Mihai Paun, Positivity of quotients of holomorphic tensor fields (Lecture I)

I will present the main results and techniques in the preprint arxiv:2502.02183 (joint with J. Cao). This was motivated by the recent preprint by W. Ou, cf. arXiv:2501.1808 (especially by the algebraicity criteria for foliations on compact Kähler manifolds in this article).

The first lecture is dedicated to the proof of the aforementioned algebraicity criteria. Two main techniques are coming into the picture here. One is the usual substitute for the absence of ample line bundles on compact Kähler manifolds, i.e. the Monge-Ampère equation (combined with the mass concentration method). The second one concerns basic properties of Lelong numbers of closed positive currents.

In the second lecture I will survey a few results concerning the positivity of direct images, and the finally I will explain the proof of the main results (i.e. the complete generalisation to the Kähler case of the results obtained in collaboration with F. Campana).