On flops and canonical metrics

Abstract

This article is concerned with an observation for proving non-existence of canonical Kahler metrics. The idea is to use a rather explicit type of degeneration that applies in many situations. Namely, in a variation on a theme introduced by Ross-Thomas, we consider flops of the deformation to the normal cone. This yields a rather widely applicable notion of stability that is still completely explicit and readily computable, but with wider scope. We describe some applications, among them, a proof of one direction of the Calabi conjecture for asymptotically logarithmic del Pezzo surfaces. 1. Motivation and results A variety is slope stable in the sense of Ross-Thomas if, roughly, it is K-stable with respect to degenerations to the normal cone of its subvarieties. This notion has been studied extensively by a number of authors and has yielded many non-existence results for canonical metrics on projective Kahler manifolds. Our main purpose in this article is to introduce a slight variation on this theme by considering a somewhat more involved notion of stability that involves additional flops on the degeneration to the normal cone but that is still geometric and computable, and is partly inspired by the work of Li-Xu. This gives many new non-existence results, and most notably allows us to resolve one direction of the Calabi conjecture for asymptotically logarithmic del Pezzo surfaces. 1.1. Existence theorem for KEE metrics. Kahler-Einstein edge (KEE) metrics are a nat- ural generalization of Kahler-Einstein metrics: they are smooth metrics on the complement of a divisor, and have a conical singularity of angle 2πβ transverse to that 'complex edge'. When β = 1, of course, this is just an ordinary Kahler-Einstein metric, that extends smoothly across the divisor. One can think of the metric as being 'bent' at an angle 2πβ along the divisor. In the case of Riemann surfaces, KEE metrics are just the familiar constant curvature metrics with isolated cone singularities, that have been studied since the late 19th century, e.g., by Picard (25). A basic question, whose origins trace back to Tian's 1994 lectures in the setting of nonpositive curvature (32), extended in Donaldson's 2009 lectures to the setting of anticanonical divisors on Fano manifolds (10), and further extended in our previous work (5) (see also the survey (27, §8)), is the following: Problem 1.1. Under what analytic conditions on the triple (X,D,β) does a KEE metric exist on the Kahler manifold X bent at an angle 2πβ along the divisor D ⊂ X? This is partly motivated by Troyanov's solution in the Riemann surface case (36), and was settled by Jeffres-Mazzeo-Rubinstein (sufficient condition)and Darvas-Rubinstein (sufficient and necessary conditions) in higher dimensions for smooth D (17, 7). These results give an analytic criterion characterizing existence, once the cohomological condition