Calabi–Yau metrics with conical singularities along line arrangements

Abstract

Given a finite collection of lines $L_j \subset \mathbb{CP}^2$ together with real numbers $0 \lt \beta_j \lt 1$ satisfying natural constraint conditions, we show the existence of a Ricci–flat Kähler metric $g_{RF}$ with cone angle $2\pi\beta_j$ 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 $g_{RF}$ as a logarithmic Euler characteristic with points weighted according to the volume density of the metric.