Two remarks on asymptotically log Fano pairs

TL;DR

This paper discusses asymptotically log Fano pairs, their role in K-stability, and summarizes joint work on classifying two-dimensional cases known as asymptotically log del Pezzo pairs.

Contribution

It provides a classification of two-dimensional asymptotically log Fano pairs, advancing understanding in the context of K-stability and log Calabi Yau approximations.

Findings

Classification of 2D asymptotically log Fano pairs achieved

Insights into their role in K-stability

Connection to log Calabi Yau pairs clarified

Abstract

Asymptotically log Fano pairs were introduced by Cheltsov and Rubinstein, generalising a definition of Maeda. They have received attention in the last decade within the theory of K-stability, as they approximate log Calabi Yau pairs while staying in the log Fano setting. In this note, written for the ZAG Proceedings, we summarise our talk on 1st September 2020 (Day of Knowledge) given on Zoom during the 24-hour ZAG Marathon. In the talk, we reported on joint work with P. Cascini and Y. Rubinstein, on the classification of the two-dimensional case, known as asymptotically log del Pezzo pairs.