Bigness of the Admissible Canonical Bundle

The admissible canonical bundle of a curve over a number field was first introduced by Shou-Wu Zhang and used to study the Bogomolov conjecture. In this talk, I will introduce a family version of the admissible canonical bundle, talk about its bigness, and deduce the uniform Bogomolov conjecture from the bigness. Previously, the uniform Bogomolov conjecture was proved by Dimitrov-Gao-Habegger and Kuhne, and used together with Vojta's inequality to prove a uniform Mordell conjecture.

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