Successive minima and asymptotic slopes in Arakelov Geometry
| Autor: | François Ballaÿ |
|---|---|
| Přispěvatelé: | Beijing International Center for Mathematical Research, Peking University [Beijing] |
| Rok vydání: | 2020 |
| Předmět: |
adelic line bundles and divisors
Divisor successive minima hermitian vector bundles 01 natural sciences Upper and lower bounds Combinatorics Mathematics - Algebraic Geometry 0103 physical sciences FOS: Mathematics Projective space Number Theory (math.NT) 0101 mathematics Global field Algebraic Geometry (math.AG) Projective variety Mathematics 14G40 (Primary) 11G50 11H06 (Secondary) Algebra and Number Theory Conjecture essential minimum Mathematics - Number Theory Okounkov bodies Height 010102 general mathematics 16. Peace & justice Hermitian matrix [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] 2010 MSC : 14G40 (primary) 11G50 11H06 (secondary) transference theorems 010307 mathematical physics [MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] Vector space |
| Zdroj: | Compositio Mathematica Compositio Mathematica, Foundation Compositio Mathematica, 2021, 157 (6), pp.1302-1339. ⟨10.1112/S0010437X21007156⟩ |
| ISSN: | 0010-437X 1570-5846 |
| DOI: | 10.48550/arxiv.2002.06026 |
| Popis: | Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\overline{D}$ be an adelic Cartier divisor on $X$. We prove a conjecture of Chen, showing that the essential minimum $\zeta_{\mathrm{ess}}(\overline{D})$ of $\overline{D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $\zeta_{\mathrm{ess}}(\overline{D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom--Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion for equality generalizing to arbitrary projective varieties a result of Burgos Gil, Philippon and Sombra concerning toric metrized divisors on toric varieties. When applied to a projective space $X = \mathbb{P}_K^d$, our main result has several applications to the study of successive minima of hermitian vector spaces. We obtain an absolute transference theorem with a linear upper bound, answering a question raised by Gaudron. We also give new comparisons between successive slopes and absolute minima, extending results of Gaudron and R\'emond. Comment: 34 pages. Minor revisions in the introduction, results unchanged |
| Databáze: | OpenAIRE |
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