| 概要 | The Bogomolov conjecture is known as a conjecture
insisting that a closed subvarity of
an abelian variety should not have a lot of small
points unless it is an exceptional kind of subvariety.
It is well-known that this conjecture has already been
proved in the arithmetic setting but has not yet been
established in the geometric setting.
In this talk, we will formulate the geometric
Bogomolov conjecture for abelian
varieties, and give some partial answers when
there is a place at which the closed
subvariety is sufficiently degenerate in some sense.
In the proof, the equidistribution theory plays an
important role, simlarly to the arithmetic case.
From the techinical aspect in our discussion, we have to
study the minimal dimension of the components of a
canonical measure on the tropicalization of the closed subvariety.
Then we can apply the tropical version of the
equidistribution theory by Gubler to obtain our result. |