We construct and classify, in the case of two complex dimensions, the
possible tangent cones at points of limit spaces of non-collapsed sequences of
K\"ahler-Einstein metrics with cone singularities.Comment: Reference to Panov's Polyhedral Kahler Manifolds adde

de Borbon, Martin

Complex Manifolds

English

arXiv

The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q

Singularities of plane complex curves and limits of K\"ahler metrics with cone singularities. I: Tangent Cones

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