Lectures on the Arithmetic Riemann-Roch Theorem. (AM-127) [Deluxe Edition] [Paperback]

Gerd Faltings (Author)

Book Description

ISBN-10: 0691025444 | ISBN-13: 978-0691025445 | Publication Date: February 19, 1992 | Edition: Limited Ed

The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory.

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This treatise provides a new approach to the arithmetic Riemann-Roch problem, and a widely algebraic-geometric method to solve it. -- Zentralblatt für Mathematik

Product Details

• Paperback: 118 pages
• Publisher: Princeton University Press; Limited Ed edition (February 19, 1992)
• Language: English
• ISBN-10: 0691025444
• ISBN-13: 978-0691025445
• Product Dimensions: 8.9 x 6 x 0.4 inches
• Shipping Weight: 6.4 ounces (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #2,423,093 in Books (See Top 100 in Books)

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4.0 out of 5 stars Deep but important, March 14, 2004

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This review is from: Lectures on the Arithmetic Riemann-Roch Theorem (Annals of Mathematics Studies) (Hardcover)

The Riemann-Roch theorem has come a long way since its origins in the work of Bernhard Riemann 154 years ago. Riemann was attempting to establish the existence of complex functions on multiply-connected surfaces with no boundary. A surface that is (2p + 1)-connected for a positive integer p can be represented as a 4p-sided polygon after making 2p cuts. Riemann showed using the Dirichlet principle that there are p linearly-independent functions defined inside this polygon that are everywhere holomorphic. The differentials of these functions are also holomorphic integrands. By specifying D points at which a function can have simple poles, and by constraining the functions to change only by a constant amount at the cuts, then Riemann showed that one can obtain functions with only simple poles and constant "jumps" by taking a sum of p linearly independent functions with no poles with functions of the form 1/z at one of the specified points and a constant term. Non-constant meromorphic functions thus exist when p + D + 1 - 2p >= 2, or D > p, which is now called the Riemann inequality. Hence there is a linear space of complex functions of dimension >= D + 1 - p, which contains non-constant functions when D + 1 - p > 1. Roch was Riemann's student and interpreted the quantity D + 1 - p as the dimension of the space of holomorphic integrands. If a function has D simple poles, and if there are Q linearly independent integrands that vanish at these poles, then the function depends on D - p + q + 1 arbitrary constants.

The work of Riemann and Roch is readily seen to be related to the genus of the surface, if viewed in the light of the polygon of 4p sides. The modern view of the Riemann-Roch theorem in fact is naturally viewed as a generalization of a formula for the Euler characteristic, the latter of which involves the genus of a Riemann surface. The "classical" Riemann-Roch theorem is stated in terms of divisors on a Riemann surface X of genus g and reads as r(-D) - i(D) = d(D) - g + 1, where D is a fixed divisor on the surface, r(-D) is the dimension of meromorphic functions of divisors >= -D on X, and i(D) is the dimension of the space of meromorphic 1-forms of divisors >= D on X. Many other statements have been given, one being in terms of holomorphic bundles defined by D over X, where one computes the Euler characteristic of the sheaf of germs of holomorphic sections of the bundle. Another is in the context of holomorphic bundles over nonsingular complex projective varieties, where the Euler characteristic of the sheaf of holomorphic sections of the bundle is given in terms of a formula involving the first Chern class of the variety. The Euler characteristic has of course also been computed in terms of the index of Dirac operators, and so it is not surprising to find that the Riemann-Roch theorem has an analytical formulation also.

This book proves a Riemann-Roch theorem for arithmetic varieties, and the author does so via the formalism of Dirac operators and consequently that of heat kernels. In the first lecture the reader will see the "classical" Riemann-Roch theorem in an even more general context then that mentioned above: that of smooth morphisms of regular schemes. Using familiar constructions involving the K-groups and Chow groups to make the calculations more manageable, the author proves the Riemann-Roch theorem for regular schemes by reducing it to the case for projective bundles. Flag schemes are used in the proof, and the strategy used here repeats itself in the proof of the arithmetic Riemann-Roch theorem. Throughout the book the author outlines the necessary background for the eventual proof of the arithmetic Riemann-Roch theorem. This includes a discussion of how to define Chern classes for arithmetic vector bundles, as well as the background in analysis via a discussion of heat kernels and Laplacians on Riemannian manifolds. The detailed discussion of analysis was done, according to the author, so as to avoid the methods of stochastic integration, which he felt might not be familiar to the average reader with a background in algebraic geometry.