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

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.