Paper · Mar 25, 2017

The nondegenerate generalized Kähler Calabi-Yau problem

Vestislav Apostolov, Jeffrey Streets

math.DGarXiv:1703.08650v2
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Abstract

We formulate a Calabi-Yau type conjecture in generalized Kähler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kähler structures generalizing the notion of Kähler class, we conjecture unique solvability of Gualtieri's Calabi-Yau equation within this class. We establish the uniqueness, and moreover show that all such solutions are actually hyper-Kähler metrics. We furthermore establish a GIT framework for this problem, interpreting solutions of this equation as zeros of a moment map associated to a Hamiltonian action and finding a Kempf-Ness functional. Lastly we indicate the naturality of generalized Kähler-Ricci flow in this setting, showing that it evolves within the given Hamiltonian deformation class, and that the Kempf-Ness functional is monotone, so that the only possible fixed points for the flow are hyper-Kähler metrics. On a hyper-Kähler background, we establish global existence and weak convergence of the flow.

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Submitted
Mar 25, 2017
Last revised
Mar 12, 2021 (v2)
Subjects
math.DG, math.AP

Comments: Final version, to appear Crelle's Journal

Ricci flowKähler–Ricci flowRicci-flat & Calabi–YauKähler geometryStability & uniquenessLong-time existence & convergence

Its neighbourhood

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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 Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces (Jeffrey Streets, 2016)2Streets 2016 Classification of generalized Kähler-Ricci solitons on complex surfaces (Jeffrey Streets and Yury Ustinovskiy, 2019)8Streets 2019 Deformation classes in generalized Kähler geometry (Matthew Gibson and Jeffrey Streets, 2020)6Gibson 2020 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)3Apostolov 2021 Variational structure and uniqueness of generalized Kähler-Ricci solitons (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)4Apostolov 2021 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)1Apostolov 2022 Formal structure of scalar curvature in generalized Kähler geometry (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2024)5Apostolov 2024
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  1. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al.

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Long-time existence & convergence · Same authors Vestislav Apostolov, Jeffrey Streets · Shared terms “generalized Kähler”, “Kähler structures”, “global existence”

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

    Jeffrey Streets

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Long-time existence & convergence · Same author Jeffrey Streets · Shared terms “nondegenerate generalized”, “generalized Kähler”, “nondegenerate Poisson”

  3. Generalized Kähler-Ricci flow on toric Fano varieties

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same authors Vestislav Apostolov, Jeffrey Streets · Shared terms “generalized Kähler”, “weak convergence”, “global existence”

  4. Variational structure and uniqueness of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler geometry, Stability & uniqueness · Same authors Vestislav Apostolov, Jeffrey Streets · Shared terms “generalized Kähler”, “Hamiltonian action”

  5. Formal structure of scalar curvature in generalized Kähler geometry

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler geometry, Stability & uniqueness · Same authors Vestislav Apostolov, Jeffrey Streets · Shared terms “generalized Kähler”, “Hamiltonian”, “moment map”

  6. Deformation classes in generalized Kähler geometry

    Matthew Gibson and Jeffrey Streets

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

  7. Generalized Ricci Flow

    Mario Garcia-Fernandez and Jeffrey Streets

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”

  8. 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 structure”, “Kähler structures”

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) 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) 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.