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251

Index

Abel's Theorem, 119 Chern class, 110

Abelian variety, 108 closed point, 37

- dual, 110 compatible- (principally) polarized, 108 - Poisson brackets, 25

- reducible, 109 - integrable Hamiltonian systems, 62

Abel-Jacobi map, 119 completable a.c.i. system, 131

adjunction formula, Ill complete algebra, 47

affine Lie-Poisson group, 30 completion, 48

a.c.i. system, 131 complex torus, 108

- algebraic completely integrable system, 131

- completable, 131 Darboux

- irreducible, 132 - coordinates, 66

- (polarization) type of, 132 - Theorem, 66

analytic Poisson manifold, 65 deformation property, 25

ample line bundle, 105 degrees of freedom, 49

dimension, 37, 49

base space, 49 divisor

Ber-hlivanidis-van Moerbeke system, 181 - canonical, 103

bi-Hamiltonian - elementary, 108

- hierarchy, 63 - degree of, 99

- integrable system, 62 - group, 99

- vector field, 62 - pole, 99

branch point, 106 - zero, 99

double Lie algebra, 140

canonical dual Abelian variety, 110

- brackets, 66 Garnier potential, 196

- coordinates, 66 general- divisor, 103 - fiber, 37

- coordinates, 66 - level set, 37

- line bundle, 103 - point, 37

Casimir

- decomposition, 38 Hamiltonian, 19

- function, 19, 65 - derivation, 19

- level set of, 38 - vector field, 19, 65

253

Index

H6non-Heiles K-3 surface, 121

- hierarchy, 233 Kodaira

- potential, 230, 232, 233 - Serre duality, 103

hyperelliptic - vanishing Theorem, 113

- case, 83 Kummer

- curve, 106 - generalized surface, 121

- involution, 107 - intermediate surface, 121

hierarchy - surface, 121

- bi-Hamiltonian, 63 - variety, 121

- H6non-Heiles, 233 Krull dimension, 37

Hodge form, 105

Laurent solutions, 136

indicial equations, 136 Lax

integrable - equation, 141

algebra, 49, 69 - equation with spectral parameter, 141

bi-Hamiltonian system, 62 - representation, 142

Hamiltonian system, 49, 69 - type, 141

multi-Hamiltonian system, 62 level set

vector field, 49, 69 - general, 37

integral - of the Casimirs, 38

- closure, 47 - of the integrable system, 50

- curve, 68 Lie derivative, 20

- first, 1 Lie-Poisson

invariant of affine Poisson variety - affine group, 30

- polynomial, 41 - structure, 23

- matrix, 41 - modified structure, 25

involutive line bundle

- algebra, 47 - ample, 105

- Hamiltonian system, 47 - canonical, 103

irreducible a.c.i. system, 132 - holomorphic, 100

isogeny, 109 - very ample, 105

lower balances, 136

Jacobian, 114

Jacobi maximal

- inversion Theorem, 120 - algebra of Casimirs, 39

- Mumford system, 143 - spectrum, 37

- surface, 121 momentum map, 50

- variety, 114

254

Index

morphism - ideal, 29

- of affine Poisson varieties, 26 - isomorphism, 27, 68

- of integrable Hamiltonian systems, 54, 70 - manifold, 65

- of Poisson spaces, 68 - matrix, 21

multipliers, 109 - morphism, 26, 68

multi-Hamiltonian - product bracket, 30

- integrable system, 62 - reducible, 32

- vector field, 62 - space, 65

Mumford. system - standard structure, 23

- even, 163, 179 - structure, 19, 65

- odd, 161, 177 - subalgebra, 31

- trivial structure, 22

node, 122 - vector field, 20

Noether's formula, 113 Poincax6

- dual, 111

Painlev6 analysis, 136 - reducibility theorem, 109

parameter pseudo-differential operator- map, 38 - algebra of, 145

- space, 38 - monic, 145

phase space, 49 - normalized, 145

period - order of, 145

- half, 121 polarized Abelian vaxiety, 108

- matrix of A periods, 115 polarization- matrix of B periods, 115 - principal, 108

- i-th period, 115 - type, 108

- lattice, 114 - type of a.c.i. system, 132

Picard group, 100 principal balance, 136

Poisson principally polarized Abelian variety, 108

- action, 31

- algebra, 19 quasi-automorphism, 61

- affine brackets, 25

- affine subvariety, 27 ramification point, 87

- algebra, 19 rank

- analytic manifold, 65 - of Poisson structure, 22, 69

- bracket, 19, 65 - rank decomposition, 41

- canonical structure, 23 reciprocity laws, 115

- cohomology, 20 real level sets, 85

- constant structure, 23 reducible Abelian variety, 109

255

Index

regular Poisson structure, 22 Toda, lattice

Pdemann - generafized, 141

- conditions, 108 - three body, 235

- constant, 120 trope, 122

- Hurwitz formula, 106 trivial

- Roch Theorem, 103, 112 - Poisson structure, 22

- theta divisor, 110 - subsystem, 58

- theta function, 110

vector field

Sato Grassmannian, 145 - bi-Hamiltonian, 62

Schouten bracket, 20 - Hamiltonian, 19, 65

Schotky problem, 118 - integrable, 49, 69

seven-dimensionaJ system, 181 - KP, 152

spectral - linear, 131

- parameter, 141 - multi-Hamiltoinian, 62

- curve, 142 - Poisson, 20

symplectic - super-integrable, 49

- basis, 115 virtual genus, 112

- manifold, 65 Volterra group, 145

- two-form, 65

- decomposition, 67 Weierstrass point, 107

- foliation, 67 weight homogeneous, 191

super-integrable vector field, 49

Yang-Baxter equation

type of a.c.i. system, 132 - classical, 140

theta - modified classical, 140

- curve, 122

- function, 110

- function with characteristics, 118

- divisor, 110

256