The Kobayashi-Hitchin correspondence for nef and big classes

TL;DR

This paper proves a comprehensive version of the Kobayashi-Hitchin correspondence for nef and big classes, extending it to singular settings and establishing conditions for the existence and uniqueness of adapted Hermitian-Yang-Mills metrics.

Contribution

It introduces the notion of adapted currents and metrics, proving the correspondence for nef and big classes, including in singular contexts, and establishing projective flatness under certain conditions.

Findings

Proves the Kobayashi-Hitchin correspondence for nef and big classes.

Establishes the existence and uniqueness of adapted Hermitian-Yang-Mills metrics.

Shows projective flatness of certain polystable bundles when Bogomolov-Gieseker equality holds.

Abstract

This paper provides a complete proof of the Kobayashi-Hitchin correspondence for nef and big classes. We introduce the notion of an adapted closed positive $(1,1)$-current $T$ lying in a nef and big class $\alpha$, and that of a $T$-adapted Hermitian-Yang-Mills metric of a holomorphic vector bundle. Then we prove that a holomorphic vector bundle $E$ over a compact K\"{a}hler manifold $X$ is slope polystable with respect to a nef and big class $\alpha$ if and only if $E$ admits a $T$-adapted Hermitian-Yang-Mills metric for every adapted current $T$ in $\alpha$. Furthermore, we also establish the uniqueness of a $T$-adapted Hermitian-Yang-Mills metric on $E$ when it exists. Our main theorem above immediately implies that the Kobayashi-Hitchin correspondence holds even in singular settings. In particular, this singular Kobayashi-Hitchin correspondence applies to reflexive sheaves over compact normal K\"{a}hler varieties with log terminal singularities endowed with singular K\"{a}hler-Einstein metrics. Furthermore, the singular Kobayashi-Hitchin correspondence proves that the graded sheaf associated to a Jordan-H${\rm \ddot{o}}$lder filtration of a semistable sheaf for a nef and big class $\alpha$ admits a $T$-adapted Hermitian-Yang-Mills metric. As an application of the Kobayashi-Hitchin correspondence, we show that if a holomorphic vector bundle ${E}$ is slope polystable with respect to a nef and big class $\alpha$ and $E$ attains the equality of the Bogomolov-Gieseker inequality with respect to $\alpha$, then ${E}$ is projectively flat on the ample locus of $\alpha$. All these results are new even if compact K\"{a}hler manifolds are projective and nef and big classes are the 1st Chern class of nef and big line bundles. One of key features of our approach is that the adapted current $T$ need not be strictly positive and its singularities need not be described explicitly.