The Height of CM Points on Orthogonal Shimura Varieties and Colmez’s Conjecture

These are the notes for a 6 h course given by the author during the summer school “Géométrie d’Arakelov”, organized by the Institut Fourier (Grenoble) in the summer of 2017. They are based on the paper “Faltings’ Heights of Abelian Varieties with Complex Multiplication” by myself, Eyal Goren, Ben Howard and Keerthi Madapusi Pera and on notes by myself and Eyal Goren. No new results are presented. The goal is to describe the strategy to reduce the proof of an averaged version of Colmez’s conjecture to a conjecture of Bruinier, Kudla and Yang, an instance of what is known as the Kudla’s program.