Asymptotically log Fano varieties

TL;DR

This paper introduces and classifies asymptotically log Fano varieties, explores their properties in dimension two, and formulates an asymptotic version of Calabi's conjecture for del Pezzo surfaces, advancing understanding of Kähler--Einstein edge metrics.

Contribution

It defines asymptotically log Fano varieties, provides a complete classification in dimension two, and formulates an asymptotic Calabi conjecture with initial progress on its proof.

Findings

Complete classification of two-dimensional strongly asymptotically log smooth log Fano varieties.

Demonstrated existence and non-existence results for Kähler--Einstein edge metrics.

Generalized Matsushima's result on automorphism groups.

Abstract

Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimensional strongly asymptotically log smooth log Fano varieties. Based on this classification we formulate an asymptotic logarithmic version of Calabi's conjecture for del Pezzo surfaces for the existence of K\"ahler--Einstein edge metrics in this regime. We make some initial progress towards its proof by demonstrating some existence and non-existence results, among them a generalization of Matsushima's result on the reductivity of the automorphism group of the pair, and results on log canonical thresholds of pairs. One by-product of this study is a new conjectural picture for the small angle regime and limit which reveals a rich structure in the asymptotic regime, of which a folklore conjecture concerning the case of a Fano manifold with an anticanonical divisor is a special case.