Abstract
In this note, we rst establish a connection between the weight in the geo- metric invariant theory and the height introduced by Cornalba and Harris (CH) and Zhang (Z). Then we give two applications. First, it provides a converse of Cornalba and Harris's result, which can be treated as a function eld analogue of Zhang's theorem over number eld. In particular, this connection gives a numerical interpretation of the moral stability = positivity that was advocated by Viehweg. Second, we relate these to the study the positivity of CM-line bundle introduced by Tian and the determinant line bundle introduced by Donaldson.
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