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Existence and geometry of Hermitian metrics with constant second scalar curvature

Liangdi Zhang

TL;DR

The paper develops a systematic study of Hermitian metrics with constant second scalar curvature for the Bismut and Chern connections on compact complex manifolds. It proves existence results for the second Bismut scalar curvature under a topological condition and establishes existence and, in favorable cases, uniqueness results for constant second Chern scalar curvature within fixed Hermitian conformal classes, together with geometric consequences on Kodaira dimension and (anti-)canonical bundles. A key result shows that an Einstein-type condition for the second Chern curvature on pluriclosed Gauduchon metrics enforces constancy of S_C^(2), with rigidity outcomes leading to Kähler–Einstein behavior in certain dimensions. The paper also provides non-Kähler examples with constant Chern scalar curvatures, illustrating the scope beyond the Kähler setting. Collectively, these results deepen the了解 of scalar curvature problems in Hermitian geometry and link curvature conditions to global complex-geometric invariants.

Abstract

We study Hermitian metrics with constant second scalar curvature on compact manifolds. We first consider a Yamabe-type problem for the second Bismut scalar curvature under a natural topological condition, and then analyze elliptic equations arising from constant second Chern scalar curvature within a fixed Hermitian conformal class and derive geometric consequences. Finally, under an Einstein-type condition on the second Chern curvature, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, which in certain cases further implies the existence of a Kähler-Einstein metric.

Existence and geometry of Hermitian metrics with constant second scalar curvature

TL;DR

The paper develops a systematic study of Hermitian metrics with constant second scalar curvature for the Bismut and Chern connections on compact complex manifolds. It proves existence results for the second Bismut scalar curvature under a topological condition and establishes existence and, in favorable cases, uniqueness results for constant second Chern scalar curvature within fixed Hermitian conformal classes, together with geometric consequences on Kodaira dimension and (anti-)canonical bundles. A key result shows that an Einstein-type condition for the second Chern curvature on pluriclosed Gauduchon metrics enforces constancy of S_C^(2), with rigidity outcomes leading to Kähler–Einstein behavior in certain dimensions. The paper also provides non-Kähler examples with constant Chern scalar curvatures, illustrating the scope beyond the Kähler setting. Collectively, these results deepen the了解 of scalar curvature problems in Hermitian geometry and link curvature conditions to global complex-geometric invariants.

Abstract

We study Hermitian metrics with constant second scalar curvature on compact manifolds. We first consider a Yamabe-type problem for the second Bismut scalar curvature under a natural topological condition, and then analyze elliptic equations arising from constant second Chern scalar curvature within a fixed Hermitian conformal class and derive geometric consequences. Finally, under an Einstein-type condition on the second Chern curvature, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, which in certain cases further implies the existence of a Kähler-Einstein metric.
Paper Structure (7 sections, 22 theorems, 183 equations)

This paper contains 7 sections, 22 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conformal transformation formulas and scalar curvatures
  4. A Yamabe-type problem for the Bismut connection
  5. Second Chern scalar curvature and geometric consequences
  6. Weak second Hermitian--Einstein metrics
  7. Non-Kähler examples of constant Chern scalar curvatures

Key Result

Theorem 1.1

Let $M$ be a compact complex manifold of complex dimension $n\ge 2$. If $b_1(M)=0$, then for any Hermitian metric $\omega$ on $M$, there exists a Hermitian metric $\widetilde{\omega} \in \{\omega\}$ with constant second Bismut scalar curvature.

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Lemma 2.1: see e.g. Gau84
  • ...and 24 more