# Some Hermitian problems in complex non-K\"ahler geometry

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

created by daniele on 25 Feb 2020

12 jul 2019
-- 11:00

Hradec Kralove

**Abstract.**

We consider several problems concerning existence of ``special'' Hermitian metrics on complex manifolds, moving our attention from the K\"ahler to the non-K\"ahler setting.

On the one side, we focus on the role of the {\em Chern connection} when looking for Hermitian metrics with special curvature properties.
Information on its curvature is encoded in the first Bott-Chern class in {\em Bott-Chern cohomology}.
The {\em 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 {\em Chern-Einstein metrics}. Note the plural ``notions'', due to the lack of {\em symmetries of the curvature tensor} of the Chern connection.

On the other side, the problem of existence of special metrics ({\itshape e.g.} balanced metrics in the sense of Michelsohn) under cohomological assumptions ({\itshape 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.