Blt Azumaya algebras and moduli of maximal orders
Rajesh S. Kulkarni, Max Lieblich

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.
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