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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.

Algebraic geometry --- Algebraïsche meetkunde --- Geometry [Algebraic ] --- Géométrie algébrique --- Meetkunde [Algebraïsche ] --- Riemann-Roch theorema's --- Riemann-Roch thoerems --- Theoremes de Riemann-Roch --- Geometry, Algebraic. --- Riemann-Roch theorems. --- Theorems, Riemann-Roch --- Algebraic functions --- Geometry, Algebraic --- Geometry --- Addition. --- Adjoint. --- Alexander Grothendieck. --- Algebraic geometry. --- Analytic torsion. --- Arakelov theory. --- Asymptote. --- Asymptotic expansion. --- Asymptotic formula. --- Big O notation. --- Cartesian coordinate system. --- Characteristic class. --- Chern class. --- Chow group. --- Closed immersion. --- Codimension. --- Coherent sheaf. --- Cohomology. --- Combination. --- Commutator. --- Computation. --- Covariant derivative. --- Curvature. --- Derivative. --- Determinant. --- Diagonal. --- Differentiable manifold. --- Differential form. --- Dimension (vector space). --- Divisor. --- Domain of a function. --- Dual basis. --- E6 (mathematics). --- Eigenvalues and eigenvectors. --- Embedding. --- Endomorphism. --- Exact sequence. --- Exponential function. --- Generic point. --- Heat kernel. --- Injective function. --- Intersection theory. --- K-group. --- Levi-Civita connection. --- Line bundle. --- Linear algebra. --- Local coordinates. --- Mathematical induction. --- Morphism. --- Natural number. --- Neighbourhood (mathematics). --- Parameter. --- Projective space. --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Riemannian manifold. --- Riemann–Roch theorem. --- Self-adjoint operator. --- Smoothness. --- Sobolev space. --- Stochastic calculus. --- Summation. --- Supertrace. --- Theorem. --- Transition function. --- Upper half-plane. --- Vector bundle. --- Volume form.

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Surfaces [Algebraic ] --- Degenerations --- Abelian varieties --- Compactifications --- Moduli theory --- Surfaces, Algebraic --- #KVIV:BB --- 512.74 --- 512.74 Algebraic groups. Abelian varieties --- Algebraic groups. Abelian varieties --- Degenerations of algebraic surfaces --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Topology --- Varieties, Abelian

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Surfaces [Algebraic ] --- Degenerations

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This volume contains selected papers authored by speakers and participants of the 2013 Arbeitstagung, held at the Max Planck Institute for Mathematics in Bonn, Germany, from May 22-28. The 2013 meeting (and this resulting proceedings) was dedicated to the memory of Friedrich Hirzebruch, who passed away on May 27, 2012. Hirzebruch organized the first Arbeitstagung in 1957 with a unique concept that would become its most distinctive feature: the program was not determined beforehand by the organizers, but during the meeting by all participants in an open discussion. This ensured that the talks would be on the latest developments in mathematics and that many important results were presented at the conference for the first time. Written by leading mathematicians, the papers in this volume cover various topics from algebraic geometry, topology, analysis, operator theory, and representation theory and display the breadth and depth of pure mathematics that has always been characteristic of the Arbeitstagung.

Group theory --- Algebraic geometry --- Differential geometry. Global analysis --- Topology --- Geometry --- Operator theory --- Mathematics --- landmeetkunde --- analyse (wiskunde) --- differentiaal geometrie --- statistiek --- wiskunde --- geometrie --- topologie