  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.
