Abstract: We present some steps in the proof of this celebrated theorem ('60 --- '90 of the last century).

Abstract: We define locally conformally Kähler (lcK) spaces with possible singularities and talk about a few recent results obtained on them, chiefly the existence of a type of Vaisman Theorem about the compatibility of an lcK and a Kähler structure. 

Abstract: Vaisman manifolds are non-Kähler manifolds closely related to projective geometry. When their first Chern class vanishes, they exhibit different behaviour, depending on the sign of a refined characteristic class. I will describe this behaviour in comparison to the Kähler context. In particular, I will discuss the existence of canonical Vaisman metrics, their automorphism group and their small deformations. 

Abstract: TBA

Abstract: Resonance varieties are algebro-geometric objects that emerged from the geometric group theory. They are naturally associated to vector subspaces in second exterior powers and carry natural scheme structures that can be non-reduced. In algebraic geometry, they made an unexpected appearance in connection with syzygies of canonical curves. In this talk, based on a joint work with G. Farkas, C. Raicu and A. Suciu, I report on some recent results concerning the geometry of resonance schemes in the vector bundle case.

Abstract: Kodaira's canonical bundle formula applies to fibrations with general fiber of Kodaira dimension zero. I will present a generalization of this formula to the case when the Kodaira dimension of the general fiber is non-negative, due to Shokurov (part of it conjectural). We show in the end that the moduli part is semipositive, modulo standard conjectures. Based on joint work with Cascini, Shokurov, Spicer.

Abstract: We want to find under which conditions Vaisman theorem is still true for compact Kahler spaces which are locally reducible and globally irreducible. Based on a joint work (in progress) with Miron Stanciu.

Abstract: A central problem in geometry is to classify complex manifolds, at least in low dimension. Uniformization theorem and Enriques-Kodaira classification are wonderful results of the scientific community of the last two centuries. Great results are also known for algebraic threefolds, in terms of the minimal model program. Unknown is still the wide landscape of complex, possibly non-Kähler manifolds. Relying on existence of special metrics can be of help in attacking the problem. 

In this talk, we survey some results obtained with Maurizio Parton and Victor Vuletescu on the classification of compact complex threefolds with locally conformally Kähler metrics, and we collect some related questions.

Vuli contributed significantly to my motivation and understanding of many beautiful pieces of Mathematics: thanks and la mulți ani!