1. Introduction

Many properties of a projective algebraic variety can be encoded by convex cones, such as the ample cone and the cone of curves. This is especially useful when these cones have only finitely many edges, as happens for Fano varieties. For a broader class of varieties which includes Calabi-Yau varieties and many rationally connected varieties, the Kawamata-Morrison cone conjecture predicts the structure of these cones. I like to think of this conjecture as what comes after the abundance conjecture. Roughly speaking, the cone theorem of Mori, Kawamata, Shokurov, Kollár, and Reid describes the structure of the curves on a projective variety X on which the canonical bundle Kx has negative degree; the abundance conjecture would give strong information about the curves on which Kx has degree zero; and the cone conjecture fully describes the structure of the curves on which Kx has degree zero.

We give a gentle summary of the proof of the cone conjecture for algebraic surfaces, with plenty of examples [Totaro 2010]. For algebraic surfaces, these cones are naturally described using hyperbolic geometry, and the proof can also be formulated in those terms.

Example 7.3 shows that the automorphism group of a K3 surface need not be commensurable with an arithmetic group. This answers a question by Barry Mazur [1993, Section 7].

Thanks to John Christian Ottem, Artie Prendergast-Smith, and Marcus Zibrowius for their comments.

2. The main trichotomy

Let X be a smooth complex projective variety. There are three main types of varieties. (Not every variety is of one of these three types, but minimal model theory relates every variety to one of these extreme types.)