Some remarks on regular foliations with numerically trivial canonical class

TL;DR

This paper investigates regular foliations with numerically trivial canonical class on complex projective manifolds, providing classifications, addressing special cases, and confirming a conjecture in certain scenarios.

Contribution

It offers new classifications of such foliations, applies an algebraicity criterion, and verifies the generalized Bondal conjecture in specific instances.

Findings

Classification of codimension two regular foliations with trivial canonical class

Validation of the algebraicity criterion for leaves of algebraic foliations

Confirmation of the generalized Bondal conjecture in particular cases

Abstract

In this article, we first describe codimension two regular foliations with numerically trivial canonical class on complex projective manifolds whose canonical class is not numerically effective. Building on a recent algebraicity criterion for leaves of algebraic foliations, we then address regular foliations of small rank with numerically trivial canonical class on complex projective manifolds whose canonical class is pseudo-effective. Finally, we confirm the generalized Bondal conjecture formulated by Beauville in some special cases.