We associate to certain filtrations of a graded linear series of a big line bundle a concave function on its Okounkov body, whose law with respect to the Lebesgue measure describes the asymptotic distribution of the jumps of the filtration. As a consequence, we obtain a Fujita-type approximation theorem in this general filtered setting. We then specialize these results to the filtrations by minima in the usual context of Arakelov geometry (and for more general adelically normed graded linear series), thereby obtaining in a simple way a natural construction of an arithmetic Okounkov body, the existence of the arithmetic volume as a limit and an arithmetic Fujita approximation theorem for adelically normed graded linear series. We also obtain an easy proof of the existence of the sectional capacity previously obtained by Lau, Rumely and Varley.
ARITHMETIC FUJITA APPROXIMATION H CHEN A.-We prove an arithmetic analogue of Fujita's approximation theorem in Arakelov geometry, conjectured by Moriwaki, by using measures associated to R-filtrations. R.-On démontre un analogue arithmétique du théorème d'approximation de Fujita en géométrie d'Arakelov-conjecturé par Moriwaki-par les mesures associées aux R-filtrations.
Résumé. -On reformule la théorie des polygones de Harder-Narasimhan par le langage des R-filtrations. En utlisant une variante du lemme de Fekete et un argument combinatoire des monômes, onétablit la convergence uniforme des polygones associés a une algèbre graduée munie des filtrations. Cela conduità l'existence de plusieur invariants arithmétiques dont un cas très particulier est la capacité sectionnelle. Deux applications de ce résultat dans la géométrie d'Arakelov sont abordées : le théorème de Hilbert-Samuel arithmétique ainsi que l'existence et l'interprétation géométrique de la pente maximale asymptotique.
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