This note is an elaborated version of the talk with the same title I gave in the January 2006 workshop at the HRI. The talk was aimed at (complex) algebraic geometers with little background in Hodge theory. In the talk, I have tried to give as much details from Hodge theory so as to allow the audience to follow the main arguments. Much of these details are presented here as well. I try to avoid...

This expository article introduces some conjectures and theorems related to the Faltings heights of abelian varieties with complex multiplication. The topics include the Colmez conjecture, the averaged Colmez conjecture, and the André–Oort conjecture.

1) The Nullstellensatz and the R-Nullstellensatz. 2) Chevalley’s Theorem: the image of a constructible set is constructible. 3) (Grothendieck, Ax) An injective polynomial map from C to C is surjective. 4) Hilbert’s 17th problem: a positive semidefinite rational function f ∈ R(t1, . . . , tn) is a sum of squares. 5) Polynomially compact operators have invariant subspaces. (Let V be a complex Hil...

To a symmetric, relatively ample line bundle on an abelian scheme one can associate a linear combination of the determinant bundle and the relative canonical bundle, which is a torsion element in the Picard group of the base. We improve the bound on the order of this element found by Faltings and Chai. In particular, we obtain an optimal bound when the degree of the line bundle d is odd and the...

We generalize the construction of dual BGG complexes in Faltings–Chai and Mokrane–Tilouine–Polo (from the case of Siegel modular varieties) to all smooth integral models of PEL-type Shimura varieties.