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Concave transforms of compactified S-metrized divisors

Debam Biswas, Yulin Cai

Abstract

We associate a concave transform to any compactified S-metrized divisor on a quasi-projective variety over an adelic curve. Then we show a Hilbert-Samuel type formula for relatively nef compactified S-metrized YZ-divisors.

Concave transforms of compactified S-metrized divisors

Abstract

We associate a concave transform to any compactified S-metrized divisor on a quasi-projective variety over an adelic curve. Then we show a Hilbert-Samuel type formula for relatively nef compactified S-metrized YZ-divisors.

Paper Structure

This paper contains 35 sections, 35 theorems, 323 equations.

Table of Contents

  1. Introduction
  2. Okounkov bodies in algebraic geometry
  3. Concave transforms in local case for projective varieties
  4. Concave transforms in Arakelov theory for projective varieties
  5. Arakelov theory for line bundles with singular metrics
  6. Main results
  7. Adelic curves and adelic vector bundles
  8. Adelic Curves
  9. Adelic vector bundles
  10. Purification
  11. Harder-Narasimhan Filtration
  12. Graded algebra of adelic vector bundles
  13. Over global fields
  14. Review of compactified $S$-metrized divisors
  15. Compactified (geometric) divisors
  16. ...and 20 more sections

Key Result

Theorem A

Let $\overline{V_\bullet}=\{\overline{V_m}\}_{m\in\mathbb{N}}$ be the graded $K$-algebra of adelic vector bundles associated to $\overline{D}$ defined in def:volume. If $D$ is big, then and with equality if $\widehat{\mu}_{\min}^{\mathrm{asy}}(\overline{D})>-\infty$, where $d\lambda$ is the standard Lebesgue measure on $\Delta(D)\subseteq \mathbb{R}^d$.

Theorems & Definitions (107)

  • Theorem A: \ref{['theorem:concavemain']}
  • Theorem B: \ref{['thm:Hilbert-Samuel formula']}
  • Theorem C: \ref{['theorem:equidsitributionforfinitenergy']}
  • Definition 2.1.1: chen2020arakelov § 3.1
  • Remark 2.1.2
  • Definition 2.3.2
  • Remark 2.3.3
  • Lemma 2.3.4
  • proof
  • Definition 2.5.2: chen2020arakelov Definition 6.3.24
  • ...and 97 more