6 sep 2018 -- 14:30
Universidad Nacional de Córdoba (Argentina)
Abstract.
In the tentative to move from the Kähler to the non-Kähler setting, we consider several problems concerning Hermitian metrics on complex manifolds with {\em special} curvature properties and/or characterized by cohomological conditions.
We start by studying an analogue of the Yamabe problem for Hermitian manifolds. More precisely, we prove the existence of Hermitian metrics having {\em constant scalar curvature with respect to the Chern connection} when the expected curvature is non-positive, and we point out the difficulties in the positive curvature case. This problem relates also to several notions of {\em Chern-Einstein metrics}.
We also investigate {\em symmetries of the curvature tensor} of ``canonical'' connections of Hermitian manifolds. In particular, we focus on $6$-dimensional {\em Calabi-Yau solvmanifolds}, namely, with trivial canonical bundle.
We finally provide some considerations in the {\em locally conformally K\"ahler case}, here including the investigation of lcK metrics induced by immersion into Hopf manifolds.
The talk is base on joint works with: Simone Calamai, Cristiano Spotti; Antonio Otal, Raquel Villacampa, Luis Ugarte; Michela Zedda.