The Poincar\'e Problem for a foliated surface

TL;DR

This paper introduces new birational invariants for foliations on surfaces, establishes inequalities and criteria for their classification, and explores bounds on their volume, advancing understanding of foliations' geometric properties.

Contribution

It defines three birational Chern number invariants for foliations, derives inequalities for foliations of general type, and provides criteria for transcendentality and volume bounds.

Findings

Chern numbers vanish for non-general type foliations unless induced by a genus 1 fibration.

Slope inequality established for algebraically integral foliations of general type.

Noether type inequalities proved, leading to criteria for transcendental foliations and volume bounds.

Abstract

Let $\mathcal F$ be a foliation on a smooth projective surface $S$ over the complex number $\mathbb{C}$. We introduce three birational non-negative invariants $c_1^2(\mathcal F)$, $c_2(\mathcal F)$ and $\chi(\mathcal F)$, called the Chern numbers. If the foliation $\mathcal F$ is not of general type, the first Chern number $c_1^2(\mathcal F)=0$, and $c_2(\mathcal F)=\chi(\mathcal F)=0$ except when $\mathcal F$ is induced by a non-isotrivial fibration of genus $g=1$. If $\mathcal F$ is of general type, we obtain a slope inequality when $\mathcal F$ is algebraically integral. As a corollary, $\mathcal F$ is always transcendental if the slope is less than $2$. On the other hand, we also prove three sharp Noether type inequalities if $\mathcal F$ is of general type. As applications, we obtain a criterion for foliations to be transcendental using Noether type inequalities, and we also give a partial positive answer to the question on the lower bound on the volume of a foliation of general type.