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On the Kobayashi-Hitchin correspondence for Kähler currents

Satoshi Jinnouchi

TL;DR

This work extends the Kobayashi-Hitchin correspondence to Kähler currents representing nef and big classes. By solving a Monge-Ampère equation with controlled singularities, the authors construct a nef+big current $T$ whose analytic structure supports a $T$-admissible Hermitian-Yang-Mills metric that exists for $\alpha^{n-1}$-slope polystable bundles. The existence is established via a three-step program leveraging Uhlenbeck compactness, limiting objects $E_{\infty}$ and nontrivial morphisms, and a rank-extension argument to conclude $\Psi_{\infty}$ is an isomorphism. The results bridge stability theory, singular Kähler geometry, and gauge theory, with applications to Gromov-Hausdorff limits and minimal model program contexts.

Abstract

In this paper, we show that if a holomorphic vector bundle is slope polystable with respect to a Kähler class, then it admits a Hermitian-Yang-Mills metric with respect to a suitable Kähler current with singularities in higher codimension which represents the Kähler class. Most parts of the proof remains valid for closed positive $(1,1)$-currents representing a nef and big class.

On the Kobayashi-Hitchin correspondence for Kähler currents

TL;DR

This work extends the Kobayashi-Hitchin correspondence to Kähler currents representing nef and big classes. By solving a Monge-Ampère equation with controlled singularities, the authors construct a nef+big current whose analytic structure supports a -admissible Hermitian-Yang-Mills metric that exists for -slope polystable bundles. The existence is established via a three-step program leveraging Uhlenbeck compactness, limiting objects and nontrivial morphisms, and a rank-extension argument to conclude is an isomorphism. The results bridge stability theory, singular Kähler geometry, and gauge theory, with applications to Gromov-Hausdorff limits and minimal model program contexts.

Abstract

In this paper, we show that if a holomorphic vector bundle is slope polystable with respect to a Kähler class, then it admits a Hermitian-Yang-Mills metric with respect to a suitable Kähler current with singularities in higher codimension which represents the Kähler class. Most parts of the proof remains valid for closed positive -currents representing a nef and big class.

Paper Structure

This paper contains 12 sections, 29 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. closed positive $(1,1)$-currents and big cohomology classes
  4. nonpluripolar products and positive products
  5. solvability of Monge-Ampere equations in big classes and regularity results
  6. Slope stability and the admissible HYM connections
  7. Existence of admissible HYM metrics
  8. choice of positive currents
  9. Proof of Theorem \ref{['toward nef and big class']}
  10. Uhlenbeck limit of the HYM connections
  11. nontrivial limit of $\Phi_i$
  12. proof of the main result

Key Result

Theorem 1.2

Let $(Y,\omega_Y)$ be a compact Kähler manifold. Let $\omega$ be the solution to the complex Monge-Ampère equation $\omega^n=e^F\omega_Y^n$, where $F$ satisfies the following conditions: Then, if a holomorphic vector bundle $F$ on $Y$ is $\{\omega\}^{n-1}$-slope polystable, then $F$ admits an $\omega$-admissible HYM metric.

Theorems & Definitions (58)

  • Definition 1.1
  • Theorem 1.2: = Theorem \ref{['main thm thm']}
  • Remark 1.3
  • Theorem 1.5: =Theorem \ref{['limiting objects']}
  • Lemma 1.6: =Corollary \ref{['L-infty estimate of limit object']}, Proposition \ref{['uniform esti of diff']}
  • Definition 2.1
  • Definition 2.2: Bou04
  • Proposition 2.3: Bou04
  • Proposition 2.4: Bou04
  • Lemma 2.5
  • ...and 48 more