Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Continuity of volumes on arithmetic varieties


Author: Atsushi Moriwaki
Journal: J. Algebraic Geom. 18 (2009), 407-457
DOI: https://doi.org/10.1090/S1056-3911-08-00500-6
Published electronically: May 13, 2008
MathSciNet review: 2496453
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Abstract: We introduce the volume function for $ C^{\infty}$-hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence, we have the arithmetic Hilbert-Samuel formula for a nef $ C^{\infty}$-hermitian invertible sheaf. We also give other applications, for example, a generalized Hodge index theorem, an arithmetic Bogomolov-Gieseker's inequality, etc.


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Atsushi Moriwaki
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502, Japan
Email: moriwaki@math.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/S1056-3911-08-00500-6
Received by editor(s): January 22, 2007
Received by editor(s) in revised form: September 14, 2007
Published electronically: May 13, 2008

American Mathematical Society