Lehmann–Suwa residues of codimension one holomorphic foliations and applications

Let $\mathscr{F}$ 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 $\mathscr{F}$ 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 $\mathscr{F}$.

Keywords

residues formula, holomorphic foliations, Levi-flat hypersurfaces

2010 Mathematics Subject Classification

32S65, 32V40

Full Text (PDF format)

Received 9 March 2019

Accepted 10 December 2019

Published 18 February 2021