Electronic ISBN:  9781470402815 
Product Code:  MEMO/145/690.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 145; 2000; 130 ppMSC: Primary 14; 11; Secondary 31;
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 selfintersection number of a metrized line bundle, and the existence of the sectional capacity is equivalent to an arithmetic HilbertSamuel 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 welldefined. Finally, we show that sectional capacities for arbitrary norms can be wellapproximated using objects of finite type.ReadershipGraduate students and research mathematicians interested in algebraic geometry.

Table of Contents

Chapters

Introduction

1. The standard hypotheses

2. The definition of the sectional capacity

3. Reductions

4. Existence of the monic basis for very ample line bundles

5. Zaharjuta’s construction

6. Local capacities

7. Existence of the global sectional capacity

8. A positivity criterion

9. Base change

10. Pullbacks

11. Products

12. Continuity, Part I

13. Continuity, Part II

14. Local capacities of sets

15. Approximation theorems


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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 selfintersection number of a metrized line bundle, and the existence of the sectional capacity is equivalent to an arithmetic HilbertSamuel 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 welldefined. Finally, we show that sectional capacities for arbitrary norms can be wellapproximated using objects of finite type.
Graduate students and research mathematicians interested in algebraic geometry.

Chapters

Introduction

1. The standard hypotheses

2. The definition of the sectional capacity

3. Reductions

4. Existence of the monic basis for very ample line bundles

5. Zaharjuta’s construction

6. Local capacities

7. Existence of the global sectional capacity

8. A positivity criterion

9. Base change

10. Pullbacks

11. Products

12. Continuity, Part I

13. Continuity, Part II

14. Local capacities of sets

15. Approximation theorems