Paper · Feb 9, 2022

Kähler stability of symplectic forms

Jeffrey Streets, Gang Tian

math.DGWider flowsarXiv:2202.04564
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Abstract

Using dynamical stability of symplectic curvature flow, we show that on a compact Calabi-Yau manifold, any small symplectic deformation of a Kähler form remains Kähler.

Comments: to appear in J. Geom. Anal. special issue: "Analysis and Geometry of Complete Manifolds, a special issue in honor of Peter Li"

Other curvature flowsRicci-flat & Calabi–YauKähler geometryStability & uniqueness

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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 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)1Apostolov 2022 Static solutions to symplectic curvature flow in dimension four (Gavin Ball, 2023)2Ball 2023 Formal structure of scalar curvature in generalized Kähler geometry (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2024)4Apostolov 2024 Ricci Flow on ALF manifolds (Dain Kim and Tristan Ozuch, 2025)3Kim 2025 Integrable Deformations and Stability of the Ricci Flow (Maxwell Stolarski and Alex Waldron, 2026)7Stolarski 2026 Topology of low-dimensional generalized Ricci solitons and string backgrounds (Jeffrey Streets, 2026)8Streets 2026 Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds (Jiyuan Han, 2026)6Han 2026 Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation (Randall R. Van Why, 2026)5Van Why 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The generalized Kähler Calabi-Yau problem

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

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Same author Jeffrey Streets

  2. Static solutions to symplectic curvature flow in dimension four

    Gavin BallWider flows PDF

    Topics Other curvature flows, Kähler geometry · Shared terms “symplectic curvature”

  3. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry, Stability & uniqueness

  4. Formal structure of scalar curvature in generalized Kähler geometry

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy PDF

    Topics Kähler geometry, Stability & uniqueness · Same author Jeffrey Streets · Shared terms “symplectic”, “Calabi”

  5. Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation

    Randall R. Van Why PDF

    Topics Kähler geometry · Shared terms “symplectic deformation”

  6. Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds

    Jiyuan Han PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry

  7. Integrable Deformations and Stability of the Ricci Flow

    Maxwell Stolarski and Alex Waldron PDF

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness · Shared terms “dynamical stability”

  8. Topology of low-dimensional generalized Ricci solitons and string backgrounds

    Jeffrey Streets PDF

    Topics Ricci-flat & Calabi–Yau · Same author Jeffrey Streets

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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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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. 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)