The cone conjecture for Calabi-Yau pairs in dimension two

By B. Totaro. 2009

74 citations recorded by the source corpus.

We prove the Morrison-Kawamata cone conjecture for klt Calabi-Yau pairs in dimension 2. That is, for a large class of rational surfaces as well as K3 surfaces and abelian surfaces, the action of the automorphism group of the surface on the convex cone of ample divisors has a rational polyhedral fundamental domain. More concretely: there many be infinitely many curves with negative self-intersection on the surface, but all such curves fall into finitely many orbits under the automorphism group of the surface. The proof uses the geometry of groups acting on hyperbolic space. We deduce a characterization of the surfaces in this class with finitely generated Cox ring.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e6af28a4d1f678656e620e5bea4b22c5b1a42ffd

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In Geometry and Topology

Citations in the corpus

Lectures on K3 Surfaces (2016, background): “The more general case of klt Calabi–Yau pairs in dimension two is the subject of Totaro’s [600].”
A survey of Torelli and monodromy results for holomorphic-symplectic varieties (2011, background): “4), the Enriques surfaces case by Namikawa ([Nam], Theorem 1.4), the case of abelian and hyperelliptic surfaces by Kawamata ([Ka], Theorem 2.1), the case of two-dimensional Calabi-Yau pairs by Totaro [Tot], and the case of abelian varieties by Prendergast-Smith [Pre]. A version of the conjectures fo”
A note on the cone conjecture for K3 surfaces in positive characteristic (2011, result): “While the authors are optimistic that several of Totaro’s more general results [21] should extend using the methods here, higher-dimensional results remain mysterious even in characteristic 0.”