Degenerating conic Kähler–Einstein metrics to the normal cone

Biquard, Olivier; Guenancia, Henri

doi:10.2140/gt.2026.30.247

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Degenerating conic Kähler–Einstein metrics to the normal cone

Olivier Biquard and Henri Guenancia

Geometry & Topology 30 (2026) 247–306

DOI: 10.2140/gt.2026.30.247

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, $\frac{1}{α} (- K_{X})$ with $α > 1$ . It is well known that Kähler–Einstein metrics on $X$ with conic singularities along $D$ may exist only if the angle $2 π β$ is bigger than some positive limit value $2 π β_{*}$ . 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 $β \to β_{*}$ and, in particular, we exhibit the appearance of the Tian–Yau metric of $X ∖ D$ .

Keywords

Kähler–Einstein metrics, degeneration to the normal cone, conical singularities

Mathematical Subject Classification

Primary: 32Q20

References

Publication

Received: 29 August 2024

Revised: 4 April 2025

Accepted: 2 July 2025

Published: 19 January 2026

Proposed: Gang Tian

Seconded: Simon Donaldson, Aaron Naber

Authors

Olivier Biquard
Sorbonne Université Université Paris Cité, CNRS, IMJ-PRG F-75005 Paris France

Henri Guenancia
Univ. Bordeaux CNRS, Bordeaux INP, IMB, UMR 5251 F-33400 Talence France

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Volume 30, issue 1 (2026)