Arakelov type inequalities for Hodge bundles

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Arakelov-type inequalities for Hodge bundles Chris Peters Department of Mathematics University of Grenoble I UMR 5582 CNRS-UJF 38402-Saint-Martin d'Heres, France June 20 2000 Prepublication de l'Institut Fourier no 511 (2000) 0. Introduction The inequalities from the title refer back to Arakelov's article [Arakelov]. The main result of that paper is: Theorem. Fix a complete curve C of genus > 1 and a finite set S of points on C. There are at most finitely many non-isotrivial families of curves of given genus over C that are smooth over C \ S. The proof consists of two parts. First one proves that there are only finitely many such families (this is a boundedness statement) by bounding the degree d of the relative canonical bundle in terms of the genus p of C, the genus g of the fiber and the cardinality of the set S : 0 ≤ d ≤ (2p? 2 + _S) g 2 . The second part consists of establishing rigidity for a non-isotrivial family. It follows upon iden- tifying the deformation space of the family with the H1 of the inverse of the relative canonical bundle, which is shown to be ample. Kodaira vanishing then completes the proof. This approach can be carried out for other situations as well.

  • unipotent local

  • can also

  • higgs bundle

  • over

  • higgs bundles

  • hodge metric

  • called higgs bundle

  • compact curve


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§0.
Arakelov-type for Hodge
inequalities bundles
Chris Peters Department of Mathematics University of Grenoble I UMR 5582 CNRS-UJF 38402-Saint-MartindHe`res,France
June 20 2000
o Pr´epublicationdelInstitutFouriern511(2000) http://www-fourier.ujf-grenoble.fr/prepublications.html
Introduction
The inequalities from the title refer back to Arakelov’s article [Arakelov]. The main result of that paper is:
Theorem.Fix a complete curveCof genus>1and a finite setSof points onCare. There at most finitely many non-isotrivial families of curves of given genus overCthat are smooth over C\S.
The proof consists of two parts. First one proves that there are only finitely many such families (this is aboundednessstatement) by bounding the degreedof the relative canonical bundle in terms of the genuspofC, the genusgof the fiber and the cardinality of the setS: g 0d(2p2 + #S). 2 The second part consists of establishingrigidityIt follows upon iden-for a non-isotrivial family. 1 tifying the deformation space of the family with theHof the inverse of the relative canonical bundle, which is shown to be ample. Kodaira vanishing then completes the proof. This approach can be carried out for other situations as well. In fact [Faltings] deals with the case of abelian varieties and shows that boundedness always holds and that for rigidity one has to impose further conditions besides non-isotriviality. Subsequently the rigidity statement has been generalized in [Peters90] and using his result, the case of K3-surfaces, resp. Abelian varieties could be treated completely by Saito and Zucker in [Saito-Zucker], resp. by Saito in [Saito].
Keywords:Variations of Hodge structure, Higgs bundles, Higgs field, period map, Hodge bundles. AMS Classification:14D07, 32G20
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