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On Vafa-Witten equations over Kaehler manifolds

Xuemiao Chen

TL;DR

The paper analyzes the analytic properties of the Vafa-Witten equations on compact Kähler manifolds, focusing on compactness phenomena for solutions and the spectral-cover framework. By employing a Weitzenböck-type identity with an auxiliary metric on $K_X$ and extending Mochizuki–Simpson-type spectral-estimates to higher dimensions, it establishes uniform $C^0$ bounds for nilpotent Higgs fields, develops Uhlenbeck compactness for sequences with bounded spectral covers, and characterizes limits via spectral data. It then treats rank-two cases in depth, introducing $\mathbb{Z}_2$ holomorphic $n$-forms and proving Taubes-style convergence results, while providing a general compactness scheme for unbounded spectral covers and intrinsic descriptions of limiting objects. The work yields analytic interpretations of Vafa-Witten invariants on Kähler surfaces, clarifies the moduli of $\mathsf{SU}(2)$ monopoles, and offers concrete stability criteria and compactness results that illuminate the interplay between gauge theory and complex geometry on higher-dimensional Kähler manifolds.

Abstract

In this paper, we study the analytic properties of solutions to the Vafa-Witten equation over a compact Kaehler manifold. Simple obstructions to the existence of nontrivial solutions are identified. The gauge theoretical compactness for the $\mathbb{C}^*$ invariant locus of the moduli space is shown to behave similarly as the Hermitian-Yang-Mills connections. More generally, this holds for solutions with uniformly bounded spectral covers such as nilpotent solutions. When spectral covers are unbounded, we manage to take limits of the renormalized Higgs fields which are intrinsically characterized by the convergence of the associated spectral covers. This gives a simpler proof for Taubes' results on rank two solutions over Kaehler surfaces together with a new complex geometric interpretation. The moduli space of $SU(2)$ monopoles and some related examples are also discussed in the final section.

On Vafa-Witten equations over Kaehler manifolds

TL;DR

The paper analyzes the analytic properties of the Vafa-Witten equations on compact Kähler manifolds, focusing on compactness phenomena for solutions and the spectral-cover framework. By employing a Weitzenböck-type identity with an auxiliary metric on and extending Mochizuki–Simpson-type spectral-estimates to higher dimensions, it establishes uniform bounds for nilpotent Higgs fields, develops Uhlenbeck compactness for sequences with bounded spectral covers, and characterizes limits via spectral data. It then treats rank-two cases in depth, introducing holomorphic -forms and proving Taubes-style convergence results, while providing a general compactness scheme for unbounded spectral covers and intrinsic descriptions of limiting objects. The work yields analytic interpretations of Vafa-Witten invariants on Kähler surfaces, clarifies the moduli of monopoles, and offers concrete stability criteria and compactness results that illuminate the interplay between gauge theory and complex geometry on higher-dimensional Kähler manifolds.

Abstract

In this paper, we study the analytic properties of solutions to the Vafa-Witten equation over a compact Kaehler manifold. Simple obstructions to the existence of nontrivial solutions are identified. The gauge theoretical compactness for the invariant locus of the moduli space is shown to behave similarly as the Hermitian-Yang-Mills connections. More generally, this holds for solutions with uniformly bounded spectral covers such as nilpotent solutions. When spectral covers are unbounded, we manage to take limits of the renormalized Higgs fields which are intrinsically characterized by the convergence of the associated spectral covers. This gives a simpler proof for Taubes' results on rank two solutions over Kaehler surfaces together with a new complex geometric interpretation. The moduli space of monopoles and some related examples are also discussed in the final section.
Paper Structure (19 sections, 48 theorems, 187 equations)

This paper contains 19 sections, 48 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear algebra at one point
  4. Stability
  5. Vafa-Witten equation
  6. Monopoles
  7. The Weitzenböck formula with consequences
  8. Useful estimates from the spectral cover
  9. $C^0$ estimates on the Higgs fields from the control of spectral cover
  10. Estimates from the splitting associated to the spectral data
  11. Applied to a sequence when the spectral covers become unbounded
  12. Uhlenbeck compactness for solutions with uniformly bounded spectral covers
  13. Compactness for rank two solutions with trace free Higgs fields
  14. $\mathbb Z_2$ Holomorphic $n$-forms
  15. Sequential limits
  16. ...and 4 more sections

Key Result

Proposition 1.1

Suppose there exists a solution $(A,\phi)$ with $\phi\neq 0$ to the Vafa-Witten equation. Then If $\deg K_X=0$ holds, then and i.e., $X$ is a Calabi-Yau manifold The Kähler metric $\omega$ is not necessarily a Calabi-Yau metric. Here $\nabla$ denote the connection on $End(E)\otimes K_X$ induced by $A$ and the Chern connection on $K_X$ given by $H_{K_X}$.

Theorems & Definitions (99)

  • Proposition 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Proposition 2.1
  • ...and 89 more