Integrable Systems in the Realm of Algebraic Geometry

by Pol Vanhaecke

Overview

2. Divisors and line bundles ........................ 99. 2.1. Divisors .............................. 99. 2.2. Line bundles ............................ 100. 2.3. Sections of line bundles ....................... 101. 2.4. The Riemann-Roch Theorem ..................... 103. 2.5. Line bundles and embeddings in projective space ............ 105. 2.6. Hyperelliptic curves ......................... 106. 3. Abelian varieties ............................ 108. 3.1. Complex tori and Abelian varieties .................. 108. 3.2. Line bundles on Abelian varieties ................... 109. 3.3. Abelian surfaces .......................... 111. 4. Jacobi varieties ............................. 114. 4.1. The algebraic Jacobian ....................... 114. 4.2. The analytic/transcendental Jacobian ................. 114. 4.3. Abel's Theorem and Jacobi inversion ................. 119. 4.4. Jacobi and Kummer surfaces ..................... 121. 5. Abelian surfaces of type (1,4) ....................... 123. 5.1. The generic case .......................... 123. 5.2. The non-generic case ........................ 124. V. Algebraic completely integrable Hamiltonian systems ........ 127. 1. Introduction .............................. 127. 2. A.c.i. systems ............................. 129. 3. Painlev analysis for a.c.i, systems .................... 135. 4. The linearization of two-dkmensional a.e.i, systems ............. 138. 5. Lax equations ............................. 140. VI. The Mumford systems ..................... 143. 1. Introduction .............................. 143. 2. Genesis ................................ 145. 2.1. The algebra of pseudo-differential operators .............. 145. 2.2. The matrix associated to two commuting operators ........... 146. 2.3. The inverse construction ....................... 150. 2.4. The KP vector fields ........................ 152. ix 3. Multi-Hamiltonian structure and symmetries ................ 155. 3.1. The loop algebra 9(q ........................ 155. 3.2. Reducing the R-brackets and the vector field ............. 157. 4. The odd and the even Mumford systems .................. 161. 4.1. The (odd) Mumford system ..................... 161. 4.2. The even Mumford system ...................... 163.

Product Details

ISBN-13:	9783540423379
Publisher:	Springer Berlin Heidelberg
Publication date:	07/31/2001
Series:	Lecture Notes in Mathematics , #1638
Edition description:	Second Edition 2001
Pages:	264
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

Introduction.- Integrable Hamiltonian systems on affine Poisson varietie: Affine Poisson varieties and their morphisms; Integrable Hamiltonian systems and their morphisms; Integrable Hamiltonian systems on other spaces.- Integrable Hamiltonian systems and symmetric products of curves: The systems and their integrability; The geometry of the level manifolds .- Interludium: the geometry of Abelian varieties: Divisors and line bundles; Abelian varieties; Jacobi varieties; Abelian surfaces of type (1,4).- Algebraic completely integrable Hamiltonian systems: A.c.i. systems; Painlev analysis for a.c.i. systems; The linearization of two-dimensional a.c.i. systems; Lax equations.- The Mumford systems: Genesis; Multi-Hamiltonian structure and symmetries; The odd and the even Mumford systems; The general case .- Two-dimensional a.c.i. systems and applications: The genus two Mumford systems; Application: generalized Kummersurfaces; The Garnier potential; An integrable geodesic flow on SO(4);...