Paper · Sep 7, 2018

On Gauduchon connections with Kähler-like curvature

Daniele Angella, Antonio Otal, Luis Ugarte, Raquel Villacampa

math.DGarXiv:1809.02632v2
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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 -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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Submitted
Sep 7, 2018
Last revised
Nov 26, 2019 (v2)
Subjects
math.DG, math.CV
Journal
Commun. Anal. Geom. 30 (2022), no. 5, 961--1006
DOI
10.4310/CAG.2022.v30.n5.a2

Comments: to appear in Commun. Anal. Geom

Ricci flowGeneralized & G₂ flowsKähler geometryHomogeneous spaces

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  1. Hermitian metrics on complex non-Kähler manifolds

    Daniele Angella

    Topics Generalized & G₂ flows, Kähler geometry · Same author Daniele Angella · Shared terms “Hermitian metrics”, “Chern connection”, “analytic”

  2. Leafwise flat forms on Inoue-Bombieri surfaces

    Daniele Angella and Valentino Tosatti

    Topics Generalized & G₂ flows · Same author Daniele Angella · Shared terms “Gauduchon”, “Chern”

  3. Geometric formalities along the Chern-Ricci flow

    Daniele Angella and Tommaso Sferruzza

    Topics Generalized & G₂ flows · Same author Daniele Angella · Shared terms “Chern”, “Hermitian”

  4. On the Calabi estimate of geometric flows of Hermitian metrics

    Marco Gallo and Luigi Vezzoni

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “Hermitian metrics”, “Chern”

  5. Griffiths positivity for Bismut curvature and its behaviour along Hermitian Curvature Flows

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    Topics Generalized & G₂ flows, Homogeneous spaces, Kähler geometry · Shared terms “Bismut”, “Hermitian”, “trivial canonical”

  6. Chern-Ricci flow on Kato surfaces

    Daniele Angella and Mauricio Corrêa

    Topics Generalized & G₂ flows · Same author Daniele Angella · Shared terms “Hermitian metrics”, “Chern”

  7. Hermitian Curvature flow on unimodular Lie groups and static invariant metrics

    Ramiro A. Lafuente, Mattia Pujia and Luigi Vezzoni

    Topics Generalized & G₂ flows, Homogeneous spaces, Kähler geometry · Shared terms “Hermitian metrics”, “complex structure”, “pluriclosed”

  8. Chern-Ricci flow and t-Gauduchon Ricci-flat condition

    Eder M. Correa, Giovane Galindo and Lino Grama

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “Gauduchon”, “Hermitian metrics”, “Chern”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

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