On Kodaira dimension of maximal orders

Nathan Grieve, Colin Ingalls

Published 2016-11-30Version 1

We define and study a concept of Kodaira dimension for a maximal order $\Lambda$ in $\Sigma$ a $\KK$-central simple algebra. Here $\KK$ is the function field of $X$ a normal projective variety over an algebraically closed field of characteristic zero. Our point of view is to define this concept, which we denote by $\kappa(X,\alpha)$, in terms of the growth of the log pair $(X,\Delta_\alpha)$ determined by the ramification of $\alpha \in \Br(\KK)$ the class of $\Sigma$ in the Brauer group of $\KK$. We then study the nature of $\kappa(X,\alpha)$ in various geometric and algebraic situations. For instance, we show that it does not decrease under embeddings of $\KK$-central division algebras. Further, we formulate and establish a notion of birational invariance for $\kappa(X,\alpha)$ under the hypothesis that the log pair $(X,\Delta_\alpha)$ is $\QQ$-Gorenstein. On the other hand, we also establish a Reimann-Hurwitz type theorem for those finite morphisms which have target space $X$ and source the normalization of $X$ in a finite field extension of $\KK$. We then show how this result can be used to study the nature of Kodaira dimensions for those extensions of division algebras which induce a Galois extension of their centres.