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Algebraic geometry --- Arithmetical algebraic geometry. --- Riemann-Roch theorems. --- Géométrie algébrique arithmétique. --- Arithmetical algebraic geometry --- Riemann-Roch theorems --- Theorems, Riemann-Roch --- Algebraic functions --- Geometry, Algebraic --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory
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Algebraic geometry --- Geometry, Algebraic --- Riemann-Roch theorems --- 512.73 --- Theorems, Riemann-Roch --- Algebraic functions --- Geometry --- Cohomology theory of algebraic varieties and schemes --- Geometry, Algebraic. --- Riemann-Roch theorems. --- 512.73 Cohomology theory of algebraic varieties and schemes --- Rieman-Roch (Theorema's van). --- Géométrie algébrique --- Rieman-Roch (Théorèmes de). --- Meetkunde (Algebraïsche).
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B.L. van der Waerden: Démonstration algébrique du théorème de Riemann-Roch.- F. Severi: Del teorema di Riemann-Roch per curve, superficie e varietà. Le origini storiche e lo stato attuale.- F. Hirzebruch: Arithmetic genera and the theorem of Riemann-Roch.
Algebra. --- Algebraic varieties. --- Geometry. --- Geometry, Algebraic. --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Riemann-Roch theorems --- Geometry, Algebraic --- Theorems, Riemann-Roch --- Mathematics. --- Algebraic geometry. --- Algebraic Geometry. --- Algebraic functions --- Geometry, algebraic. --- Algebraic geometry
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This book is devoted to computing the index of elliptic PDEs on non-compact Riemannian manifolds in the presence of local singularities and zeros, as well as polynomial growth at infinity. The classical Riemann–Roch theorem and its generalizations to elliptic equations on bounded domains and compact manifolds, due to Maz’ya, Plameneskii, Nadirashvilli, Gromov and Shubin, account for the contribution to the index due to a divisor of zeros and singularities. On the other hand, the Liouville theorems of Avellaneda, Lin, Li, Moser, Struwe, Kuchment and Pinchover provide the index of periodic elliptic equations on abelian coverings of compact manifolds with polynomial growth at infinity, i.e. in the presence of a "divisor" at infinity. A natural question is whether one can combine the Riemann–Roch and Liouville type results. This monograph shows that this can indeed be done, however the answers are more intricate than one might initially expect. Namely, the interaction between the finite divisor and the point at infinity is non-trivial. The text is targeted towards researchers in PDEs, geometric analysis, and mathematical physics.
Global analysis (Mathematics). --- Manifolds (Mathematics). --- Mathematical analysis. --- Analysis (Mathematics). --- Topology. --- Global Analysis and Analysis on Manifolds. --- Analysis. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- 517.1 Mathematical analysis --- Mathematical analysis --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Differential equations, Elliptic. --- Riemann-Roch theorems. --- Theorems, Riemann-Roch --- Algebraic functions --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear --- Differential equations, Partial
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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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