Table of Contents

I Metrics and Chern Forms.- §1. Néron Functions and Divisors.- §2. Metrics on Line Sheaves.- §3. The Chern Form of a Metric.- §4. Chern Forms in the Case of Riemann Surfaces.- II Green’s Functions on Rlemann Surface.- §1. Green’s Functions.- §2. The Canonical Green’s Function.- §3. Some Formulas About the Green’s Function.- §4. Coleman’s Proof for the Existence of Green’s Function.- §5. The Green’s Function on Elliptic Curves.- III Intersection on an Arithmetic Surface.- §1. The Chow Groups.- §2. Intersections.- §3. Fibral Intersections.- §4. Morphisms and Base Change.- §5. Néron Symbols.- IV Hodge Index Theorem and the Adjunction Formula.- §1. Arakelov Divisors and Intersections.- §2. The Hodge Index Theorem.- §3. Metrized Line Sheaves and Intersections.- §4. The Canonical Sheaf and the Residue Theorem.- §5. Metrizations and Arakelov’s Adjunction Formula.- V The Faltings Reimann-Roch Theorem.- §1. Riemann-Roch on an Arithmetic Curve.- §2. Volume Exact Sequences.- §3. Faltings Riemann-Roch.- §4. An Application of Riemann-Roch.- §5. Semistability.- §6. Positivity of the Canonical Sheaf.- VI Faltings Volumes on Cohomology.- §1. Determinants.- §2. Determinant of Cohomology.- §3. Existence of the Faltings Volumes.- §4. Estimates for the Faltings Volumes.- §5. A Lower Bound for Green’s Functions.- Appendix by Paul Vojta Diophantine Inequalities and Arakelov Theory.- §1. General Introductory Notions.- §2. Theorems over Function Fields.- §3. Conjectures over Number Fields.- §4. Another Height Inequality.- §5. Applications.- References.- Frequently Used Symbols.

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