Asymptotically log del Pezzo surfaces

Abstract: Asymptotically log Fano varieties are a type of log smooth log pairs of varieties of Fano pairs introduced by Cheltsov and Rubinstein when studying the existence of Kaehler-Einstein metrics with conical singularities of maximal angle. From an MMP point of view they are strictly log canonical and as such, they do not belong to a finite number of families. However, one may hope to give a fairly explicit classification for them in low dimensions. An asymptotically log Fano variety, has an associated convex object known as the body of ample angles. Cheltsov and Rubinstein classified strongly asymptotically log del Pezzo surfaces. These are two-dimensional asymptotically log Fano varieties for which the body of ample angles is maximal around the origin. This apparently technical condition has striking consequences both for the structure and birational geometry of these surfaces, making all minimal asymptotically log del Pezzo surfaces to have rank at most two. The latter condition is what allowed Cheltsov and Rubinstein to give a full classification of asymptotically log del Pezzo surfaces. In this talk, we introduce these notions while attacking the more general problem of classifying asymptotically log del Pezzo surfaces. We further show that the body of ample angles is in fact a convex polytope.

algebraic geometry

Audience: researchers in the topic

( video )

Fano Varieties and Birational Geometry