Lehmann-Suwa residues of codimension one holomorphic foliations and applications

TL;DR

This paper computes Lehmann-Suwa residues for certain holomorphic foliations on complex manifolds, relates them to Baum-Bott residues, and applies these results to conditions for singularities in Levi-flat hypersurfaces.

Contribution

It provides explicit formulas for Lehmann-Suwa residues and links them to Baum-Bott residues, extending previous results and applying to Levi-flat hypersurface singularities.

Findings

Lehmann-Suwa residues are expressed as multiples of integration currents.

A formula relating Baum-Bott and Lehmann-Suwa residues is established.

Conditions for dicritical singularities in Levi-flat hypersurfaces are derived.

Abstract

Let $\mathcal{F}$ be a singular codimension one holomorphic foliation on a compact complex manifold $X$ of dimension at least three such that its singular set has codimension at least two. In this paper, we determine Lehmann-Suwa residues of $\mathcal{F}$ as multiples of complex numbers by integration currents along irreducible complex subvarieties of $X$. We then prove a formula that determines the Baum-Bott residue of simple almost Liouvillian foliations of codimension one, in terms of Lehmann-Suwa residues, generalizing a result of Marco Brunella. As an application, we give sufficient conditions for the existence of dicritical singularities of a singular real-analytic Levi-flat hypersurface $M\subset X$ tangent to $\mathcal{F}$.