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Non-Archimedean volumes of metrized nef line bundles

Sébastien Boucksom, Walter Gubler, Florent Martin

Abstract

Let $L$ be a line bundle on a proper, geometrically reduced scheme $X$ over a non-trivially valued non-Archimedean field $K$. Roughly speaking, the non-Archimedean volume of a continuous metric on the Berkovich analytification of $L$ measures the asymptotic growth of the space of small sections of tensor powers of $L$. For a continuous semipositive metric on $L$ in the sense of Zhang, we show first that the non-Archimedean volume agrees with the energy. The existence of such a semipositive metric yields that $L$ is nef. A second result is that the non-Archimedean volume is differentiable at any semipositive continuous metric. These results are known when $L$ is ample, and the purpose of this paper is to generalize them to the nef case. The method is based on a detailed study of the content and the volume of a finitely presented torsion module over the (possibly non-noetherian) valuation ring of $K$.

Non-Archimedean volumes of metrized nef line bundles

Abstract

Let be a line bundle on a proper, geometrically reduced scheme over a non-trivially valued non-Archimedean field . Roughly speaking, the non-Archimedean volume of a continuous metric on the Berkovich analytification of measures the asymptotic growth of the space of small sections of tensor powers of . For a continuous semipositive metric on in the sense of Zhang, we show first that the non-Archimedean volume agrees with the energy. The existence of such a semipositive metric yields that is nef. A second result is that the non-Archimedean volume is differentiable at any semipositive continuous metric. These results are known when is ample, and the purpose of this paper is to generalize them to the nef case. The method is based on a detailed study of the content and the volume of a finitely presented torsion module over the (possibly non-noetherian) valuation ring of .

Paper Structure

This paper contains 22 sections, 47 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathbb{Q}$-line bundles and positivity in the real Picard group
  4. Lattices and content
  5. Models and metrics
  6. Continuous semipositive metrics and Monge--Ampère measures
  7. Energy
  8. Non-Archimedean volumes
  9. Volumes on torsion schemes
  10. Hilbert--Samuel theory
  11. Global sections in the non-Noetherian case
  12. Asymptotics
  13. Asymptotic cohomological functions on the real Picard group
  14. Volume formulas for nef models and semipositive metrics
  15. An asymptotic volume formula
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $L$ be a line bundle on a reduced proper scheme $X$ over a non-Archimedean field $K$. If $\phi_1, \phi_2$ are two continuous semipositive metrics on $L^{{\mathrm{an}}}$, then

Theorems & Definitions (117)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Definition 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 107 more