Toric Sasaki-Einstein metrics with conical singularities

Abstract

We show that any toric Kähler cone with smooth compact cross-section admits a family of Calabi-Yau cone metrics with conical singularities along its toric divisors. The family is parametrized by the Reeb cone and the angles are given explicitly in terms of the Reeb vector field. The result is optimal, in the sense that any toric Calabi-Yau cone metric with conical singularities along the toric divisor (and smooth elsewhere) belongs to this family. We also provide examples and interpret our results in terms of Sasaki-Einstein metrics.

Moment polytopes on Sasaki manifolds and volume minimization

“…By the proof of Proposition 1.1, we have only to show Φ(q) = 0 for the affine map Φ defined (4.3). This proof is motivated by the computations in [10]. First of all, by Donaldson's expression of the obstruction in [11]…”

“…dx is the average (Kähler geometers') scalar curvature, which is equal to m(m+1) for ω ∈ 2πc B 1 (S)/(m+1). As shown in [10], Lemma 3.8, σ ξ is expressed using the distinguished point q by (4.7)…”

“…Therefore there are two possible extensions of these studies, one on the Kähler cone and the other on the Kähler local orbit spaces of the Reeb flow. In [10], de Borbon and Legendre used the volume minimization argument to prove the existence on toric Kähler cone manifolds of Ricci-flat Kähler cone metrics with cone angle along the boundary invariant divisors without assuming the Calabi-Yau condition of the Kähler cone. This Calabi-Yau condition will be explained in the paragraph after Proposition 1.1 below.…”

“…The existence of −γ is also known in algebraic geometry of toric varieties, see [9], Theorem 4.2.8. The paper [10] also gives an account from the broader view points of what they call angle cone.…”