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A local version of the arithmetic Hodge index theorem over quasiprojective varieties

Marc Abboud

Abstract

We define a local intersection number for metrised line bundles over quasiprojective varieties with compact support and show the local arithmetic Hodge index theorem for this intersection number. As a consequence we obtain a uniqueness result for the Monge-Ampère equation over quasiprojective varieties within a certain class of solutions both in the archimedean and non-archimedean setting.

A local version of the arithmetic Hodge index theorem over quasiprojective varieties

Abstract

We define a local intersection number for metrised line bundles over quasiprojective varieties with compact support and show the local arithmetic Hodge index theorem for this intersection number. As a consequence we obtain a uniqueness result for the Monge-Ampère equation over quasiprojective varieties within a certain class of solutions both in the archimedean and non-archimedean setting.

Paper Structure

This paper contains 20 sections, 20 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Local intersection number for metrised line bundles over quasiprojective varieties
  3. An arithmetic Hodge index theorem over quasiprojective varieties
  4. Arithmetic divisors and metrised line bundles
  5. Terminology
  6. Divisors and line bundles
  7. Berkovich spaces
  8. Arithmetic divisors and metrised line bundles
  9. Model Green functions
  10. Positivity
  11. Chambert-Loir measures
  12. Local intersection number
  13. Arithmetic divisors and metrised line bundles over quasiprojective varieties.
  14. Chambert-Loir measures
  15. Definition of the intersection number
  16. ...and 5 more sections

Key Result

Theorem A

Let $U$ be a normal quasiprojective variety over a complete field $\mathbf{K}_v$. Let $\widehat{\mathop{\mathrm{Pic}}\nolimits} (U)_{int}$ be the set of integrable metrised line bundles over $U$ and let $\widehat{\mathop{\mathrm{Pic}}\nolimits}_c (U)$ be the set of integrable compactly supported ver defined by eq:def-intersection-number. It is linear in the first variable and multilinear and symme

Theorems & Definitions (32)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 2.1
  • Example 2.2
  • Proposition 2.3: gublerLocalHeightsSubvarieties1998 Lemma 7.8
  • Proposition 2.4: gublerLocalHeightsSubvarieties1998 Theorem 7.12
  • Proposition 2.5: Corollaire 6.4.4 of chambert-loirFormesDifferentiellesReelles2012
  • Definition 2.6
  • Theorem 2.7: Theorem 1.2 of guoIntegrationFormulaChern2025
  • ...and 22 more