Algebraic surfaces and hyperbolic geometry
type: Item
instance of: scholarly article
zbMATH Open document ID: 1253.14040
author: Burt Totaro
publication date: 2012-10-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1008.3825
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 32Q99
zbMATH DE Number: 6092086
zbMATH Keywords: algebraic surfaces
zbMATH Keywords: line bundles
MaRDI profile type: Publication
arXiv ID: 1008.3825
arXiv classification: math.AG
arXiv classification: math.GR
title: Algebraic surfaces and hyperbolic geometry
review text: The author surveys the Morrison-Kawamata cone conjecture both for Calabi-Yau varieties and Calabi-Yau pairs. Many explicit examples are provided in order to clarify definitions and stated facts.NEWLINENEWLINEIn the two dimensional case, the solution of such a conjecture is stated [\textit{H. Sterk}, Math. Z. 189, 507--513 (1985; Zbl 0545.14032)], \textit{Y. Namikawa} [Math. Ann. 270, 201--222 (1985; Zbl 0536.14024)] and \textit{Y. Kawamata} [Int. J. Math. 8, No. 5, 665--687 (1997; Zbl 0931.14022)]. An outline of the proof of this result is also given. The author provides an extension of the above result to the case of Calabi-Yau pairs, obtained by the author [Duke Math. J. 154, No. 2, 241--263 (2010; Zbl 1203.14046)], and an outline of the proof is also given.NEWLINENEWLINEThe author also provides conditions in order to construct \(K3\) surfaces whose group of automorphisms is a discrete group not commensurable to an arithmetic group. Explicit examples are provided. In the last example, the author considers an explicit smooth rational surface (obtained by blowing-up of \(12\) points of the complex projective plane) whose group of automorphisms is \(\Gamma=\mathrm{PGL}(2,\rho)\), where \(\rho=e^{2 \pi i/3}\). The group \(\Gamma\) acts on the hyperbolic \(3\)-space as a discrete group and it contains an index \(24\) subgroup which uniformizes the complement of the well-known figure-eight knot.NEWLINENEWLINEThe paper is nicely written and very friendly to non-specialists.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14002].
reviewed by: Rubén A. Hidalgo
description: scientific article; zbMATH DE number 6092086