This chapter essentially consists of three parts. In the first part (§ 4.1), we discuss
conjectures and results in the literature which give bounds, or formulae in terms of L-
functions, for “Faltings heights.” We also mention various applications of such conjectures
and results. The second part (§ 4.2 and § 4.3) is devoted to the work of Yuan-Zhang [60] in
which they proved the averaged Colmez conjecture. Here we discuss the main ideas and
concepts used in their proof, and we explain in detail various constructions in [60]. In the …

I his chapter essentially consists of three parts. In the first part (94.1), we discuss conjectures
and results in the literature which give bounds, or formulae in terms of^-functions, for"
Fallings heights." We also mention various applications of such conjectures and results. The
second part (§ 4.2 and § 4.3) is devoted to the work of Yuan-Zhang [60] in which they proved
the averaged Colmez conjecture. Here we discuss the main ideas and concepts used in
their proof, and we explain in detail various constructions in [60]. In the third part (§ 4.4), we …