Paper · Sep 2, 2020

The Gibbons-Hawking ansatz in generalized Kähler geometry

Jeffrey Streets, Yury Ustinovskiy

math.DGarXiv:2009.00778v3
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Abstract

We derive a local ansatz for generalized Kähler surfaces with nondegenerate Poisson structure and a biholomorphic action which generalizes the classic Gibbons-Hawking ansatz for invariant hyperKähler manifolds, and allows for the choice of one arbitrary function. By imposing the generalized Kähler-Ricci soliton equation, or equivalently the equations of type IIB string theory, the construction becomes rigid, and we classify all complete solutions with the smallest possible symmetry group.

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Submitted
Sep 2, 2020
Last revised
Feb 7, 2022 (v3)
Subjects
math.DG, math-ph, math.CV
DOI
10.1007/s00220-022-04329-6

Comments: 61 pages, 3 figures, the version to appear in Communications in Mathematical Physics

Ricci solitonsKähler–Ricci solitonsRicci-flat & Calabi–YauKähler geometryPhysics

Its neighbourhood

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 Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces (Jeffrey Streets, 2016)6Streets 2016 Classification of generalized Kähler-Ricci solitons on complex surfaces (Jeffrey Streets and Yury Ustinovskiy, 2019)2Streets 2019 Deformation classes in generalized Kähler geometry (Matthew Gibson and Jeffrey Streets, 2020)7Gibson 2020 Generalized Kähler-Ricci flow on toric Fano varieties (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)8Apostolov 2021 Variational structure and uniqueness of generalized Kähler-Ricci solitons (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)3Apostolov 2021 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)5Apostolov 2022 Formal structure of scalar curvature in generalized Kähler geometry (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2024)4Apostolov 2024 Toric geometry of generalized Kähler-Ricci solitons (Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al., 2025)1Apostolov 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Toric geometry of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al.

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “Hawking ansatz”, “Gibbons Hawking”

  2. Classification of generalized Kähler-Ricci solitons on complex surfaces

    Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “Poisson structure”

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

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”

  4. Formal structure of scalar curvature in generalized Kähler geometry

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”

  5. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al.

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “hyperkähler manifolds”

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

    Jeffrey Streets

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

  7. Deformation classes in generalized Kähler geometry

    Matthew Gibson and Jeffrey Streets

    Topics Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “symmetry group”, “Poisson”

  8. Generalized Kähler-Ricci flow on toric Fano varieties

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler geometry · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.