A note on polarizations of finitely generated fields

In our previous paper, we established Northcott's theorem for height functions over finitely generated fields. Unfortunately, Northcott's theorem on finitely generated fields does not hold in general. Actually, it depends on the choice of a polarization. In this short note, we will propose a weaker condition of the polarization to guarantee Northcott's theorem. We will also show the generalization of conjectures of Bogomolov and Lang in under the weaker condition.