Calabi–Yau metrics with conical singularities along line arrangements

Given a finite collection of lines L_j ⊂ CP² together with real numbers 0 < βj < 1 satisfying natural constraint conditions, we show the existence of a Ricci-flat Kähler metric gRF with cone angle 2πβ_j along each line L_j asymptotic to a polyhedral Kähler cone at each multiple point. Moreover, we discuss a Chern-Weil formula that expresses the energy of gRF as a logarithmic Euler characteristic with points weighted according to the volume density of the metric.