates where $D = \{f=0\}$ and the vector field $v$ satisfies $v(f) = kf$, the authors derive explicit Grothendieck residue formulas.
  • For the logarithmic residue, the formula involves the trace of the Jacobian of the induced vector field on $D$ ($J_D(v)$) and the logarithmic derivative $k = v(f)/f$.
  • The variational residue is expressed via a transgression form involving the difference between the ordinary and logarithmic connection forms.
• Globalization: They utilize Stokes' theorem on the complement of small balls around singularities (links) to relate the sum of local residues to global Chern numbers.

3. Key Contributions

A. Theoretical Framework: Excess Logarithmic Residues

The paper introduces the concept of excess logarithmic residues as a distinct invariant. Unlike previous works that treated logarithmic residues as a direct analogue of classical ones, this work isolates the difference term. This allows for a precise measurement of how the tangency to a divisor alters the topological contribution of singularities.

B. Global Residue Formula (Theorem 1.1 / Theorem 5.2)

The authors prove a fundamental global identity. For a foliation $\mathcal{F}$ with isolated singularities tangent to $D$:
$$\sum_{p \in \text{Sing}(\mathcal{F})} \widehat{\text{Res}}^{\log}_{c_1^n}(\mathcal{F}, D, p) = \int_X c_1(N^{\log}_{\mathcal{F}/X})^n$$
and for the variational term:
$$\sum_{p} \text{Var}_{c_1^n}(\mathcal{F}, D, p) = \int_X \left( c_1(N_{\mathcal{F}})^n - c_1(N^{\log}_{\mathcal{F}/X})^n \right)$$
Similar formulas hold for mixed terms involving powers of the divisor class $[D]$.

C. Application to the Poincaré Problem

The Poincaré problem asks for bounds on the degree $m$ of an invariant hypersurface $D$ in terms of the degree $d$ of the foliation $\mathcal{F}$ on $\mathbb{P}^n$.

• Result: The authors derive a Poincaré-type bound: If the total logarithmic residue is non-negative, then:
  $$m \leq d + n$$
• Significance: This provides a new criterion for the existence of invariant hypersurfaces based on the non-negativity of local logarithmic residues, generalizing previous results by Carnicer, Brunella, and others.

D. Dynamical Test for Log Canonicity (Theorem 8.1)

In dimension two, the authors establish a deep connection between foliation dynamics and singularity theory.

• Setup: Consider a normal $\mathbb{Q}$-Gorenstein surface $Y$ with a functorial log resolution $\pi: X \to Y$ and exceptional divisor $D$. A foliation on $Y$ lifts to a logarithmic foliation on $X$.
• Main Theorem: The vector of exceptional logarithmic residues ($I$) along the exceptional components, when multiplied by the inverse of the exceptional intersection matrix ($M$), recovers the log discrepancy vector ($b = a+1$):
  $$b = -M^{-1} I$$
• Implication: This provides a dynamical and foliated test for log canonicity. A singularity is log canonical if and only if the components of the vector $-M^{-1}I$ are non-negative.

4. Key Results Summary

1. Local Expressions: Explicit Grothendieck residue formulas are derived for $\widehat{\text{Res}}^{\log}$ and $\text{Var}$ in terms of the Jacobian matrix of the vector field restricted to the divisor and the logarithmic derivative $v(f)/f$.
2. Global Identity: The sum of local excess residues equals the integral of the difference of Chern classes of the ordinary and logarithmic normal sheaves.
3. Poincaré Bound: Non-negativity of the total logarithmic residue implies the bound $m \leq d+n$ for invariant hypersurfaces in $\mathbb{P}^n$.
4. Singularity Detection: For surface singularities, the residue vector of a lifted foliation encodes the log discrepancies, allowing one to determine if a singularity is log canonical purely through the dynamics of the lifted foliation.

5. Significance

• Unification: The paper unifies classical Baum–Bott theory, logarithmic residue theory, and the study of singularities of algebraic varieties.
• New Invariants: It introduces "excess" residues as a new tool to study the interaction between foliations and divisors, moving beyond simple tangency conditions.
• Algorithmic Potential: The result in Section 8 offers a potential algorithmic method to compute log discrepancies (and thus test for log canonicity) by analyzing the local behavior of a lifted foliation, bridging complex dynamics and birational geometry.
• Refinement of Poincaré Problem: It refines the understanding of the Poincaré problem by linking the degree bound directly to the positivity of local logarithmic invariants, offering a more nuanced condition than previous global bounds.

In conclusion, this work provides a rigorous analytical framework for understanding how the presence of a divisor modifies the topological invariants of a foliation, with powerful applications ranging from bounding invariant hypersurfaces to characterizing the nature of surface singularities.