Mathematical Research Letters

Volume 19 (2012)

Number 4

Height and GIT weight

Pages: 909 – 926

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n4.a14

Author

Xiaowei Wang (Department of Math, The Chinese University of Hong Kong, Shatin, NT, Hong Kong)

Abstract

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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