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

TL;DR

This paper establishes the existence of asymptotically conical Ricci-flat Kähler metrics with cone singularities along a smooth complex curve in 2, potentially serving as blow-up limits of certain Kähler Einstein metrics.

Contribution

It proves an existence theorem for such metrics in 2 with cone singularities, advancing understanding of geometric limits in Kähler geometry.

Findings

Existence of asymptotically conical Ricci-flat Kähler metrics with cone singularities.

Metrics arise as blow-up limits of Kähler Einstein metrics with cone singularities.

Provides a new class of explicit geometric structures in complex differential geometry.

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.