# Some Hermitian problems in complex non-Kähler geometry

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Daniele Angella
(Dipartimento di Matematica e Informatica "Ulisse Dini", Università di Firenze)

created by chrysikos on 06 Jul 2019

12 jul 2019
-- 13:00

University of Hradec Králové, Faculty of Science

**Abstract.**

We consider several problems concerning existence of “special” Hermitian metrics on complex manifolds, moving our attention from the Kähler to the non-Kähler setting. On the one side, we focus on the role of the Chern connection when looking for Hermitian metrics with special curvature properties. Information on its curvature is encoded in the first Bott-Chern class in Bott-Chern cohomology. The Chern-Yamabe problem acts as an analogue of the Yamabe problem for Hermitian manifolds, and concerns Hermitian metrics having constant scalar curvature with respect to the Chern connection in a conformal class. When the expected curvature is non-positive, some results can be shown; and difficulties
arise in the positive curvature case. This problem relates also to several notions of Chern-Einstein metrics. Note the plural “notions”, due to the lack of symmetries of the curvature tensor of the Chern connection. On the other side, the problem of existence of special metrics (e.g. balanced metrics in the sense of Michelsohn) under cohomological assumptions (e.g. $\partial\overline{\partial}$-Lemma) exhibits difficulties when attacked with analytic techniques and pde’s.
The talk is based on and inspired by joint works and collaboration with Simone Calamai, Antonio Otal, Cristiano Spotti, Nicoletta Tardini, Adriano Tomassini, Luis Ugarte, Raquel Villacampa.