Introduction to Arakelov Theory

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.