Asymptotically conical Ricci flat Kähler metrics on $\mathbb{C}^2$ with cone singularities along a complex curve

Abstract: We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.