Degenerating conic Kähler-Einstein metrics to the normal cone

2024/07/01 by Biquard, Olivier, Guenancia, Henri
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.01150

Abstract

Let X be a Fano manifold of dimension at least 2 and D be a smooth divisor in a multiple of the anticanonical class, \frac1α(-K_X) with α>1. It is well-known that Kähler-Einstein metrics on X with conic singularities along D may exist only if the angle 2πβ is bigger than some positive limit value 2πβ_*. Under the hypothesis that the automorphisms of D are induced by the automorphisms of the pair (X,D), we prove that for β>β_* close enough to β_*, such Kähler-Einstein metrics do exist. We identify the limits at various scales when β→β_* and, in particular, we exhibit the appearance of the Tian-Yau metric of X∖ D.

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