Remarks on Relative Canonical Bundles and Algebraicity Criteria for Foliations in K\"ahler context

TL;DR

This paper advances understanding of the positivity of relative canonical bundles in Kähler geometry, proves an algebraicity criterion, and extends uniruledness criteria for foliations, building on recent work by W. Ou.

Contribution

It provides new results on the positivity of relative canonical bundles in Kähler settings, proves Ou's algebraicity criterion, and extends uniruledness criteria for foliations.

Findings

Progress on positivity of relative canonical bundles in Kähler context

Proof of Ou's algebraicity criterion

Extension of uniruledness criterion for foliations

Abstract

In this note, motivated by the recent preprint of W. Ou, we pursue three main objectives. The first is to make progress towards the positivity of the relative canonical bundle in the K\"ahler setting. In the second part, we provide a proof of Ou's algebraicity criterion. Finally, based on the two previous parts, we slightly extend his uniruledness criterion.