Convergence des polygones de Harder-Narasimhan

Abstract

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.

Okounkov bodies of filtered linear series

“…with the direct images of λ (by G V • and by G V • ). A special case of this general phenomenon appears in the example computed in [Che07]…”

“…We first compare our results with some of the firstnamed author's previous results. In [Che07,Che08] the author constructed the asymptotic measure of a big line bundle L endowed with Arakelov-geometric data L as above. This measure describes the asymptotic distribution of the jumping values of the Harder-Narasimhan filtration (see [Che07] Section 2.2).…”

“…In [Che07,Che08] the author constructed the asymptotic measure of a big line bundle L endowed with Arakelov-geometric data L as above. This measure describes the asymptotic distribution of the jumping values of the Harder-Narasimhan filtration (see [Che07] Section 2.2). Now a comparison of the latter filtration with the filtration by minima considered above as in [Che08] shows that the asymptotic measure coincides with the limit measure we obtain in Theorem 1.9, i.e.…”

Distribution of logarithmic spectra of the equilibrium energy

“…Let G Φ : ∆(Γ) → R ∪ {−∞} be the map sending x to sup{t ∈ R : x ∈ ∆(Γ t Φ )}. This is a real concave function on (9) ∆(Γ) • . The function G Φ is therefore continuous on ∆(Γ) • .…”

“…which is an analogue of the arithmetic Hilbert-Samuel theorem for Hermitian invertible bundles on a projective arithmetic variety. Given the results in [3] and the convergences proved in [9], it is natural to wonder whether the eigenvalue distribution of the metric L 2 ϕ,µ with respect to L 2 ψ,µ is wellbehaved asymptotically in more general situations. Unlike the arithmetic case, it does not seem possible to reformulate these spectra functorially, which is a key step in the proof of the convergence results in [9].…”