On the Kodaira dimension of maximal orders

Let k be an algebraically closed field of characteristic zero and K a finitely generated field over k. Let Sigma be a central simple K-algebra, X a normal projective model of K and A a sheaf of maximal O-x-orders in Sigma. There is a ramification Q-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 K. 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 bcanonical 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. In proving our main result, we establish existence of equivariant b-terminal resolutions of G-b-log pairs and we also find two variants of the Riemann-Hurwitz formula. The first variant applies to effective embeddings of central simple algebras with fixed centres while the second applies to the pullback of a central simple algebra by a Galois extension of its centre. We also give two different local characterizations of effective embeddings. The first is in terms of complete local invariants, while the second uses Galois cohomology. (C) 2021 Elsevier Inc. All rights reserved.