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Arakelov Geometry over Adelic Curves

  • Book
  • © 2020

Overview

Authors:
  1. Huayi Chen

    1. Institut de Mathématiques de Jussieu – Paris Rive Gauche, Université de Paris, Paris, France

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  2. Atsushi Moriwaki

    1. Department of Mathematics, Kyoto University, Kyoto, Japan

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  • Introduces a new mathematical theory having strong links with several research domains
  • Opens new research topics with original research results; attracts attention from researchers and graduate students
  • Presents in detail the background and the foundation of an Arakelov theory over adelic curves

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2258)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xviii
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  2. Metrized vector bundles: local theory

    • Huayi Chen, Atsushi Moriwaki
    Pages 1-106

  3. Local metrics

    • Huayi Chen, Atsushi Moriwaki
    Pages 107-165

  4. Adelic curves

    • Huayi Chen, Atsushi Moriwaki
    Pages 167-204

  5. Vector bundles on adelic curves: global theory

    • Huayi Chen, Atsushi Moriwaki
    Pages 205-296

  6. Slopes of tensor product

    • Huayi Chen, Atsushi Moriwaki
    Pages 297-326

  7. Adelic line bundles on arithmetic varieties

    • Huayi Chen, Atsushi Moriwaki
    Pages 327-402

  8. Nakai-Moishezon’s criterion

    • Huayi Chen, Atsushi Moriwaki
    Pages 403-426

  9. Back Matter

    Pages 427-452
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Keywords

About this book

The purpose of this book is to build the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for research on arithmetic geometry in several directions. By adelic curve is meant a field equipped with a family of absolute values parametrized by a measure space, such that the logarithmic absolute value of each non-zero element of the field is an integrable function on the measure space. In the literature, such construction has been discussed  in various settings which are apparently transversal to each other. The authors first formalize the notion of adelic curves and discuss in a systematic way its algebraic covers, which are important in the study of height theory of algebraic points beyond Weil–Lang’s height theory. They then establish a theory of adelic vector bundles on adelic curves, which considerably generalizes the classic geometry of vector bundles or that of Hermitian vector bundles over an arithmetic curve. They focus on ananalogue of the slope theory in the setting of adelic curves and in particular estimate the minimal slope of tensor product adelic vector bundles. Finally, by using the adelic vector bundles as a tool, a birational Arakelov geometry for projective variety over an adelic curve is developed. As an application, a vast generalization of Nakai–Moishezon’s criterion of positivity is proven in clarifying the arguments of geometric nature from several fundamental results in the classic geometry of numbers. 


Assuming basic knowledge of algebraic geometry and algebraic number theory, the book is almost self-contained. It is suitable for researchers in arithmetic geometry as well as graduate students focusing on these topics for their doctoral theses.



Authors and Affiliations

  • Institut de Mathématiques de Jussieu – Paris Rive Gauche, Université de Paris, Paris, France

    Huayi Chen

  • Department of Mathematics, Kyoto University, Kyoto, Japan

    Atsushi Moriwaki

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