Non-Archimedean volumes of metrized nef line bundles

TL;DR

This paper extends the understanding of non-Archimedean volumes of line bundles to nef cases, showing their equivalence with energy and differentiability at semipositive metrics, generalizing previous ample cases.

Contribution

It generalizes key properties of non-Archimedean volumes from ample to nef line bundles, including their relation to energy and differentiability at semipositive metrics.

Findings

Non-Archimedean volume equals energy for semipositive metrics.

Non-Archimedean volume is differentiable at semipositive metrics.

Results extend known properties from ample to nef line bundles.

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 $K$.