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Long-time behavior of the Hermitian-Yang-Mills flow on non-Kähler manifolds

Zeng Chen, Chao Li, Chuanjing Zhang, Xi Zhang

TL;DR

This work analyzes the Hermitian-Yang-Mills flow on compact Hermitian (non-Kähler) manifolds, establishing monotonicity and convergence properties for eigenvalues of the mean curvature and linking them to Harder-Narasimhan data in the Gauduchon setting. By developing refined energy estimates and Uhlenbeck-type compactness arguments in the non-Kähler context, the authors show that the flow’s long-time limit is governed by the Harder-Narasimhan type of the initial bundle, culminating in an isomorphism between the limiting reflexive sheaf and the graded Harder-Narasimhan-Seshadri object. They also extend the Atiyah-Bott-Bando-Siu program to non-Kähler manifolds, prove positivity results for Chern forms in the ample case, and provide practical tools to compute HN data for tensor, symmetric, and exterior powers. The results yield both qualitative and quantitative descriptions of the flow’s asymptotics and establish a robust bridge between analytic flow limits and algebro-geometric invariants in the non-Kähler setting.

Abstract

In this paper, we study the long-time behavior of the Hermitian-Yang-Mills flow over compact Hermitian manifolds. We obtain the monotonicity of lower bound and upper bound of the eigenvalues of the mean curvature along the Hermitian-Yang-Mills flow. In the Gauduchon case, we show that the eigenvalues of the mean curvature converge to geometric invariants determined by the Harder-Narasimhan type. Furthermore, we generalize the Atiyah-Bott-Bando-Siu question to the non-Kähler case.

Long-time behavior of the Hermitian-Yang-Mills flow on non-Kähler manifolds

TL;DR

This work analyzes the Hermitian-Yang-Mills flow on compact Hermitian (non-Kähler) manifolds, establishing monotonicity and convergence properties for eigenvalues of the mean curvature and linking them to Harder-Narasimhan data in the Gauduchon setting. By developing refined energy estimates and Uhlenbeck-type compactness arguments in the non-Kähler context, the authors show that the flow’s long-time limit is governed by the Harder-Narasimhan type of the initial bundle, culminating in an isomorphism between the limiting reflexive sheaf and the graded Harder-Narasimhan-Seshadri object. They also extend the Atiyah-Bott-Bando-Siu program to non-Kähler manifolds, prove positivity results for Chern forms in the ample case, and provide practical tools to compute HN data for tensor, symmetric, and exterior powers. The results yield both qualitative and quantitative descriptions of the flow’s asymptotics and establish a robust bridge between analytic flow limits and algebro-geometric invariants in the non-Kähler setting.

Abstract

In this paper, we study the long-time behavior of the Hermitian-Yang-Mills flow over compact Hermitian manifolds. We obtain the monotonicity of lower bound and upper bound of the eigenvalues of the mean curvature along the Hermitian-Yang-Mills flow. In the Gauduchon case, we show that the eigenvalues of the mean curvature converge to geometric invariants determined by the Harder-Narasimhan type. Furthermore, we generalize the Atiyah-Bott-Bando-Siu question to the non-Kähler case.
Paper Structure (12 sections, 38 theorems, 237 equations)

This paper contains 12 sections, 38 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Stability and the Harder-Narasimhan filtration.
  4. The Hermitian-Yang-Mills flow
  5. Proofs of Theorem \ref{['thm2']}, \ref{['thmuni']} and \ref{['thm3']}
  6. Some applications
  7. Application to the calculation of the Harder-Narasimhan type
  8. A proof of Theorem \ref{['ample']}
  9. The Atiyah-Bott-Bando-Siu question on some compact non-Kähler manifolds
  10. The HN-type of the limit
  11. The non-zero holomorphic map
  12. Proof of Theorem \ref{['thm1.9']}

Key Result

Theorem 1.1

Let $(M, \omega )$ be a compact Hermitian manifold, $(E, \bar{\partial }_{E})$ be a holomorphic vector bundle over $M$. Assume $H(t)$ is a smooth solution of the Hermitian-Yang-Mills flow (Flow) on $(E, \bar{\partial }_{E})$. Then $\hat{\lambda }_L(H(t), \omega )$ is increasing and $\hat{\lambda }_U

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: LiZZ
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8: NZ2
  • Theorem 1.9
  • Remark 1.10
  • ...and 45 more