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Zbl 0915.14014
Gasbarri, Carlo
Canonical heights on the moduli space of stable vector bundles over an algebraic curve. (Hauteurs canoniques sur l'espace de modules des fibrés stables sur une courbe algébrique.) (French)
[J] Bull. Soc. Math. Fr. 125, No.4, 457-491 (1997). ISSN 0037-9484

Consider a number field $K$ with ring of integers $O_K$ and a smooth $K$-curve $X_K$ of positive genus which extends to a smooth proper scheme $X$ over $B=\text{Spec }O_K$ having a section $B\to X$. Given a vector bundle $\cal E$ of rank $r>1$ and degree $d$ over $X$, Arakelov theory teaches one how to metrise it in a canonical fashion and how to define Chern classes ${\widehat c}_i(\widehat{\cal E})$ for the metrised bundle $\widehat{\cal E}$. Miyaoka, Moriwaki and Soulé have shown independently that if the restriction of $\cal E$ to $X_K$ is stable, then the real number $2r{\widehat c}_2(\widehat{\cal E})-(r-1)({\widehat c}_1(\widehat{\cal E}),{\widehat c}_1(\widehat{\cal E}))$ is always non-negative [see e.g. {\it C. Soulé}, Invent. Math. 116, No. 1-3, 577-599 (1994; Zbl 0834.14013)]. In the present paper, the author shows that in the case when $r$ and $d$ are coprime, the above expression can be used to define a height function on the (fine) moduli space ${\cal M}_{X_K}(r, {\cal F}_K)$ of stable vector bundles over $X_K$ of rank $r$, degree $d$ and determinant isomorphic to a fixed line bundle ${\cal F}_K$, by a procedure analogous to the Faltings-Hriljac interpretation of the canonical height over the jacobian of $X_K$. He also includes a detailed account of a direct approach to the construction of the moduli space of stable vector bundles over the arithmetic surface $X$, needed for obtaining a $B$-model of ${\cal M}_{X_K}(r, {\cal F}_K)$ over which the Arakelov-theoretic construction can be carried through.
[T.Szamuely (Budapest)]

MSC 2000:: *14G40 Arithmetic varieties and schemes
14H10 Families, algebraic moduli (curves)
14D20 Algebraic moduli problems

Keywords: height functions; Arakelov theory; arithmetic surfaces; fine moduli spaces; vector bundles on curves

Citations: Zbl 0834.14013

Cited in: Zbl 0955.14018

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