Paper · May 6, 2020

Deformation classes in generalized Kähler geometry

Matthew Gibson, Jeffrey Streets

math.DGarXiv:2005.03062
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Abstract

We introduce natural deformation classes of generalized Kähler structures using the Courant symmetry group. We show that these yield natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. Lastly we show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.

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Submitted
May 6, 2020
Subjects
math.DG
Ricci flowKähler–Ricci flowKähler geometry

Its neighbourhood

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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 Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces (Jeffrey Streets, 2016)2Streets 2016 Global viscosity solutions of generalized Kahler-Ricci flow (Jeffrey Streets, 2016)5Streets 2016 The nondegenerate generalized Kähler Calabi-Yau problem (Vestislav Apostolov and Jeffrey Streets, 2017)4Apostolov 2017 Classification of generalized Kähler-Ricci solitons on complex surfaces (Jeffrey Streets and Yury Ustinovskiy, 2019)1Streets 2019 Generalized Ricci Flow (Mario Garcia-Fernandez and Jeffrey Streets, 2020)7Garcia-Ferna 2020 Generalized Kähler-Ricci flow on toric Fano varieties (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)6Apostolov 2021 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)3Apostolov 2022 Formal structure of scalar curvature in generalized Kähler geometry (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2024)8Apostolov 2024
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  1. Classification of generalized Kähler-Ricci solitons on complex surfaces

    Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “Poisson”, “Kähler structures”

  2. Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces

    Jeffrey Streets

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “Poisson”

  3. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al.

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “Kähler structures”, “extension”

  4. The nondegenerate generalized Kähler Calabi-Yau problem

    Vestislav Apostolov and Jeffrey Streets

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “Kähler structures”, “Lastly”

  5. Global viscosity solutions of generalized Kahler-Ricci flow

    Jeffrey Streets

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”

  6. Generalized Kähler-Ricci flow on toric Fano varieties

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “extension”

  7. Generalized Ricci Flow

    Mario Garcia-Fernandez and Jeffrey Streets

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “extension”, “preserves”

  8. Formal structure of scalar curvature in generalized Kähler geometry

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “Courant”, “extension”

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) 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)

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. 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.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.