Towards classification of codimension 1 foliations on threefolds of general type

TL;DR

This paper classifies certain codimension 1 foliations on threefolds of general type, focusing on those with canonical singularities, non-big canonical class, and non-trivial algebraic parts, advancing understanding of their structure.

Contribution

It provides a classification of codimension 1 foliations with specific singularity and algebraic properties on threefolds of general type, including non-trivial algebraic parts and transcendental cases.

Findings

Classification of foliations with canonical singularities and $ u(K_{})<3$

Description of transcendental foliations with non-big canonical class

Results hold for manifolds of any dimension under certain smoothness assumptions

Abstract

We aim to classify codimension 1 foliations $\mathscr{F}$ with canonical singularities and $\nu(K_{\mathscr{F}}) < 3$ on threefolds of general type. We prove a classification result for foliations satisfying these conditions and having non-trivial algebraic part. We also describe purely transcendental foliations $\mathscr{F}$ with the canonical class $K_{\mathscr{F}}$ being not big on manifolds of general type in any dimension, assuming that $\mathscr{F}$ is non-singular in codimension $2$.