Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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D. Angella - A. Otal German - L. Ugarte - R. Villacampa

On Gauduchon connections with K\"ahler-like curvature

created by daniele on 15 Jan 2019
modified on 19 Mar 2023


Published Paper

Inserted: 15 jan 2019
Last Updated: 19 mar 2023

Journal: Commun. Anal. Geom.
Volume: 30
Number: 5
Pages: 961--1006
Year: 2022
Doi: 10.4310/CAG.2022.v30.n5.a2

ArXiv: 1809.02632 PDF


We study Hermitian metrics with a Gauduchon connection being "Kähler-like", namely, satisfying the same symmetries for curvature as the Levi Civita and Chern connections. In particular, we investigate 6-dimensional solvmanifolds with invariant complex structures with trivial canonical bundle and with invariant Hermitian metrics. The results for this case give evidence for two conjectures that are expected to hold in more generality: first, if the Bismut connection is Kähler-like, then the metric is pluriclosed; second, if another Gauduchon connection, different from Chern or Bismut, is Kähler-like, then the metric is Kähler. As a further motivation, we show that the Kähler-like condition for the Levi Civita connection assures that the Ricci flow preserves Hermitianity along analytic solutions.

Tags: SIR2014-AnHyC

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