Asymptotically log Fano varieties

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 Kähler--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.

v2: added references

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Cited by in corpus (19)

• Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
• Existence and deformations of Kahler-Einstein metrics on smoothable Q-Fano varieties
• Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces
• Smooth and singular Kahler-Einstein metrics
• Degenerating Kähler-Einstein cones, locally symmetric cusps, and the Tian-Yau metric
• Alpha invariants and K-stability for general polarisations of Fano varieties
• Polystable log Calabi-Yau varieties and Gravitational instantons
• Small angle limits of Hamilton's footballs
• Tian's properness conjectures: an introduction to Kahler geometry
• High-dimensional convex sets arising in algebraic geometry
• On flops and canonical metrics
• On the uniform K-stability for some asymptotically log del Pezzo surfaces
• Classification of strongly asymptotically log del Pezzo flags and surfaces
• Delta invariants of smooth cubic surfaces
• On log K-stability for asymptotically log Fano varieties
• On K-polystability for log del Pezzo pairs of Maeda type
• On the body of ample angles of asymptotically log Fano varieties
• Small angle limits of negatively curved Kahler-Einstein metrics with crossing edge singularities
• Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics