# Hermitian geometry of complex manifolds

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

created by daniele on 28 Jan 2020

27 jan 2020
-- 14:15

Uniwersytet Jagielloński, Kraków

**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 that can be translated and attacked as analytic pdes.

In this context, we study 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}. The plural here refers to the lack of {\em symmetries of the curvature tensor} of ``canonical'' connections of Hermitian manifolds.
To the broad subject, the Chern-Ricci flow plays an useful role: the problem of uniform {\em convergence of the normalized Chern-Ricci flow} can be dealt with on Inoue-Bombieri surfaces with Gauduchon metrics.

The talk is based on joint collaborations with: Simone Calamai, Antonio Otal, Cristiano Spotti, Valentino Tosatti, Luis Ugarte, Raquel Villacampa.