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

TL;DR

This paper explores the existence of Hermitian-Yang-Mills metrics on sheaves over K"ahler-Einstein varieties, establishes conditions for the Miyaoka-Yau equality on K-stable varieties, and provides a counterexample in vector bundle stability.

Contribution

It introduces new conditions for admissible Hermitian-Yang-Mills metrics, characterizes varieties satisfying the Miyaoka-Yau equality, and constructs a novel example of vector bundle semistability behavior.

Findings

Stable sheaves admit admissible Hermitian-Yang-Mills metrics.

K-stable varieties with big anti-canonical divisors satisfy the Miyaoka-Yau equality.

Existence of a rank 3 bundle semistable for nef and big line bundle but not for ample ones.

Abstract

This short note includes three results: $(1)$ If a reflexive sheaf $\mathcal{E}$ on a log terminal K\"{a}hler-Einstein variety $(X,\omega)$ is slope stable with respect to a singular K\"{a}hler-Einstein metric $\omega$, then $\mathcal{E}$ admits an $\omega$-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-\'{e}tale 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.