Algebraic integrability of foliations with numerically trivial canonical bundle

TL;DR

This paper proves a flatness criterion for certain sheaves, leading to algebraic leaves in stable foliations with trivial canonical bundle, and establishes a Beauville-Bogomolov decomposition for minimal models.

Contribution

It introduces a flatness criterion for reflexive sheaves under stability conditions, advancing the understanding of foliations with trivial canonical class.

Findings

Establishes algebraicity of leaves for stable foliations with trivial canonical bundle.

Proves a flatness criterion for reflexive sheaves on mildly singular varieties.

Supports the Beauville-Bogomolov decomposition for minimal models.

Abstract

Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent work of Druel and Greb-Guenancia-Kebekus this establishes the Beauville-Bogomolov decomposition for minimal models with trivial canonical class.