Calabi-Yau metrics with conical singularities along line arrangements

Abstract

Given a weighted line arrangement in the projective plane, with weights satisfying natural constraint conditions, we show the existence of a Ricci-flat Kähler metric with cone singularities along the lines asymptotic to a polyhedral Kähler cone at each multiple point. Moreover, we discuss a Chern-Weil formula that expresses the energy of the metric as a `logarithmic' Euler characteristic with points weighted according to the volume density of the metric.

arXiv commentFinal version. Accepted in the Journal of Differential Geometry