The talk is based on a joint work with Gregory Edwards (University of Notre-Dame). We produce Ricci-flat Kähler metrics, in two complex dimensions, with cone singularities along three or more intersecting complex lines. We concentrate in the case when the angles strictly do not satisfy the Troyanov condition. As a result, we identify the tangent cone at the intersection point.
We first produce an approximate solution, with the desired singular behaviour. The main work is on inverting the Laplacian of such singular metrics.