For big compactified S-metrized divisors on quasi-projective varieties over adelic curves, concave transforms are constructed and shown to satisfy arithmetic volume and Hilbert-Samuel formulas.
Abstract divisorial spaces and arithmetic intersection numbers
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abstract
Yuan and Zhang introduced arithmetic intersection numbers for adelic line bundles on quasi-projective varieties over a number field. Burgos and Kramer generalized this approach allowing more singular metrics at archimedean places. We introduce abstract divisorial spaces as a tool to generalize these arithmetic intersection numbers to the setting of a proper adelic base curve in the sense of Chen and Moriwaki. We also allow more singular metrics at non-archimedean places using relative mixed energy there as well.
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Concave transforms of compactified S-metrized divisors
For big compactified S-metrized divisors on quasi-projective varieties over adelic curves, concave transforms are constructed and shown to satisfy arithmetic volume and Hilbert-Samuel formulas.