Abstract 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.
Let [Formula: see text] and [Formula: see text] be two Lie algebras over an arbitrary field [Formula: see text], and let [Formula: see text] be the semidirect sum of [Formula: see text] by [Formula: see text]. In this paper, we give the structure of derivation algebra of [Formula: see text]; then as a consequence we illustrate the structure and dimension derivation algebra of Heisenberg Lie algebras.
In this paper, a simple Lie algebra, referred to as the completed Witt Lie algebra, is introduced. Its derivation algebra and automorphism group are completely described. As a by-product, it is obtained that the first cohomology group of this Lie algebra with coefficients in its adjoint module is trivial. Furthermore, we completely determine the conjugate classes of this Lie algebra under its automorphism group, and also obtain that this Lie algebra does not contain any nonzero ad -locally finite element.
Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones