Paper · Feb 12, 2021

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

Giuseppe Barbaro

math.DGarXiv:2102.06637v2
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Abstract

In this note we study a positivity notion for the curvature of the Bismut connection; more precisely, we study the notion of Bismut-Griffiths-positivity for complex Hermitian non-Kähler manifolds. Since the Kähler-Ricci flow preserves and regularizes the usual Griffiths positivity we investigate the behaviour of the Bismut-Griffiths-positivity under the action of the Hermitian curvature flows. In particular we study two concrete classes of examples, namely, linear Hopf manifolds and six-dimensional Calabi-Yau solvmanifolds with holomorphically-trivial canonical bundle. From these examples we identify some HCFs which do not preserve Bismut-Griffiths-non-negativity.

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Submitted
Feb 12, 2021
Last revised
Mar 24, 2021 (v2)
Subjects
math.DG, math.CV
DOI
10.1016/j.geomphys.2021.104323
Ricci flowKähler–Ricci flowGeneralized & G₂ flowsRicci-flat & Calabi–YauKähler geometryHomogeneous spaces

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper The Hermitian curvature flow on manifolds with non-negative Griffiths curvature (Yury Ustinovskiy, 2016)5Ustinovskiy 2016 Lie-Algebraic Curvature Conditions preserved by the Hermitian Curvature Flow (Yury Ustinovskiy, 2017)2Ustinovskiy 2017 On Gauduchon connections with Kähler-like curvature (Daniele Angella, Antonio Otal, Luis Ugarte et al., 2018)1Angella 2018 Hermitian Curvature Flow on compact homogeneous spaces (Francesco Panelli and Fabio Podestà, 2019)7Panelli 2019 On the Structure of Hermitian Manifolds with Semipositive Griffiths Curvature (Yury Ustinovskiy, 2019)6Ustinovskiy 2019 Bismut Ricci flat manifolds with symmetries (Fabio Podestà and Alberto Raffero, 2022)3Podestà 2022 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)8Apostolov 2022 Cohomogeneity one solutions of the IIB system (Lorenzo Foscolo and Mario Garcia-Fernandez, 2026)4Foscolo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On Gauduchon connections with Kähler-like curvature

    Daniele Angella, Antonio Otal, Luis Ugarte et al.

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

  2. Lie-Algebraic Curvature Conditions preserved by the Hermitian Curvature Flow

    Yury Ustinovskiy

    Topics Generalized & G₂ flows · Shared terms “Hermitian curvature”, “Griffiths positivity”, “HCF”

  3. Bismut Ricci flat manifolds with symmetries

    Fabio Podestà and Alberto Raffero

    Topics Generalized & G₂ flows, Ricci-flat & Calabi–Yau, Homogeneous spaces · Shared terms “Bismut”

  4. Cohomogeneity one solutions of the IIB system

    Lorenzo Foscolo and Mario Garcia-Fernandez

    Topics Generalized & G₂ flows, Ricci-flat & Calabi–Yau, Homogeneous spaces · Shared terms “behaviour along”, “Bismut”, “holomorphically trivial”

  5. The Hermitian curvature flow on manifolds with non-negative Griffiths curvature

    Yury Ustinovskiy

    Topics Generalized & G₂ flows · Shared terms “Griffiths”, “Hermitian curvature”, “complex Hermitian”

  6. On the Structure of Hermitian Manifolds with Semipositive Griffiths Curvature

    Yury Ustinovskiy

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “Griffiths”, “Hermitian curvature”, “HCF”

  7. Hermitian Curvature Flow on compact homogeneous spaces

    Francesco Panelli and Fabio Podestà

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “Hermitian curvature”

  8. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al.

    Topics Generalized & G₂ flows, Ricci-flat & Calabi–Yau, Kähler–Ricci flow · Shared terms “Bismut”

Read around it

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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  3. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.