New gap principle for semiabelian varieties using globally valued fields

TL;DR

This paper establishes a new gap principle for semiabelian varieties over globally valued fields, reducing the Bogomolov conjecture to the abelian case, and proves an unconditional result in positive characteristic for certain cases.

Contribution

It introduces a novel gap principle for semiabelian varieties and reduces the Bogomolov conjecture to the abelian case over globally valued fields, including positive characteristic.

Findings

Reduces Bogomolov conjecture for semiabelian varieties to abelian varieties over GVFs.

Proves an unconditional gap principle in positive characteristic for semiabelian varieties with elliptic abelian quotients.

Establishes equivalence between Gao-Ge-Kühne gap principle and Bogomolov conjecture over GVFs.

Abstract

Hrushovski observed that the new gap principle of Gao-Ge-K\"uhne is essentially equivalent to the Bogomolov conjecture over arbitrary globally valued fields of characteristic $0$. Building on this observation, we prove a new gap principle for semiabelian varieties by reducing the Bogomolov conjecture for semiabelian varieties to the Bogomolov conjecture for abelian varieties over arbitrary GVFs. This reduction remains valid in positive characteristic; however, the corresponding Bogomolov conjecture for abelian varieties is not yet known in that setting. We prove an unconditional new gap principle in positive characteristic for semiabelian varieties whose abelian quotient is an elliptic curve.