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Positivity in Arakelov Geometry over Adelic Curves

Hilbert-Samuel Formula and Equidistribution Theorem

  • Book
  • Sep 2024


  • Provides new results on the arithmetic positivities of Arakelov geometry over adelic curves
  • Highlights connections with several classical topics in Arakelov geometry and Diophantine geometry
  • Includes detailed proofs and explanations

Part of the book series: Progress in Mathematics (PM, volume 355)

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About this book

This monograph presents new research on Arakelov geometry over adelic curves, a novel theory of arithmetic geometry developed by the authors. It explores positivity conditions and establishes the Hilbert-Samuel formula and the equidistribution theorem in the context of adelic curves. Connections with several classical topics in Arakelov geometry and Diophantine geometry are highlighted, such as the arithmetic Hilbert-Samuel formula, positivity of line bundles, equidistribution of small subvarieties, and theorems resembling the Bogomolov conjecture. Detailed proofs and explanations are provided to ensure the text is accessible to both graduate students and experienced researchers.


  • Arakelov geometry
  • adelic curves
  • Hilbert-Samuel formula
  • adelic line bundle
  • arithmetic intersection
  • positivity conditions
  • Fujita approximation

Authors and Affiliations

  • Institute for Theoretical Sciences & School of Science, Westlake University, Hangzhou, China

    Huayi Chen

  • School of General Education, Chubu University, Kasugai, Japan

    Atsushi Moriwaki

About the authors

Huayi Chen has obtained his doctoral degree in mathematics from Ecole Polytechnique in 2006. He has been associate professor at Université Paris Diderot from 2008 to 2012, professor at Université Grenoble Alpes from 2012 to 2016, and then professor at Université Paris Cité from 2016 to 2023. From 2024, he works at Westlake University as chair professor in mathematics. His research interests include Arakelov geometry, geometry of numbers and convex geometry.


Atsushi Moriwaki is a mathematician specializing in algebraic geometry and arithmetic geometry, especially Arakelov geometry. He earned his Ph.D. in Mathematics from Kyoto University in 1991 and went on to serve as a professor at Kyoto University from 2003 to 2023. In 2023, he was appointed as a professor emeritus at Kyoto University and started a new role as a professor at Chubu University.

Bibliographic Information

  • Book Title: Positivity in Arakelov Geometry over Adelic Curves

  • Book Subtitle: Hilbert-Samuel Formula and Equidistribution Theorem

  • Authors: Huayi Chen, Atsushi Moriwaki

  • Series Title: Progress in Mathematics

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024

  • Hardcover ISBN: 978-3-031-61667-9Due: 20 September 2024

  • Softcover ISBN: 978-3-031-61670-9Due: 20 September 2025

  • eBook ISBN: 978-3-031-61668-6Due: 20 September 2024

  • Series ISSN: 0743-1643

  • Series E-ISSN: 2296-505X

  • Edition Number: 1

  • Number of Pages: IX, 253

  • Number of Illustrations: 7 b/w illustrations

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