Sharp bounds on the height of some arithmetic Fano varieties

Abstract

In the framework of Arakelov geometry one can define the height of a polarized arithmetic variety equipped with an hermitian metric over its complexification. When the arithmetic variety is Fano, the complexification is K-semistable and the metrics are normalized in a natural way, we find in this thesis a universal upper bound on the height in a number of cases. For example for the canonical integral model of toric varieties of low dimension (in paper 1) and for general diagonal hypersurfaces (in paper 2). The bound is sharp with equality for the projective space over the integers equipped with a Fubini-Study metric. These results provide positive cases of a conjectural general bound that we introduce, which can be seen as an arithmetic analog of Fujita’s sharp upper bound on the anti-canonical degree of an n-dimensional K-semistable Fano variety in [11]. An extension of the toric result to arbitrary dimension hinges on a conjectural sharp bound for the second largest anti-canonical degree of a toric K-semistable Fano variety in a given dimension. A version of the conjecture for log-Fano pairs is also introduced (in paper 2), which is settled in low dimensions for toric log-pairs and for simple normal crossings hyperplane divisors in projective space. Along the way we define a canonical height of a K-semistable arithmetic (log) Fano variety, making a connection with positively curved (log) Kähler-Einstein metrics.