05696nam 22007455 450 991014459840332120250724090653.03-540-44576-510.1007/3-540-44576-5(CKB)1000000000233209(SSID)ssj0000324045(PQKBManifestationID)12133614(PQKBTitleCode)TC0000324045(PQKBWorkID)10305168(PQKB)10126227(DE-He213)978-3-540-44576-0(MiAaPQ)EBC3071851(PPN)155223097(BIP)7312785(EXLCZ)99100000000023320920121227d2001 u| 0engurnn#008mamaatxtccrIntegrable Systems in the Realm of Algebraic Geometry /by Pol Vanhaecke2nd ed. 2001.Berlin, Heidelberg :Springer Berlin Heidelberg :Imprint: Springer,2001.1 online resource (XII, 264 p.)Lecture Notes in Mathematics,1617-9692 ;1638Bibliographic Level Mode of Issuance: Monograph3-540-42337-0 Includes bibliographical references and index.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);...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.Lecture Notes in Mathematics,1617-9692 ;1638DynamicsGlobal analysis (Mathematics)Manifolds (Mathematics)Geometry, AlgebraicMathematical physicsDynamical SystemsGlobal Analysis and Analysis on ManifoldsAlgebraic GeometryTheoretical, Mathematical and Computational PhysicsDynamics.Global analysis (Mathematics)Manifolds (Mathematics)Geometry, Algebraic.Mathematical physics.Dynamical Systems.Global Analysis and Analysis on Manifolds.Algebraic Geometry.Theoretical, Mathematical and Computational Physics.516.35314K20mscVanhaecke Polauthttp://id.loc.gov/vocabulary/relators/aut61070MiAaPQMiAaPQMiAaPQBOOK9910144598403321Integrable systems in the realm of algebraic geometry258921UNINA