Paper · Apr 22, 2024

Formal structure of scalar curvature in generalized Kähler geometry

Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy

math.DGarXiv:2404.14595v2
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Abstract

Building on works of Boulanger and Goto, we show that Goto's scalar curvature is the moment map for an action of generalized Hamiltonian automorphisms of the associated Courant algebroid, constrained by the choice of an adapted volume form. We derive an explicit formula for Goto's scalar curvature, and show that it is constant for generalized Kähler-Ricci solitons. Restricting to the generically symplectic type case, we realize the generalized Kähler class as the complexified orbit of the Hamiltonian action above. This leads to a natural extension of Mabuchi's metric and -energy, implying a conditional uniqueness result. Finally, in this setting we derive a Calabi-Matsushima-Lichnerowicz obstruction and a Futaki invariant.

Comments: final version, to appear CAMS

Ricci solitonsKähler–Ricci solitonsKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability

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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 2021, 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 Variational structure and uniqueness of generalized Kähler-Ricci solitons (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)1Apostolov 2021 On geometry of steady toric Kähler-Ricci solitons (Yury Ustinovskiy, 2022)7Ustinovskiy 2022 Moment map, convex function and extremal point (King Leung Lee, Jacob Sturm and Xiaowei Wang, 2022)8Lee 2022 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)3Apostolov 2022 Weighted K-stability for a class of non-compact toric fibrations (Charles Cifarelli, 2023)6Cifarelli 2023 From Kähler Ricci solitons to Calabi-Yau Kähler cones (Vestislav Apostolov, Abdellah Lahdili and Eveline Legendre, 2024)4Apostolov 2024 Toric geometry of generalized Kähler-Ricci solitons (Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al., 2025)2Apostolov 2025 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)5Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Variational structure and uniqueness of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy PDF

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

  2. Toric geometry of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al. PDF

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

  3. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al. PDF

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

  4. From Kähler Ricci solitons to Calabi-Yau Kähler cones

    Vestislav Apostolov, Abdellah Lahdili and Eveline Legendre PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Same author Vestislav Apostolov · Shared terms “Mabuchi”, “Calabi”, “obstruction”

  5. Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

    Liding Huang and Xiaohua Zhu PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “generalized Kähler”, “symplectic type”, “Mabuchi”

  6. Weighted K-stability for a class of non-compact toric fibrations

    Charles Cifarelli PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Scalar curvature · Shared terms “Futaki invariant”, “Mabuchi”, “Calabi”

  7. On geometry of steady toric Kähler-Ricci solitons

    Yury Ustinovskiy PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Same author Yury Ustinovskiy · Shared terms “Calabi”

  8. Moment map, convex function and extremal point

    King Leung Lee, Jacob Sturm and Xiaowei Wang PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “moment map”, “Hamiltonian action”, “symplectic”

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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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

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  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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