Log Baum--Bott Residues for foliations by curves

Technical Summary: Log Baum–Bott Residues for Foliations by Curves

Problem Statement
The paper addresses the extension of the classical Baum–Bott theory, which relates the characteristic numbers of a manifold to the residues of singularities of holomorphic foliations, to the setting of logarithmic geometry. Specifically, the authors investigate one-dimensional holomorphic foliations (foliations by curves) that are logarithmic along a free divisor $D$ on a complex manifold $X$. The central problem is to define and compute "log Baum–Bott residues" for singularities located on the divisor $D$, and to establish a global residual formula that incorporates these logarithmic indices. Furthermore, the work seeks to apply these results to singular varieties via log resolutions and to explore implications for the Zariski–Lipman conjecture and Poincaré's Problem.

Methodology
The authors employ a combination of local algebraic geometry, sheaf theory, and global topological methods:

1. Logarithmic Vector Fields and Frames: The study utilizes the sheaf of logarithmic vector fields $TX(-\log D)$. For a foliation $\mathcal{F}$ logarithmic along $D$, local representatives are sections of this sheaf. The authors define a specific matrix $M_{\log}(\vartheta)$ associated with a local logarithmic vector field $\vartheta$, constructed using a local frame of $TX(-\log D)$ and the Lie bracket structure of the frame.
2. Residue Definitions: Generalizing the Grothendieck residue, the authors define the Log Baum–Bott residue $\text{Res}^{\log}_{\varphi}(\mathcal{F}, D, p)$ for a homogeneous symmetric polynomial $\varphi$ of degree $n$. This is defined via the Grothendieck residue of $\varphi(M_{\log}(\vartheta))$ with respect to the local coordinates of the vector field.
3. Cohomological Framework: The proofs rely on the relative Čech-de Rham cohomology and the Alexander duality. By constructing a specific covering of the singular set and utilizing the Bott vanishing theorem for connections on the normal bundle of the foliation relative to the logarithmic tangent bundle, the authors relate global characteristic classes to local residues.
4. Resolution of Singularities: To handle singular varieties, the authors utilize functorial log resolutions $\pi: (X, D) \to (Y, \emptyset)$. They lift the foliation to the resolution and apply the logarithmic Baum–Bott theorem on the smooth manifold $X$ to derive formulas for the singular variety $Y$.
5. Index Comparisons: The paper establishes relationships between the new logarithmic residues and existing indices, including the Aleksandrov logarithmic index, the GSV-index (Gómez-Mont, Seade, Verjovsky), and the Camacho–Sad index, particularly in the context of surfaces.

Key Contributions and Results

• The Log Baum–Bott Theorem (Theorem 1.2): The primary result is a residual formula for a one-dimensional holomorphic foliation $\mathcal{F}$ on a compact complex manifold $X$ logarithmic along a free divisor $D$. For a homogeneous symmetric polynomial $\varphi$ of degree $n = \dim X$:
  $$\int_X \varphi(TX(-\log D) - T\mathcal{F}) = \sum_{S_\lambda \not\subset D} BB_\varphi(\mathcal{F}, S_\lambda) + \sum_{S_\lambda \subset D} \text{Res}^{\log}_\varphi(\mathcal{F}, D, S_\lambda)$$
  where the sum runs over connected components of the singular set. This generalizes the classical theorem to the logarithmic setting and non-compact $\partial$-manifolds.

• Generalization of Aleksandrov's Index: The authors demonstrate that for isolated singularities on a free divisor, the log Baum–Bott residue generalizes the Aleksandrov logarithmic index. Specifically, when $\varphi = \det$, the residue coincides with the logarithmic index $\text{Log}(\mathcal{F}, D, p)$.

• Relations on Surfaces (Corollary 1.6): For foliations on compact complex surfaces, the difference between the classical Baum–Bott index and the log Baum–Bott index along an invariant curve $D$ is explicitly expressed in terms of the GSV-index and the Camacho–Sad (CS) index:
  $$BB_{c_1^2}(\mathcal{F}, D) - \text{Res}^{\log}_{c_1^2}(\mathcal{F}, D) = 2\text{GSV}(\mathcal{F}, D) + \text{CS}(\mathcal{F}, D)$$
  The paper notes that if $\mathcal{F}$ is a "generalized curve" along $D$ (no saddle-nodes in resolution), the log residue vanishes.

• Application to Poincaré's Problem (Corollary 1.5): The authors derive a new obstruction for Poincaré's Problem on $\mathbb{P}^n$ (for odd $n$). If a foliation has an invariant free divisor $D$ and the sum of log residues is non-negative, then $\deg(D) \leq \deg(\mathcal{F}) + n$.

• Singular Varieties and Log Resolutions (Corollary 1.7): A Baum–Bott type formula is obtained for singular normal varieties via log resolutions. The global characteristic numbers of the singular variety are expressed as a sum of classical Baum–Bott residues on the regular part and log residues on the exceptional divisor.

• Zariski–Lipman Conjecture and Saddle-Nodes (Corollary 1.9): The paper proves a weak global version of the Zariski–Lipman conjecture for compact algebraic surfaces. It establishes a "foliated smoothness criterion": If $Y$ is a compact analytic surface with a locally free tangent sheaf and a foliation induced by a twisted vector field whose singularities lie only in $\text{Sing}(Y)$, then if the foliation is a "generalized curve," $Y$ must be smooth. Conversely, if $Y$ is singular, the reduction of singularities of such a foliation must necessarily exhibit at least one saddle-node. This suggests a dynamical manifestation of the Zariski–Lipman condition.

• Constructible Functions (Section 6): The authors organize the logarithmic residues into a natural residue-valued constructible function, providing a framework that captures complex-valued invariants (like Camacho–Sad indices) which are lost in purely intersection-theoretic approaches relying on characteristic cycles with integer multiplicities.

Significance and Claims
The paper claims to provide a unified framework for studying foliations with singularities on free divisors, bridging the gap between local logarithmic indices and global topological invariants. By extending the Baum–Bott theorem to logarithmic settings and singular varieties, the authors offer new tools for analyzing the topology of foliations on non-smooth spaces.

The significance of the work lies in:

1. Generalization: Extending the Baum–Bott theorem to the logarithmic context and singular varieties, allowing for the inclusion of singularities more general than isolated complete intersection singularities (ICIS).
2. Unification: Connecting various indices (Aleksandrov, GSV, Camacho–Sad, Baum–Bott) through a single residue formalism.
3. Dynamical Constraints: Providing a novel perspective on the Zariski–Lipman conjecture by linking the algebraic condition of a locally free tangent sheaf to the dynamical necessity of saddle-nodes in the resolution of singularities.
4. Obstructions: Offering new quantitative obstructions (via log residues) to the existence of algebraic solutions for polynomial differential equations (Poincaré's Problem).

The authors maintain a modest tone regarding the Zariski–Lipman results, framing them as a "weak global version" and a "foliated smoothness criterion" that highlights the appearance of saddle-nodes in singular settings, rather than a full proof of the conjecture in all generality.