Blt Azumaya algebras and moduli of maximal orders

TL;DR

This paper investigates the moduli spaces of maximal orders in ramified division algebras over surfaces, introducing a refined moduli problem via blt Azumaya algebras that admits a compactification with a virtual fundamental class.

Contribution

It proposes a Kollár-type condition to refine the moduli of maximal orders, leading to a better-behaved moduli stack of blt Azumaya algebras with a compactification.

Findings

Existence of a refined moduli stack of blt Azumaya algebras

The refined moduli stack admits a compactification

Construction of a virtual fundamental class for the compactified moduli

Abstract

We study moduli spaces of maximal orders in a ramified division algebra over the function field of a smooth projective surface. As in the case of moduli of stable commutative surfaces, we show that there is a Koll\'ar-type condition giving a better moduli problem with the same geometric points: the stack of blt Azumaya algebras. One virtue of this refined moduli problem is that it admits a compactification with a virtual fundamental class.