Height and GIT weight

In this paper, we first establish a connection between the weight in the geometric invariant theory and the height introduced by Cornalba and Harris and Zhang. Then we give two applications. First, it provides a converse of Cornalba and Harris’s result, which can be treated as a function field analog of Zhang’s theorem over number field. 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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Published 27 December 2012