Paper · Nov 20, 2016

Hirzebruch manifolds and positive holomorphic sectional curvature

Bo Yang, Fangyang Zheng

math.DGarXiv:1611.06571v2
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Abstract

This paper is the first step in a systematic project to study examples of Kähler manifolds with positive holomorphic sectional curvature (). Previously Hitchin proved that any compact Kähler surface with must be rational and he constructed such examples on Hirzebruch surfaces . We generalize Hitchin's construction and prove that any Hirzebruch manifold admits a Kähler metric of in each of its Kähler classes. We demonstrate that the pinching behaviors of holomorphic sectional curvatures of new examples differ from those of Hitchin's which were studied in the recent work of Alvarez-Chaturvedi-Heier. Some connections to recent works on the Kähler-Ricci flow on Hirzebruch manifolds are also discussed. It seems interesting to study the space of all Kähler metrics of on a given Kähler manifold. We give higher dimensional examples such that some Kähler classes admit Kähler metrics with and some do not.

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Submitted
Nov 20, 2016
Last revised
Dec 13, 2016 (v2)
Subjects
math.DG
Journal
Ann. Inst. Fourier (Grenoble) 69 (2019), no.6, 2589-2634

Comments: In this 2nd version. we add a corollary (Corollary 1.8) and update some typos and inaccuaricies pointed by Gordon Heier

Ricci flowKähler–Ricci flowCurvature pinchingKähler geometry

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 Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach (Ryosuke Nomura, 2016)1Nomura 2016 The Kähler-Ricci flow on manifolds with negative holomorphic curvature (Freid Tong, 2018)6Tong 2018 Longtime existence of Kähler Ricci flow and holomorphic sectional curvature (Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al., 2018)3Huang 2018 Kähler manifolds and cross quadratic bisectional curvature (Lei Ni and Fangyang Zheng, 2019)2Ni 2019 Kähler-Ricci Flow preserves negative anti-bisectional curvature (Gabriel Khan and Fangyang Zheng, 2020)4Khan 2020 On almost nonpositive k-Ricci curvature (Kai Tang, 2021)8Tang 2021 On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces (Jiangtao Li, 2025)5Li 2025 Gradient Shrinking Ricci Solitons and Modified Sectional Curvature (Xiaodong Cao, Ernani Ribeiro and Hosea Wondo, 2025)7Cao 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach

    Ryosuke Nomura

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “holomorphic sectional”, “sectional curvature”, “compact Kähler”

  2. Kähler manifolds and cross quadratic bisectional curvature

    Lei Ni and Fangyang Zheng

    Topics Kähler–Ricci flow, Kähler geometry · Same author Fangyang Zheng · Shared terms “compact Kähler”

  3. Longtime existence of Kähler Ricci flow and holomorphic sectional curvature

    Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al.

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “holomorphic sectional”, “sectional curvature”

  4. Kähler-Ricci Flow preserves negative anti-bisectional curvature

    Gabriel Khan and Fangyang Zheng

    Topics Kähler–Ricci flow, Kähler geometry · Same author Fangyang Zheng

  5. On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces

    Jiangtao Li

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Hirzebruch”

  6. The Kähler-Ricci flow on manifolds with negative holomorphic curvature

    Freid Tong

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “holomorphic sectional”, “sectional curvature”

  7. Gradient Shrinking Ricci Solitons and Modified Sectional Curvature

    Xiaodong Cao, Ernani Ribeiro and Hosea Wondo

    Topics Curvature pinching, Kähler geometry · Shared terms “sectional curvature”, “Hitchin”

  8. On almost nonpositive -Ricci curvature

    Kai Tang

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “holomorphic sectional”, “sectional curvature”, “compact 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 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) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  2. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.