Paper · Apr 15, 2019

On the Structure of Hermitian Manifolds with Semipositive Griffiths Curvature

Yury Ustinovskiy

math.DGarXiv:1904.06810
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Abstract

In this paper we establish partial structure results on the geometry of compact Hermitian manifolds of semipositive Griffiths curvature. We show that after appropriate arbitrary small deformation of the initial metric, the null spaces of the Chern-Ricci two-form generate a holomorphic, integrable distribution. This distribution induces an isometric, holomorphic, almost free action of a complex Lie group on the universal cover of the manifold. Our proof combines the strong maximum principle for the Hermitian Curvature Flow (HCF), new results on the interplay of the HCF and the torsion-twisted connection, and observations on the geometry of the torsion-twisted connection on a general Hermitian manifold.

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Submitted
Apr 15, 2019
Subjects
math.DG
DOI
10.1090/tran/8101
Generalized & G₂ flowsHomogeneous spaces

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Connected papers

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)1Ustinovskiy 2016 Hermitian curvature flow on complex homogeneous manifolds (Yury Ustinovskiy, 2017)3Ustinovskiy 2017 Lie-Algebraic Curvature Conditions preserved by the Hermitian Curvature Flow (Yury Ustinovskiy, 2017)2Ustinovskiy 2017 Expanding solitons to the Hermitian curvature flow on complex Lie groups (Mattia Pujia, 2018)4Pujia 2018 Hermitian Curvature Flow on compact homogeneous spaces (Francesco Panelli and Fabio Podestà, 2019)5Panelli 2019 Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups (James Stanfield, 2020)6Stanfield 2020 Positive Hermitian Curvature Flow on special linear groups and perfect solitons (James Stanfield, 2021)8Stanfield 2021 Positive Hermitian curvature flow on 2-step nilpotent Lie groups (Ettore Lo Giudice, 2025)7Giudice 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Hermitian curvature flow on manifolds with non-negative Griffiths curvature

    Yury Ustinovskiy

    Topics Generalized & G₂ flows · Same author Yury Ustinovskiy · Shared terms “Griffiths curvature”, “torsion twisted”, “HCF”

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

    Yury Ustinovskiy

    Topics Generalized & G₂ flows · Same author Yury Ustinovskiy · Shared terms “HCF”, “Hermitian curvature”, “Hermitian manifolds”

  3. Hermitian curvature flow on complex homogeneous manifolds

    Yury Ustinovskiy

    Topics Generalized & G₂ flows, Homogeneous spaces · Same author Yury Ustinovskiy · Shared terms “HCF”, “Hermitian curvature”, “complex Lie”

  4. Expanding solitons to the Hermitian curvature flow on complex Lie groups

    Mattia Pujia

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

  5. Hermitian Curvature Flow on compact homogeneous spaces

    Francesco Panelli and Fabio Podestà

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

  6. Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups

    James Stanfield

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

  7. Positive Hermitian curvature flow on 2-step nilpotent Lie groups

    Ettore Lo Giudice

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

  8. Positive Hermitian Curvature Flow on special linear groups and perfect solitons

    James Stanfield

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

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.

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) 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)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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