Martin Orr's Blog

The Faltings height of an abelian variety over the rationals

Posted by Martin Orr on Thursday, 17 November 2011 at 15:58

The Faltings height is a real number attached to an abelian variety (defined over a number field), which is at the centre of Faltings' proof of Finiteness Theorem I. In this post all I will do is define the Faltings height of an abelian variety over \mathbb{Q}, as already this requires a lot of preliminaries on cotangent and canonical sheaves of schemes. Further complications arise over other base fields, which I will discuss next time.

For an abelian variety A over \mathbb{Q}, the Faltings height is the (logarithm of the) volume of A as a complex manifold with respect to a particular volume form, chosen using the \mathbb{Q}-structure of A. The preliminaries are needed in order to choose the volume form.

Faltings' proof of Finiteness I proceeds by showing that for any fixed number field, there are finitely many abelian varieties of bounded Faltings height. This is done by showing that the Faltings height is not far away from the classical height of a point representing the abelian variety in the moduli space \mathcal{A}_g. Then he shows that the Faltings height is bounded within an isogeny class. Both of these parts are difficult.

5 comments Tags abelian-varieties, alg-geom, faltings, maths, number-theory