Properness for scaled gauged maps

We prove properness of moduli stacks of gauged maps satisfying a stability conditition introduced by Mundet [42], Schmitt [48] and Ziltener [59]. The proof combines a git construction of Schmitt [48], properness for twisted stable maps by Abramovich-Vistoli [1], a variation of a boundedness argument due to Ciocan–Fontanine–Kim–Maulik [13], and a removal of singularities for bundles on surfaces in Colliot–Thelene–Sansuc [14].