Algebraic surfaces and hyperbolic geometry (Q2918467)

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].