Lectures on Arakelov Geometry
Arakelov theory is a new geometric approach to diophantine equations. It combines algebraic geometry, in the sense of Grothendieck, with refined analytic tools such as currents on complex manifolds and the spectrum of Laplace operators. It has been used by Faltings and Vojta in their proofs of outstanding conjectures in diophantine geometry. This account presents the work of Gillet and Soulé, extending Arakelov geometry to higher dimensions. It includes a proof of Serre's conjecture on intersection multiplicities and an arithmetic Riemann-Roch theorem. To aid number theorists, background material on differential geometry is described, but techniques from algebra and analysis are covered as well. Several open problems and research themes are also mentioned.
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Lectures on Arakelov Geometry
Arakelov theory is a new geometric approach to diophantine equations. It combines algebraic geometry, in the sense of Grothendieck, with refined analytic tools such as currents on complex manifolds and the spectrum of Laplace operators. It has been used by Faltings and Vojta in their proofs of outstanding conjectures in diophantine geometry. This account presents the work of Gillet and Soulé, extending Arakelov geometry to higher dimensions. It includes a proof of Serre's conjecture on intersection multiplicities and an arithmetic Riemann-Roch theorem. To aid number theorists, background material on differential geometry is described, but techniques from algebra and analysis are covered as well. Several open problems and research themes are also mentioned.
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Lectures on Arakelov Geometry

Lectures on Arakelov Geometry

Lectures on Arakelov Geometry

Lectures on Arakelov Geometry

Overview

Arakelov theory is a new geometric approach to diophantine equations. It combines algebraic geometry, in the sense of Grothendieck, with refined analytic tools such as currents on complex manifolds and the spectrum of Laplace operators. It has been used by Faltings and Vojta in their proofs of outstanding conjectures in diophantine geometry. This account presents the work of Gillet and Soulé, extending Arakelov geometry to higher dimensions. It includes a proof of Serre's conjecture on intersection multiplicities and an arithmetic Riemann-Roch theorem. To aid number theorists, background material on differential geometry is described, but techniques from algebra and analysis are covered as well. Several open problems and research themes are also mentioned.

Product Details

ISBN-13: 9780521477093
Publisher: Cambridge University Press
Publication date: 09/15/1994
Series: Cambridge Studies in Advanced Mathematics , #33
Edition description: New Edition
Pages: 188
Product dimensions: 5.94(w) x 8.82(h) x 0.47(d)

Table of Contents

Introduction; 1. Intersection theory on regular schemes; 2. Green currents; 3. Arithmetic Chow groups; 4. Characteristic classes; 5. The determinant of Laplace operators; 6. The determinant of the cohomology; 7. The curvature of the determinant line bundle; 8. The arithmetic Riemann–Roch–Grothendieck theorem; References; Index.
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