Characterization of generic projective space bundles and algebraicity of foliations

TL;DR

This paper investigates positivity notions for distributions on complex projective manifolds, characterizes projective space bundles, and explores algebraicity of foliation leaves, establishing bounds and classifications based on positivity invariants.

Contribution

It introduces new characterizations of projective space bundles via positivity and provides bounds and classifications for foliations' algebraic leaves based on positivity measures.

Findings

Characterization of projective space bundles using movable curve classes

Lower bounds for algebraic rank of foliations based on positivity

Classification of foliations attaining or exceeding the algebraic rank bound

Abstract

In this paper we consider various notions of positivity for distributions on complex projective manifolds. We start by analyzing distributions having big slope with respect to curve classes, obtaining characterizations of generic projective space bundles in terms of movable curve classes. We then apply this result to investigate algebraicity of leaves of foliations, providing a lower bound for the algebraic rank of a foliation in terms of invariants measuring positivity. We classify foliations attaining this bound, and describe those whose algebraic rank slightly exceeds this bound.