On the desingularisation of moduli of principal bundles

TL;DR

This paper constructs smooth, modular compactifications of moduli spaces of stable principal bundles for certain algebraic groups, providing canonical desingularizations of the semistable moduli spaces, extending previous work on SL(2).

Contribution

It introduces a new method to produce canonical desingularizations for moduli of principal bundles for specific algebraic groups, generalizing prior desingularization results.

Findings

Provides smooth modular compactifications for principal H-bundles.

Offers canonical desingularizations of semistable moduli spaces.

Extends previous desingularization techniques to new algebraic groups.

Abstract

In \cite{nr} Narasimhan and Ramanan and in \cite{desing}, Seshadri constructed desingularisations of the moduli space $M^{ss}_{_{\text{SL}(2)}}$ of semistable $\SL(2)$-bundles on a smooth projective curve $C$ of genus $g \geq 3$. Seshadri's construction was even modular and canonical. In this paper, we construct a smooth modular compactification of the moduli of stable principal $H$-bundles when $H$ is a simply connected almost simple algebraic group of type ${\tt B}_{_\ell}, {\tt D}_{_\ell}, {\tt G}_{_2}, {\tt F}_{_4}~~or~~{\tt C}_{_3}$. These spaces give canonical desingularisations of the moduli space $M^{ss}_{_H}$ of semistable principal $H$-bundles and thereby, a comprehensive generalisation of \cite{desing}.