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.
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Date
2023Author
Andreasson, Rolf
Keywords
Arakelov geometry, Kähler-Einstein metrics, toric geometry, K-stability, Fano varieties, height bounds
Publication type
licentiate thesis
Language
eng