Paper · Apr 1, 2024

Hermitian null loci

Quang-Tuan Dang

math.CVarXiv:2404.01126v2
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Abstract

We establish a transcendental generalization of Nakamaye's theorem to compact complex manifolds when the form is not assumed to be closed. We apply the recent analytic technique developed by Collins and Tosatti to show that the non-Hermitian locus of a nef and big -form, which is not necessarily closed, on a compact complex manifold equals the union of all positive-dimensional analytic subvarieties where the restriction of the form is not big (null locus). As an application, we can give an alternative proof of the Nakai–Moishezon criterion of Buchdahl and Lamari for complex surfaces and generalize this result in higher dimensions Finally, we investigate finite time non-collapsing singularities of the Chern–Ricci flow, partially answering a question raised by Tosatti and Weinkove.

Comments: v.1 comments welcome, v.2 remove section 4 in version 1, accepted to Trans. Amer. Math. Soc

Ricci flowGeneralized & G₂ flowsSingularities(Non)collapsingAnalytic minimal model program

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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 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 Chern-Ricci flow (Valentino Tosatti and Ben Weinkove, 2021)8Tosatti 2021 Pluripotential Chern-Ricci Flows (Quang-Tuan Dang, 2022)2Dang 2022 Singularities of the Chern-Ricci flow (Quang-Tuan Dang, 2023)1Dang 2023 The behavior of the second Ricci flow on complex parallelizable manifolds (Lucio Bedulli and Luigi Vezzoni, 2024)5Bedulli 2024 Hermitian metrics on complex non-Kähler manifolds (Daniele Angella, 2025)7Angella 2025 Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds (Quang-Tuan Dang, 2026)4Dang 2026 Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type (Haoyuan Sun, 2026)6Sun 2026 Convergence of the Chern-Ricci flow on complex minimal surfaces of general type (Haoyuan Sun, 2026)3Sun 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularities of the Chern-Ricci flow

    Quang-Tuan Dang PDF

    Topics Analytic minimal model program, Generalized & G₂ flows, Singularities · Same author Quang-Tuan Dang · Shared terms “partially answering”, “Tosatti”, “Chern”

  2. Pluripotential Chern-Ricci Flows

    Quang-Tuan Dang PDF

    Topics Analytic minimal model program, Generalized & G₂ flows, Singularities · Same author Quang-Tuan Dang · Shared terms “Chern”, “Hermitian”, “developed”

  3. Convergence of the Chern-Ricci flow on complex minimal surfaces of general type

    Haoyuan Sun PDF

    Topics Analytic minimal model program, Generalized & G₂ flows, (Non)collapsing · Shared terms “null locus”, “Tosatti”, “Chern”

  4. Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds

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    Topics Generalized & G₂ flows · Same author Quang-Tuan Dang · Shared terms “Hermitian”, “Chern”

  5. The behavior of the second Ricci flow on complex parallelizable manifolds

    Lucio Bedulli and Luigi Vezzoni PDF

    Topics Generalized & G₂ flows · Shared terms “compact complex”, “complex manifold”, “Hermitian”

  6. Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type

    Haoyuan Sun PDF

    Topics Analytic minimal model program, Generalized & G₂ flows, (Non)collapsing · Shared terms “Hermitian”, “null locus”, “Tosatti”

  7. Hermitian metrics on complex non-Kähler manifolds

    Daniele Angella PDF

    Topics Generalized & G₂ flows · Shared terms “compact complex”, “Hermitian”, “complex surfaces”

  8. The Chern-Ricci flow

    Valentino Tosatti and Ben Weinkove PDF

    Topics Generalized & G₂ flows · Shared terms “complex manifold”, “Chern”, “Hermitian”

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.