Adelic Line Bundles on Quasi-Projective Varieties

Xinyi Yuan, Shou-Wu Zhang

• Publisher: Princeton University Press
• Year: 2026
• Print ISBN:
  9780691271736
  9780691271729
• eBook ISBN:
  9780691278704
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

A comprehensive new theory of adelic line bundles on quasi-projective varieties over finitely generated fields

This book introduces a comprehensive theory of adelic line bundles on quasi-projective varieties over finitely generated fields, developed in both geometric and arithmetic contexts. In the geometric setting, adelic line bundles are defined as limits of line bundles on projective compactifications under the boundary topology. In the arithmetic setting, they are defined as limits of Hermitian line bundles on projective arithmetic compactifications, also under the boundary topology. After establishing these foundational definitions, the book uses the theory to explore key concepts such as intersection theory, effective sections, volumes, and positivity of adelic line bundles. It also applies these results to study height functions of algebraic points and prove an equidistribution theorem on quasi-projective varieties. This theory has broad applications in the study of numerical, dynamical, and Diophantine properties of moduli spaces, quasi-projective varieties, and varieties over finitely generated fields.

Table of contents

• Cover
• Contents
• Preface
• 1. Introduction
• 2. Adelic Divisors and Adelic Line Bundles
• 3. Interpretation by Berkovich Spaces
• 4. Intersection Theory
• 5. Volumes and Heights
• 6. Algebraic Dynamics
• A Review on Heights and Arithmetic Intersection Theory
• Bibliography
• Index