Dynamic alpha-invariants of del Pezzo surfaces with boundary

TL;DR

This paper computes the global log canonical threshold and dynamic alpha-invariant of del Pezzo surfaces, providing insights into conditions for Kahler-Einstein metrics with edge singularities and extending previous classifications.

Contribution

It extends Cheltsov's computation of log canonical thresholds to non-singular del Pezzo surfaces over arbitrary algebraically closed fields and classifies very singular anticanonical pairs.

Findings

Computed the global log canonical threshold for various del Pezzo surfaces.

Determined the dynamic alpha-invariant for all smooth del Pezzo surfaces with elliptic boundary curves.

Provided conditions under which Kahler-Einstein metrics with edge singularities exist.

Abstract

The global log canonical threshold (or Tian's alpha-invariant) plays an important role in the geometry of Fano varieties. Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to determine when a smooth Fano variety X admits a Kahler-Einstein metric. It was conjectured that the existence of such a metric is equivalent to X being K-stable, an algebraic-geometric property. A crucial step in Donaldson's programme consists on finding a Kahler-Einstein metric with edge singularities of small angle along a smooth anticanonical boundary. Jeffres, Mazzeo and Rubinstein showed that a dynamic version of the alpha-invariant could be used to find such metrics. The global log canonical threshold measures how anticanonical pairs fail to be log canonical. In this thesis we compute the global log canonical threshold of del Pezzo surfaces in various settings. We extend Cheltsov's computation of the global log canonical threshold of complex del Pezzo surfaces to non-singular del Pezzo surfaces over a ground field which is algebraically closed and has arbitrary characteristic. Then we study which anticanonical pairs fail to be log canonical, giving a classification of very singular anticanonical pairs for del Pezzo surfaces of small degree. We conjecture under which circumstances such a classification is plausible for an arbitrary Fano variety and derive consequences. As an application, we compute the dynamic alpha-invariant on smooth del Pezzo surfaces of small degree with any smooth elliptic curve as boundary. The main result of this thesis is a computation of the dynamic alpha-invariant on all smooth del Pezzo surfaces with boundary any smooth elliptic curve C. The values of the alpha-invariant depend on the choice of C. We apply our computation to find Kahler-Einstein metrics with edge singularities.