Non-Archimedean volumes of metrized nef line bundles
S\'ebastien Boucksom, Walter Gubler, Florent Martin

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 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…
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Taxonomy
Topicsadvanced mathematical theories · Geometry and complex manifolds · Algebraic Geometry and Number Theory
