Log Baum--Bott Residues for foliations by curves

Abstract

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 $(X, \mathcal{F}, D)$, where $\mathcal{F}$ 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 Poincar\'e'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.