ALE Calabi-Yau metrics with conical singularities along a compact divisor

Abstract

We construct ALE Calabi-Yau metrics with cone singularities along the exceptional set of resolutions of ${\mathbb{C}}^n/\Gamma$ with non-positive discrepancies. In particular, this includes the case of the minimal resolution of two dimensional quotient singularities for any finite subgroup $\Gamma \subset U(2)$ acting freely on the three-sphere, hence generalizing Kronheimer's construction of smooth ALE gravitational instantons. Finally, we show how our results extend to the more general asymptotically conical setting.

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• On the Yau-Tian-Donaldson conjecture for singular Fano varieties
• Schauder estimates for equations with cone metrics, I
• Construction of constant scalar curvature Kähler cone metrics
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• Kähler metrics with cone singularities along a divisor of bounded Ricci curvature

Cited by in corpus (1)

• The large scale structure of complete $4$-manifolds with nonnegative Ricci curvature and Euclidean volume growth