An analogue of the Narasimhan-Seshadri theorem and some applications

TL;DR

This paper extends the classical Narasimhan-Seshadri theorem to higher-dimensional varieties, providing new proofs, effective versions, and existence results for strongly stable bundles and principal G-bundles over different fields.

Contribution

It establishes an analogue of the Narasimhan-Seshadri theorem in higher dimensions and applies it to prove new existence and effectiveness results for stable bundles and principal G-bundles.

Findings

New proof of Balaji and Kollár's main theorem

Effective version of the stability theorem

Existence of strongly stable principal G-bundles with full holonomy

Abstract

We prove an analogue in higher dimensions of the classical Narasimhan-Seshadri theorem for strongly stable vector bundles of degree 0 on a smooth projective variety $X$ with a fixed ample line bundle $\Theta$. As applications, over fields of characteristic zero, we give a new proof of the main theorem in a recent paper of Balaji and Koll\'ar and derive an effective version of this theorem; over uncountable fields of positive characteristics, if $G$ is a simple and simply connected algebraic group and the characteristic of the field is bigger than the Coxeter index of $G$, we prove the existence of strongly stable principal $G$ bundles on smooth projective surfaces whose holonomy group is the whole of $G$.