{
  "id": "1611.10278",
  "version": "v1",
  "published": "2016-11-30T17:33:40.000Z",
  "updated": "2016-11-30T17:33:40.000Z",
  "title": "On Kodaira dimension of maximal orders",
  "authors": [
    "Nathan Grieve",
    "Colin Ingalls"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-30T17:33:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kodaira dimension",
      "maximal order",
      "log pair",
      "central simple algebra",
      "central division algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}