K\"ahler-Einstein Metrics for some quasi smooth log del Pezzo Surfaces

TL;DR

This paper investigates the existence of K"ahler-Einstein metrics and tigers on certain quasi smooth log del Pezzo surfaces, building on recent classifications and applying advanced analytical techniques.

Contribution

It provides definitive results on the existence of K"ahler-Einstein metrics and tigers for previously unresolved quasi smooth log del Pezzo surfaces.

Findings

Confirmed existence of K"ahler-Einstein metrics for new classes of surfaces.

Identified surfaces that do not admit tigers.

Extended previous classifications with new existence results.

Abstract

Recently Johnson and Koll\'ar determined the complete list of anticanonically embedded quasi smooth log del Pezzo surfaces in weighted projective 3-spaces. They also proved that many of those surfaces admit a K\"ahler-Einstein metric, and that some of them do not have tigers. The aim of this paper is to settle the question of the existence of K\"ahler-Einstein metrics and tigers for those surfaces for which the question was still open. In order to do so, we will use techniques developed earlier by Nadel and Demailly and Koll\'ar.