Integrable Systems in the realm of Algebraic Geometry

1. Introduction......Page 10
2.1. Affine Poisson varieties......Page 12
2.2. Morphisms of affine Poisson varieties......Page 19
2.3. Constructions of affine Poisson varieties......Page 21
2.4. Decompositions and invariants of affine Poisson varieties......Page 30
3.1. Integrable Hamiltonian systems on affine Poisson varieties......Page 40
3.2. Morphisms of integrable Hamiltonian systems......Page 47
3.3. Constructions of integrable Hamiltonian systems......Page 50
3.4. Compatible and multi-Hamiltonian integrable systems......Page 55
4.1. Poisson spaces......Page 58
4.2. Integrable Hamiltonian systems on Poisson spaces......Page 62
1. Introduction......Page 90
2.1. Divisors......Page 92
2.2. Line bundles......Page 93
2.3. Sections of line bundles......Page 94
2.4. The Riemann-Roch Theorem......Page 96
2.5. Line bundles and embeddings in projective space......Page 98
2.6. Hyperelliptic curves......Page 99
3.1. Complex tori and Abelian varieties......Page 101
3.2. Line bundles on Abelian varieties......Page 102
3.3. Abelian surfaces......Page 104
4.2. The analytic/transcendental Jacobian......Page 107
4.3. Abel\'s Theorem and Jacobi inversion......Page 112
4.4. Jacobi and Kummer surfaces......Page 114
5.1. The generic case......Page 116
5.2. The non-generic case......Page 117
1. Introduction......Page 119
2. A.c.i. systems......Page 121
3. Palnlevé analysis for a.c.i. systems......Page 127
4. The linearization of two-dimensional a.c.i. systems......Page 130
5. Lax equations......Page 132
1. Introduction......Page 135
2.1. The algebra of pseudo-differential operators......Page 137
2.2. The matrix associated to two commuting operators......Page 138
2.3. The inverse construction......Page 142
2.4. The KP vector fields......Page 144
3.1. The loop algebra gl_q......Page 147
3.2. Reducing the R-brackets and the vector field V......Page 149
4.1. The (odd) Mumford system......Page 153
4.2. The even Mumford system......Page 155
4.3. Algebraic complete integrability and Laurent solutions......Page 156
5. The general case......Page 160
1. Introduction......Page 166
2.1. The genus two odd Mumford system......Page 168
2.2. The genus two even Mumford system......Page 170
2.3. The Bechlivanidis-van Moerbeke system......Page 172
3.1. Genus two curves with an automorphism of order three......Page 176
3.2. The 9_4 configuration on the Jacobian of Gamma......Page 177
3.3. A projective embedding of the generalised Kummer surface......Page 181
4.1. The Garnier potential and its integrability......Page 187
4.2. Some moduli spaces of Abelian surfaces of type (1,4)......Page 193
4.3. The precise relation with the canonical Jacobian......Page 197
4.4. The relation with the canonical Jacobian made explicit......Page 202
4.5. The central Garnier potentials......Page 207
5.1. The geodesic flow on SO(4) for metric II......Page 211
5.2. Linearizing variables......Page 213
5.3. The map M-> M^3......Page 217
6.1. The cubic Hénon-Heiles potential......Page 221
6.2. The quartic Hénon-Heiles potential......Page 223
6.3. The Hénon-Heiles hierarchy......Page 224
7.1. Different forms of the Toda lattice......Page 226
7.2. A morphism to the genus 2 even Mumford system......Page 228
7.3. Toda and Abelian surfaces of type (1,3)......Page 231