Geometric Bogomolov conjecture for nowhere degenerate abelian varieties of dimension $5$ with trivial trace

We prove that the geometric Bogomolov conjecture holds for nowhere degenerate abelian varieties of dimension $5$ with trivial trace. By this result together with our previous work, we see that the conjecture holds for an abelian variety such that the difference between the dimension of its maximal nowhere degenerate abelian subvariety and that of its trace equals $5$.

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This work was initiated during my visit to the university of Regensburg in March–April 2015, which was supported by the SFB Higher Invariants. I thank ProfessorWalter Gubler for inviting me and for his hospitality. I thank him also for his comments. This work was partly supported by KAKENHI 26800012.

Received 23 September 2015

Accepted 10 May 2016

Published 29 December 2017