2024/11/06 by Maurício Corrêa, Corrêa, Maurício, Fernando Lourenço +3
Environmental Science · #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Forest ecology and management
paper · pdf · doi:10.48550/arxiv.2411.04015
openalex publication_date 2024/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a Baum--Bott type residual formula for one-dimensional holomorphic foliations, and logarithmic along free divisors. More precisely, this provides a Baum--Bott theorem for a foliated triple (X, F, D), where F is a foliation by curves and D is a free divisor on a complex manifold X. From the local point of view, we show that the log Baum--Bott residues are a generalization of the Aleksandrov logarithmic index for vector fields with isolated singularities on hypersurfaces. We also show how these new indices are related to Poincaré's Problem for foliations by curves. In the case of foliated surfaces, we show that the differences between the logarithmic residues and Baum--Bott indices along invariant curves can be expressed in terms of the GSV and Camacho--Sad indices. We also obtain a Baum--Bott type formula for singular varieties via log resolutions. Finally, we prove a weak global version of the Zariski--Lipman conjecture for compact algebraic surfaces, in the form of a foliated smoothness criterion, suggesting the appearance of saddle-nodes in the singularity reduction on singular surfaces.