Communications in Analysis and Geometry

Volume 30 (2022)

Number 5

On Gauduchon connections with Kähler-like curvature

Pages: 961 – 1006

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n5.a2

Authors

Daniele Angella (Dipartimento di Matematica e Informatica, Università di Firenze, Italy)

Antonio Otal (Centro Universitario de la Defensa, I.U.M.A., Academia General Militar, Zaragoza, Spain)

Luis Ugarte (Departamento de Matemáticas, I.U.M.A., Universidad de Zaragoza, Spain)

Raquel Villacampa (Departamento de Matemáticas, I.U.M.A., Universidad de Zaragoza, Spain)

Abstract

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 Strominger–Bismut connection is Kähler-like, then the metric is pluriclosed; second, if another Gauduchon connection, different from Chern or Strominger–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 the Hermitian condition along analytic solutions.

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The first-named author is supported by Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica”, by Project FIRB “Geometria Differenziale e Teoria Geometrica delle Funzioni”, by project SIR 2014 AnHyC “Analytic aspects in complex and hypercomplex geometry” code RBSI14DYEB, and by GNSAGA of INdAM.

The second-named, third-named, fourth-named authors are supported by the projects MTM2017-85649-P (AEI/FEDER, UE), and E22-17R “ Álgebra y Geometría”.

Received 22 September 2018

Accepted 4 November 2019

Published 17 March 2023