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Item Details
| Title:
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EXISTENCE OF THE SECTIONAL CAPACITY
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| By: |
Robert Rumely, Chi Fong Lau, Robert Varley |
| Format: |
Paperback |
| List price:
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£49.95 |
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| ISBN 10: |
0821820583 |
| ISBN 13: |
9780821820582 |
| Publisher: |
AMERICAN MATHEMATICAL SOCIETY |
| Pub. date: |
1 January, 2000 |
| Series: |
Memoirs of the American Mathematical Society No. 690 |
| Pages: |
130 |
| Description: |
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. |
| Synopsis: |
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 $K_v \otimes_K \Gamma(X,\L^{\otimes n})$, 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. |
| Illustrations: |
bibliography |
| Publication: |
US |
| Imprint: |
American Mathematical Society |
| Returns: |
Returnable |
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