Lectures on Arakelov Geometry

ISBN-13: 9780521477093 / Angielski / Miękka / 1995 / 188 str.

C. Soule; J. F. Burnol; D. Abramovich

Opis

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 Soule, 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.

Recenzje

"...has been written with great care, is very enjoyable to read and could be recommended to anyone interested in this important area." Dipendra Prasad, Mathematical Reviews

Spis treści

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.