Positivity in Arakelov Geometry over Adelic Curves Hilbert-Samuel Formula and Equidistribution Theorem

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.

Introduction.- Review and Preliminaries.- Normed Graded Linear Series over a Trivially Valued Field.- Arithmetic Volumes over a General Adelic Curve.- Hilbert-Samuel Property.- Relative Ampleness and Nefness.- Global Adelic Space of an Arithmetic Variety.- Generically Big and Pseudo-effective Adelic Line Bundles.- Global Positivity Conditions.- Appendix A: Some Slope Estimates.

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.