Log terminal orders are numerically rational

TL;DR

This paper introduces the concept of numerical rationality for noncommutative surface orders and proves that log terminal orders from the noncommutative minimal model program are numerically rational, extending classical results.

Contribution

It generalizes rational singularities to noncommutative orders and establishes their invariance under resolution, connecting noncommutative geometry with classical singularity theory.

Findings

Numerical rationality is independent of resolution.

Log terminal orders are numerically rational.

Canonical orders are examples of numerically rational orders.

Abstract

Noncommutative surfaces finite over their centres can be realised as orders over surfaces. The aim of this paper is to present a noncommutative generalisation of rational singularities, which we call numerical rationality, for such orders. We show that numerical rationality is independent of the choice of resolution. Our main result is that the log terminal orders arising from the noncommutative minimal model program, in particular, canonical orders are numerically rational. Both of these generalise well known facts about rational singularities in commutative algebraic geometry.