Algebraicity of foliations on complex projective manifolds, applications

TL;DR

This paper establishes criteria for when foliations on complex projective manifolds are algebraic, explores their properties using pseudoeffectivity, and applies these results to various conjectures and classifications in complex geometry.

Contribution

It provides a new algebraicity criterion for foliations and applies it to study pseudoeffectivity, stability, and decomposition of complex projective manifolds.

Findings

Algebraicity criterion for foliations proved.

Pseudoeffectivity linked to stability and decomposition.

Applications to Shafarevich-Viehweg conjecture and manifold classification.

Abstract

Contents 1. Algebraicity criterion: statement 2. Proof of the algebraicity criterion. 3. Pseudoeffectivity and movable classes. 4. Harder-Narasimhan filtrations and pseudo-effectivity. 5. Pseudo-effectivity of relative canonical bundles. 6. Rational curves and non-pseudoeffectivity of the canonical/cotangent bundles. 7. Birational stability of the cotangent bundle 8. Shafarevich-Viehweg conjecture. 9. Numerically trivial foliations. 10. The Beauville-Bogomolov-Yau decomposition 11. Some questions 11.1. Pseudoeffectivity of determinants. 11.2. Variations on Abundance. 11.3. Special manifolds. References