A Generalization of Principal Bundles With a Parabolic or Level Structure

TL;DR

This paper introduces a unified stability framework for principal bundles with local structures, generalizing parabolic and level structures, and constructs their moduli space using decorated tumps.

Contribution

It defines a new parameter-dependent stability notion for decorated principal bundles and constructs their coarse moduli space, extending existing theories.

Findings

Established a stability condition for decorated principal bundles.

Constructed the moduli space of decorated tumps.

Linked stable decorated principal bundles to asymptotic stability of decorated tumps.

Abstract

We define a parameter dependent notion of stability for principal bundles with a certain local decoration, which generalizes both parabolic and level structures, and construct their coarse moduli space. A necessary technical step is the construction of the moduli space of tuples of vector bundles with a global and a local decoration, which we call decorated tumps. We introduce a notion of asymptotic stability for decorated tumps and show, that stable decorated principal bundles can be described as asymptotically stable decorated tumps.