Remarks on Relative Canonical Bundles and Algebraicity Criteria for Foliations in Kähler context

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 Kähler 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.