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

We construct ALE Calabi-Yau metrics with cone singularities along the exceptional set of resolutions of ${\mathbb{C}}^n\mathrm{/}\mathrm{\Gamma }$ 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.