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The Canonical Metric on Holomorphic Pairs over Compact Non-Kähler Manifolds

Ryoma Saito

TL;DR

The paper extends the Kobayashi-Hitchin correspondence to holomorphic pairs on compact Gauduchon (non-Kähler) manifolds by solving the vortex equation via a Li-Yau style continuity method. It introduces a perturbed equation L_ε(f)=0 and shows solvability and uniform estimates, enabling a limit as ε→0 that yields a τ-Hermitian-Einstein metric for τ-stable pairs; crucially, the approach does not rely on simplicity. A key innovation is the construction of destabilizing subsheaves when stability fails, clarifying the stability-destabilty dichotomy in this non-Kähler context. The work also extends these results to Higgs bundles, providing a Higgs bundle version of the τ-Hermitian-Einstein metric result and broadening the Kobayashi-Hitchin framework to Gauduchon manifolds.

Abstract

In this paper, we prove the solvability of the vortex equation on a holomorphic vector bundle over a compact Hermitian manifold using the continuity method, and show the Kobayashi-Hitchin correspondence for holomorphic pairs. This work extends Bradlow's Kobayashi-Hitchin correspondence over compact Kähler manifolds to compact non-Kähler manifolds.

The Canonical Metric on Holomorphic Pairs over Compact Non-Kähler Manifolds

TL;DR

The paper extends the Kobayashi-Hitchin correspondence to holomorphic pairs on compact Gauduchon (non-Kähler) manifolds by solving the vortex equation via a Li-Yau style continuity method. It introduces a perturbed equation L_ε(f)=0 and shows solvability and uniform estimates, enabling a limit as ε→0 that yields a τ-Hermitian-Einstein metric for τ-stable pairs; crucially, the approach does not rely on simplicity. A key innovation is the construction of destabilizing subsheaves when stability fails, clarifying the stability-destabilty dichotomy in this non-Kähler context. The work also extends these results to Higgs bundles, providing a Higgs bundle version of the τ-Hermitian-Einstein metric result and broadening the Kobayashi-Hitchin framework to Gauduchon manifolds.

Abstract

In this paper, we prove the solvability of the vortex equation on a holomorphic vector bundle over a compact Hermitian manifold using the continuity method, and show the Kobayashi-Hitchin correspondence for holomorphic pairs. This work extends Bradlow's Kobayashi-Hitchin correspondence over compact Kähler manifolds to compact non-Kähler manifolds.

Paper Structure

This paper contains 8 sections, 26 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The existence of $\tau$-Hermitian-Einstein Metrics in $\tau$-Stable Bundles
  4. The Solvability of the Perturbed Equation
  5. The Construction of a Destabilizing Subsheaf
  6. The Solvability of the Vortex Equation
  7. The Uniqueness of $\tau$-Hermitian-Einstein Metrics
  8. Higgs Bundles Version

Key Result

Theorem 1.1

If a holomorphic pair $(E,\phi)$ over an $n$-dimensional compact Gauduchon manifold $(X,\omega_g)$ is $\tau$-stable, then it admits a unique $\tau$-Hermitian-Einstein metric.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Bradlow MR1085139
  • Definition 2.2: Bradlow MR1085139
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • ...and 36 more