Properness for scaled gauged maps

TL;DR

This paper provides an algebraic proof of the properness of moduli stacks of gauged maps, integrating various advanced techniques from algebraic geometry.

Contribution

It introduces a new algebraic proof of properness for gauged maps moduli stacks, combining multiple existing methods in a novel way.

Findings

Properness of moduli stacks of gauged maps established

Unified approach using multiple algebraic geometry techniques

Enhances understanding of stability conditions in gauged maps

Abstract

We give an algebraic proof of properness of moduli stacks of gauged maps satisfying a stability conditition introduced by Mundet and Schmitt. The proof combines a git construction of Schmitt, properness for twisted stable maps by Abramovich-Vistoli, a variation of a boundedness argument due to Ciocan-Fontanine-Kim-Maulik, and a removal of singularities for bundles on surfaces in Colliot-Th\'el\`ene-Sansuc.