Towards the Colmez Conjecture

We prove a collection of results involving Colmez's periods and the Colmez Conjecture. Using Colmez's theory of periods of CM abelian varieties, we propose a definition for the height of a partial CM-type and prove that the Colmez conjecture follows from an arithmetic period formula for surfaces. We give an explicit conjecture for the form of this period formula, which relates the height of special points on a Shimura surface with special values of $L$-functions. Further, we relate the heights of periods given by Colmez to arithmetic degree of Hermitian line bundles and thus give a formulation of Colmez's full conjecture in geometric terms.