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Admissible HYM metrics on klt KE varieties and the MY equality for big anticanonical K-stable varieties

Satoshi Jinnouchi

TL;DR

This paper proves three main results about stability and metrics on singular varieties. It shows that on a compact klt Kähler–Einstein variety, a reflexive sheaf that is stable with respect to the singular KE metric $ω$ admits a $ω$-admissible Hermitian–Yang–Mills metric, establishing a HYM correspondence in the singular setting. It further proves a structure theorem: if a projective klt variety with big $-K_X$ is K-stable and attains the Miyaoka–Yau equality, its anticanonical model $Z$ has a quasi-étale cover by $ℂP^n$, indicating a CP^n-type rigidity under equality. Finally, it provides an explicit example of a rank-3 bundle that is semistable for a nef and big class but not for any ample class, illustrating non-openness of semistability and the nuanced behavior of stability in the nef/big versus ample regime. Collectively, the results integrate tame approximations, positive products, and stability theory to illuminate HYM metrics and the birational structure of big anti-canonical varieties.

Abstract

This short note includes three results: $(1)$ If a reflexive sheaf $\mathcal{E}$ on a log terminal Kähler-Einstein variety $(X,ω)$ is slope stable with respect to a singular Kähler-Einstein metric $ω$, then $\mathcal{E}$ admits an $ω$-admissible Hermitian-Yang-Mills metric. $(2)$ If a K-stable log terminal projective variety with big anti-canonical divisor satisfies the equality of the Miyaoka-Yau inequality in the sense of \cite{IJZ25}, then its anti-canonical model admits a quasi-étale cover from $\mathbb{C}P^n$. $(3)$ There exists a holomorphic rank 3 vector bundle on a compact complex surface which is semistable for some nef and big line bundle, but it is not semistable for any ample line bundles.

Admissible HYM metrics on klt KE varieties and the MY equality for big anticanonical K-stable varieties

TL;DR

This paper proves three main results about stability and metrics on singular varieties. It shows that on a compact klt Kähler–Einstein variety, a reflexive sheaf that is stable with respect to the singular KE metric admits a -admissible Hermitian–Yang–Mills metric, establishing a HYM correspondence in the singular setting. It further proves a structure theorem: if a projective klt variety with big is K-stable and attains the Miyaoka–Yau equality, its anticanonical model has a quasi-étale cover by , indicating a CP^n-type rigidity under equality. Finally, it provides an explicit example of a rank-3 bundle that is semistable for a nef and big class but not for any ample class, illustrating non-openness of semistability and the nuanced behavior of stability in the nef/big versus ample regime. Collectively, the results integrate tame approximations, positive products, and stability theory to illuminate HYM metrics and the birational structure of big anti-canonical varieties.

Abstract

This short note includes three results: If a reflexive sheaf on a log terminal Kähler-Einstein variety is slope stable with respect to a singular Kähler-Einstein metric , then admits an -admissible Hermitian-Yang-Mills metric. If a K-stable log terminal projective variety with big anti-canonical divisor satisfies the equality of the Miyaoka-Yau inequality in the sense of \cite{IJZ25}, then its anti-canonical model admits a quasi-étale cover from . There exists a holomorphic rank 3 vector bundle on a compact complex surface which is semistable for some nef and big line bundle, but it is not semistable for any ample line bundles.
Paper Structure (8 sections, 12 theorems, 28 equations)

This paper contains 8 sections, 12 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sobolev inequality and mean value inequality
  4. Positive product
  5. Slope stability and the HYM metrics
  6. Admissible HYM metric for klt Kähler-Einstein metrics
  7. Structure theorem when MY equality holds for big anticanonical K-stable varieties
  8. Example of a semistable vector bundle for a nef line bundle on a complex surface

Key Result

Theorem 1.1

Let $X$ be a compact normal analytic variety with klt singularities. Suppose that $X$ admits a singular Kähler-Einstein metric $\omega$. Then, if a reflexive sheaf $\mathcal{E}$ on $X$ is $\{\omega\}^{n-1}$-slope stable, then $\mathcal{E}$ admits an $\omega$-admissible Hermitian-Yang-Mills metric.

Theorems & Definitions (27)

  • Theorem 1.1: =Theorem \ref{['lt HE']}
  • Theorem 1.2: =Theorem \ref{['BG ineq']}
  • Theorem 1.3: =Proposition \ref{['eg nef']}
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2: CCHTST25 Definition 1.4
  • Theorem 2.3: CCHTST25 Theorem 1.2, Proposition 3.1, GPSS23 Theorem 2.1
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6: DHY23 Lemma A.5, see also IJZ25 Proposition 3.15
  • ...and 17 more