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On Kodaira dimension and scalar curvature in almost Hermitian geometry

Xianchao Zhou

TL;DR

The paper investigates how curvature constraints influence the Kodaira dimension in almost Hermitian geometry. By developing a Gauduchon-conformal framework and utilizing Chern scalar curvatures $S_1^{\mathrm{Ch}}$ and $S_2^{\mathrm{Ch}}$, it proves that for compact almost Hermitian manifolds in the class $\mathcal{W}_2\oplus\mathcal{W}_3\oplus\mathcal{W}_4$ with $s\ge0$, the Kodaira dimension $\kappa(M,J)$ is either $-\infty$ or $0$, with the latter forcing a Kähler Calabi–Yau structure, and establishes analogous results under nonnegative mixed scalar curvature for Hermitian manifolds. The work also connects these geometric insights to Moishezon manifolds (uniruledness under $3s_J+s>0$) and provides a detailed twistor-space analysis: on the twistor space of an anti-self-dual 4-manifold, $\kappa(\mathbf Z,\mathbb J_+)= -\infty$ while $\kappa(\mathbf Z,\mathbb J_-)=0$, with canonical-line-bundle properties and balanced metrics informing curvature signs. Overall, the results extend Yau-type rigidity phenomena from Kähler to broad non-Kähler settings, linking curvature, canonical bundles, and algebraic-geometric structure in a robust framework and yielding explicit geometric examples.

Abstract

In this paper, we investigate Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds. Specifically, for a compact almost Hermitian manifold $(M, J, g)$ in the Gray-Hervella class $\mathcal{W}_2\oplus\mathcal{W}_3\oplus \mathcal{W}_4$ with nonnegative Riemannian scalar curvature, we prove that its Kodaira dimension must satisfy $κ(M, J)=-\infty$; or $κ(M, J)=0$, in which case $(M,J,g)$ is a Kähler Calabi-Yau manifold. The same conclusions also hold for compact Hermitian manifolds with an assumption of nonnegative mixed scalar curvature. As an important example, we study the twistor geometry of a compact anti-self-dual 4-manifold. In particular, for the twistor space with the Eells-Salamon almost complex structure, we show that the Kodaira dimension is zero.

On Kodaira dimension and scalar curvature in almost Hermitian geometry

TL;DR

The paper investigates how curvature constraints influence the Kodaira dimension in almost Hermitian geometry. By developing a Gauduchon-conformal framework and utilizing Chern scalar curvatures and , it proves that for compact almost Hermitian manifolds in the class with , the Kodaira dimension is either or , with the latter forcing a Kähler Calabi–Yau structure, and establishes analogous results under nonnegative mixed scalar curvature for Hermitian manifolds. The work also connects these geometric insights to Moishezon manifolds (uniruledness under ) and provides a detailed twistor-space analysis: on the twistor space of an anti-self-dual 4-manifold, while , with canonical-line-bundle properties and balanced metrics informing curvature signs. Overall, the results extend Yau-type rigidity phenomena from Kähler to broad non-Kähler settings, linking curvature, canonical bundles, and algebraic-geometric structure in a robust framework and yielding explicit geometric examples.

Abstract

In this paper, we investigate Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds. Specifically, for a compact almost Hermitian manifold in the Gray-Hervella class with nonnegative Riemannian scalar curvature, we prove that its Kodaira dimension must satisfy ; or , in which case is a Kähler Calabi-Yau manifold. The same conclusions also hold for compact Hermitian manifolds with an assumption of nonnegative mixed scalar curvature. As an important example, we study the twistor geometry of a compact anti-self-dual 4-manifold. In particular, for the twistor space with the Eells-Salamon almost complex structure, we show that the Kodaira dimension is zero.
Paper Structure (6 sections, 10 theorems, 77 equations)

This paper contains 6 sections, 10 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on almost Hermitian geometry
  3. Chern scalar curvatures and Kodaira dimension
  4. The case of nonnegative Riemannian scalar curvature
  5. The case of a nonnegative mixed scalar curvature
  6. Appendix: Scalar curvatures on twistor spaces and some consequences

Key Result

Theorem 1.1

(See Theorem 4.1). Let $(M,J,g)$ be a compact almost Hermitian manifold of complex dimension $n\geq 3$. If $(M,J,g)\in \mathcal{W}_2\oplus\mathcal{W}_3\oplus \mathcal{W}_4$ and its Riemannian scalar curvature $s\geq0$, then either (1) $\kappa(M, J)=-\infty$; or (2) $\kappa(M, J)=0$, and $(M,J,g)$ is

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Definition 3.4
  • Theorem 4.1
  • proof
  • ...and 11 more