Existence of the Sectional Capacity (eBook)

Robert Rumely (Autor)

eBook Download: PDF

130 Seiten
American Mathematical Society (Verlag)
9781470402815 (ISBN)

Infos

Let $K$ be a global field, and let $X/K$ be an equidimensional, geometrically reduced projective variety. For an ample line bundle $/overline{/mathcal L}$ on $X$ with norms $// /_v$ on the spaces of sections $K_v /otimes_K /Gamma(X,/mathcal{L}^{/otimes n})$, we prove the existence of the sectional capacity $S_/gamma(/overline{/mathcal L})$, giving content to a theory proposed by Chinburg. In the language of Arakelov Theory, the quantity $-/log(S_/gamma(/overline{/mathcal L}))$ generalizes the top arithmetic self-intersection number of a metrized line bundle, and the existence of the sectional capacity is equivalent to an arithmetic Hilbert-Samuel Theorem for line bundles with singular metrics. In the case where the norms are induced by metrics on the fibres of ${/mathcal L}$, we establish the functoriality of the sectional capacity under base change, pullbacks by finite surjective morphisms, and products. We study the continuity of $S_/gamma(/overline{/mathcal L})$ under variation of the metric and line bundle, and we apply this to show that the notion of $v$-adic sets in $X(/mathbb C_v)$ of capacity $0$ is well-defined. Finally, we show that sectional capacities for arbitrary norms can be well-approximated using objects of finite type.

Produkt-Details

• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• ISBN-13: 9781470402815 / 9781470402815