On the Kodaira dimension of maximal orders

TL;DR

This paper explores the relationship between the Kodaira dimension of maximal orders in central simple algebras and the Iitaka dimension of associated log pairs, establishing birational invariance and properties of embeddings.

Contribution

It introduces the concept of Kodaira dimension for orders, proves its birational invariance under certain conditions, and relates it to the Iitaka dimension and transcendence degree in the context of central simple algebras.

Findings

The Kodaira dimension of orders is birationally invariant with b-canonical singularities.

The Kodaira dimension cannot decrease under effective embeddings of central simple algebras.

The paper establishes a connection between GK dimension, Iitaka dimension, and transcendence degree for these orders.

Abstract

Let $\kk$ be an algebraically closed field of characteristic zero and $\KK$ a finitely generated field over $\kk$. Let $\Sigma$ be a central simple $\KK$-algebra, $X$ a normal projective model of $\KK$ and $\Lambda$ a sheaf of maximal $\Osh_X$-orders in $\Sigma$. There is a ramification $\QQ$-divisor $\Delta$ on $X$, which is related to the canonical bimodule $\omega_\Lambda$ by an adjunction formula. It only depends on the class of $\Sigma$ in the Brauer group of $\KK$. When the numerical abundance conjecture holds true, or when $\Sigma$ is a central simple algebra, we show that the Gelfand-Kirillov dimension (or GK dimension) of the canonical ring of $\Lambda$ is one more than the Iitaka dimension (or D-dimension) of the log pair $(X,\Delta)$. In the case that $\Sigma$ is a division algebra, we further show that this GK dimension is also one more than the transcendence degree of the division algebra of degree zero fractions of the canonical ring of $\Lambda$. We prove that these dimensions are birationally invariant when the b-log pair determined by the ramification divisor has b-canonical singularities. In that case we refer to the Iitaka (or D-dimension) of $(X,\Delta)$ as the Kodaira dimension of the order $\Lambda$. For this, we establish birational invariance of the Kodaira dimension of b-log pairs with b-canonical singularities. We also show that the Kodaira dimension can not decrease for an embedding of central simple algebras, finite dimensional over their centres, which induces a Galois extension of their centres, and satisfies a condition on the ramification which we call an effective embedding. For example, this condition holds if the target central simple algebra has the property that its period equals its index.