The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and application is to show the existence of Kähler-Einstein edge metrics on all but finitely many families of asymptotically log del Pezzo surfaces, partially confirming a conjecture of two of us. In an appendix we show that the basis log canonical threshold of Fujita-Odaka coincides with the greatest lower Ricci bound invariant of Tian.is also the necessary and sufficient condition for (b) if µ ≤ 0 [40, Theorem 2]; moreover, a classification (i.e., part (a)) is essentially impossible when µ ≤ 0 [32], [54, §8], and so we will restrict our attention in (a)-(b) exclusively to the case µ > 0, that we have previously called the asymptotically log Fano regime [21, Definition 1.1].Our previous work accomplished (a) in dimension 2, providing a complete classification [21, Theorem 2.1]. Furthermore, we also obtained the "necessary" portion of (b) [21,22], and this was extended to higher dimensions by Fujita [34]. The purpose of this article is to complete the "sufficient" portion of (b) in dimension 2 in all but finitely many (in fact, all but 6) of the (infinite list of) cases classified in [21].
The Calabi problem for asymptotically log Fano varietiesA special class of asymptotically log Fano varieties is as follows. This is a special case of [21, Definition 1.1].Definition 1.1. We say that a pair (X, D) consisting of a smooth projective variety X and a smooth irreducible divisor D on X is asymptotically log Fano if the divisor −K X − (1 − β)D is ample for sufficiently small β ∈ (0, 1].This definition contains the class of smooth Fano varieties (D = 0) as well as the classical notion of a smooth log Fano pair due to Maeda (β = 0) [48].One can show using a result of Kawamata-Shokurov that if (X, D) is asymptotically log Fano then |k(K X + D)| (for some k ∈ N) is free from base points and gives a morphism [21, §1] η : X → Z.The following conjecture, posed in our earlier work, gives a rather complete picture concerning (b) when D is smooth. Conjecture 1.2. [21, Conjecture 1.11] Suppose that (X, D) is an asymptotically log Fano manifold with D smooth and irreducible. There exist KEE metrics with angle 2πβ along D for all sufficiently small β if and only if η is not birational.This conjecture stipulates that the existence problem for KEE metrics in the small angle regime boils down to a simple birationality criterion. In fact, this amounts to computing a single intersection number, i.e., checking whether (K X + D) n = 0.This would be a rather far-reaching simplification as compared to checking the much harder condition of log K-stability that involves, in theory, computing the Futaki invariant of an infinite number of log test configurations, or else estimating the blct which involves, in theory, estimation of singularities of pairs that may occur after an unbounded number of blow-ups.3
