On the structure of codimension 1 foliations with pseudoeffective conormal bundle

TL;DR

This paper studies codimension 1 holomorphic foliations on projective manifolds with pseudoeffective conormal bundles, showing they are pull-backs of canonical foliations on Hilbert modular varieties, especially in the non abundant case.

Contribution

It characterizes the structure of such foliations, revealing they originate from Hilbert modular varieties, extending previous results to logarithmic foliated pairs.

Findings

Foliations with pseudoeffective conormal bundles are integrable.

Such foliations are pull-backs of canonical foliations on Hilbert modular varieties.

The results apply to logarithmic foliated pairs.

Abstract

Let $X$ a projective manifold equipped with a codimension $1$ (maybe singular) distribution whose conormal sheaf is assumed to be pseudoeffective. By a theorem of Jean-Pierre Demailly, this distribution is actually integrable and thus defines a codimension $1$ holomorphic foliation $\F$. We aim at describing the structure of such a foliation, especially in the non abundant case: It turns out that $\F$ is the pull-back of one of the "canonical foliations" on a Hilbert modular variety. This result remains valid for "logarithmic foliated pairs".