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Oda T. — Convex bodies and algebraic geometry: an introduction to the theory of toric varieties
Oda T. — Convex bodies and algebraic geometry: an introduction to the theory of toric varieties

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Название: Convex bodies and algebraic geometry: an introduction to the theory of toric varieties

Автор: Oda T.

Аннотация:

The theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces. This book is a unified up-to-date survey of the various results and interesting applications found since toric varieties were introduced in the early 1970's. It is an updated and corrected English edition of the author's book in Japanese published by Kinokuniya, Tokyo in 1985. Toric varieties are here treated as complex analytic spaces. Without assuming much prior knowledge of algebraic geometry, the author shows how elementary convex figures give rise to interesting complex analytic spaces. Easily visualized convex geometry is then used to describe algebraic geometry for these spaces, such as line bundles, projectivity, automorphism groups, birational transformations, differential forms and Mori's theory. Hence this book might serve as an accessible introduction to current algebraic geometry. Conversely, the algebraic geometry of toric varieties gives new insight into continued fractions as well as their higher-dimensional analogues, the isoperimetric problem and other questions on convex bodies. Relevant results on convex geometry are collected together in the appendix.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1988

Количество страниц: 212

Добавлена в каталог: 05.06.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\mathbb{Z}$-lattice      149 151 154 171
$\mathbb{Z}$-weighted link      54
Absolutely simple integral covex polytope      83 94 96 97
Abstract complex of faces      122 123 176 181 189
Additive subsemigroup      3
Additive subsemigroup, saturated      3
Admissible, doubly $\mathbb{Z}$-weighted triangulation      55 59 89 92
Admissible, N-weighted triangulation      55 59
Affine, half space      179 182
Affine, line bundle      144
Affine, open set      10 11
Affine, scheme      5
Alexandrov — Fenchel inequalities, the      102 187
Algebraic, cycle      105
Algebraic, subgroup of Cremona group      140—146
Algebraic, torus      4 161 169
Ample, divisor      86 94 105
Ample, invertible sheaf      83 93 130
Ample, line bundle      196
Anticanonical divisor      87 88 108
Arithmetic, genus      165
Arithmetic, genus, logarithmic      166
Arithmetic, quotient of symmetric domain      147 153 154 167 169
Arrangement of lines      98
Artin's approximation theorem      163
Automorphism group and equivariant blowing-up      140
Automorphism group, dimension of      140
Automorphism group, of toric variety      136 140 143 144
Blaschke selection theorem, den      186
Blowing-down of Fano variety      92
Blowing-up, equivariant      see “Equivariant blowing-up of Fano variety” 92
Bonnesen inequality, the      103 188
Bott's vanishing theorem      101 130
Bruenn — Minkowski inequality, the      103 187
Buchsbaum, ring      159
Buchsbaum, singularity      159 160
Buchsbaum, singularity, theorem      160
Canonical, cone      36 37
Canonical, divisor      70 71 106 118 158
Canonical, sheaf      125
Canonical, singularity      36
Caratheodory's theorem      3 175 181
Cartier divisor      52 68 73
Cartier divisor, invariant      68 72 73
Cartier divisor, invertible sheaf associated to      69
Cell decomposition      160 168 169 189
Centrally symmetric toric Fano variety      88 148
Chain of rational curves      25
CHARACTER      4
Characteristic function      157 163 178
Chern class      131 133 163 165 167
Chern class, logarithmic      165 167
Chow ring      102 108 133
Chow's lemma, equivariant      86
Chow's lemma, toric      17 85
Circumradius      103 104 188
Classification of, toric del Pezzo surfaces      88 89
Classification of, toric Fano threefolds      90
Classification of, toric varieties      42 55 59
Closed convex, cone      183
Closed convex, cone, duality for      174
Closed convex, set      179
Closure of orbit in toric variety      12 24 68 94 96
Closure of orbit in toric variety, nonsingularity of      12
Closure of orbit in toric variety, normality of      12
Cohen — Macaulay, ring      126 127 159 196 197
Cohen — Macaulay, singularity      23 118
Cohen — Macaulay, space      101 125 126 130
Cohomology ring      102 108
Compact, convex polyhedron      180 see
Compact, convex set, support function for      182
Compact, dual of symmetric domain      167
Compact, real manifold      154 160
Compact, real manifold, triangulation of      161 168
Compact, toric variety      16 17
Compact, torus      12 13 149 172
Compactness criterion, the      17
Complete, fractional ideal      70 75
Complete, linear system      73 76 94
Complete, reducibility theorem, the      4 6 10 71 122
Complex torus      169
Complex torus, dual      170
Complex torus, dual period for      170
Complex torus, period for      169
Concave function      182 see
Continued fraction, expansion      148
Continued fraction, finite      see “Finite continued fraction”
Continued fraction, periodic      52 148
Continued fraction, purely periodic      148 150 151
Continued fraction, regular      25
Contraction, for extremal ray      106—108 111
Contraction, of cycle of rational curves      152
Contraction, of divisor      157 159
Convex, body      79 179
Convex, cone      173 180
Convex, cone, closed      see “Closed convex cone”
Convex, cone, dimension of      2 174
Convex, cone, dual      2 7 105 172 173 181
Convex, cone, ideal in      177 185
Convex, cone, strongly      see “Strongly convex cone”
Convex, cone, support function for ideal in      27 155 185
Convex, hull      149 155 156 179 181
Convex, polyhedral cone      2 106 173 175 180
Convex, polyhedral cone, decomposition      184 185
Convex, polyhedral cone, dual of face of      177
Convex, polyhedral cone, duality for      174
Convex, polyhedral cone, face of      2 7 10 175
Convex, polyhedral cone, face of dual      176
Convex, polyhedral cone, nonsingular rational      15 123
Convex, polyhedral cone, projection of      177
Convex, polyhedral cone, rational      2
Convex, polyhedral set      180
Convex, polyhedral set, face of      181
Convex, polyhedral set, polar      27 181
Convex, polytope      13 48 72 76 180 181
Convex, polytope, face of      13 93 94 96 184
Convex, polytope, gauge function for      185
Convex, polytope, generating function for      194
Convex, polytope, integral      see “Integral convex polytope”
Convex, polytope, morphology for      127 182 189 194
Convex, polytope, polar      88 93 102 182 184
Convex, polytope, simple      94 97 102 182 190
Convex, polytope, simplicial      88 182 190 194
Convex, polytope, support function for      77 184
Convex, polytope, with rational vertices      87 93 196
Convex, quadrangular cone      33 36
Convex, set      179 180
Convex, set, closed      see “Closed convex set”
Convex, set, dimension of      181
Convex, set, nonnegative scalar multiple of      180 183
Convex, set, polar      156 181
Convex, set, support function for      180
Convexity condition for double $\mathbb{Z}$-weight      168
Cremona group      141
Cremona group, algebraic subgroup of      140 146
Cusp singularity      98
Cusp singularity, Hilbert modular      see “Hilbert modular cusp singularity”
Cusp singularity, Hirzebruch's resolution of      25
Cusp singularity, Tsuchihashi      see “Tsuchihashi cusp singularity”
Cycle of rational curves      44 149—152 172
Cycle of rational curves, contraction of      152
Cyclic group action, on affine plane      30 32
Cyclic group action, on affine space      36
Cyclic quotient singularity      29 30 97
Danilov's, exact sequence      127
Danilov's, spectral sequence      133
De Concini — Procesi's theorem      39
de Rham complex      129 133
Degenerate variety      115 129 147
Degeneration of, complex manifold      147
Degeneration of, Hopf surface      170
Degeneration of, parabolic Inoue surface      171
Dehn — Sommerville equalities, the      102 194
Del Pezzo surface      87
Del Pezzo surface, toric      88 92
Demazure's structure theorem      140
Dimension of, automorphism group      140
Dimension of, convex cone      2 174
Dimension of, convex set      181
Dimension of, orbit      10
Dimension of, toric variety      7
Dirichlet's unit theorem      155
Divisor      68 105
Divisor, ample      86 94 105
Divisor, anticanonical      87 88 108
Divisor, canonical      70 71 106 118 158
Divisor, Cartier      see “Cartier divisor”
Divisor, contraction of      157 159
Divisor, invariant      68 134
Divisor, very ample      94 106
Double $\mathbb{Z}$-weight      53 55 89 168
Double $\mathbb{Z}$-weight, convexity condition for      168
Doubly $\mathcal{Z}$-weighted triangulation, admissible      55 59 89 92
Du Bois singularity      160
Dual, complex torus      170
Dual, convex cone      2 7 105 172 173 181
Dual, convex polyhedral cone, face of      176
Dual, Hilbert modular cusp singularity      172
Dual, of face of convex polyhedral cone      177
Dual, open convex cone      155 178
Dual, period for complex torus      170
Dual, Tsuchihashi cusp singularity      157
Duality for, closed convex cone      174
Duality for, convex polyhedral cone      174
Duality for, finite continued fraction      29
Duality for, open convex cone      178
Duality theorem, for proper map, the      124 160
Duality theorem, the local      129
Duality theorem, the Poincare      102 197
Duality theorem, the Serre — Grothendieck      101 125 129 131 197
Dualizing complex      124 125
Dualizing complex, globally normalized      124 125 130 159
Dualizing complex, truncation of      159
Dualizing sheaf      101 118 125 126 130 158
Einstein — Kaehler metric on toric Fano variety      88 95 140
Elementary transformation      49 55 56 113
Enriques — Fano — Umemura's theorem      144
Enriques' theorem      143
Equalities, the Dehn — Sommerville      102 194
Equivariant, birational map      85
Equivariant, blowing-up      23 37 38 40 54—56 85 92 140 190
Equivariant, blowing-up, automorphism group and      140
Equivariant, blowing-up, minimal with respect to      59 142
Equivariant, blowing-up, nonsingularity of      37
Equivariant, blowing-up, normality of      37
Equivariant, Chow's lemma      86
Equivariant, compactification      18 144
Equivariant, fiber bundle      58
Equivariant, finite map      23 97
Equivariant, holomorphic map      19
Equivariant, line bundle      59 67 68 72
Equivariant, line bundle, associated to support function      68
Equivariant, proper birational map      23
Equivariant, proper map      20
Equivariant, resolution of singularity      23 101 125 159
Equivariant, theorem, the      17
Equivariant, vector bundle      71
Euler — Poincare characteristic      78 164 167 189
Euler, number      131 162 169
Euler, relation      189 194
Exceptional, curve      25 29 170
Exceptional, divisor      158
Extremal, rational curve      106 107
Extremal, ray      106—109
Extremal, ray, contraction for      106—108 111
Extremal, ray, generalized      107 111
Face, abstract complex of      122 123 176 181 189
Face, of convex polyhedral cone      2 7 10 175
Face, of convex polyhedral cone, dual of      177
Face, of convex polyhedral set      181
Face, of convex polytope      13 93 94 96 184
Face, of dual convex polyhedral cone      176
Face, partially ordered set of      176 181
Fan      2
Fan, finite complete      16 18 72 81 196
Fan, locally star closed subset of      119
Fan, map of      19
Fan, nonsingular      15
Fan, simplicial      69 87 129 135
Fan, star closed subset of      119
Fan, star subdivision of      23 38 40 49 54—56 88 89
Fan, subcomplex of      119
Fan, subdivision of      23 38 46 101 111 159 168
Fan, support function linear with respect to      59 66 67 72 196
Fan, support of      2 16 66
Fano threefold      87 144
Fano threefold, toric      see “Toric Fano threefold”
Fano variety      87 106
Fano variety, blowing-down of      92
Fano variety, blowing-up of      92
Fano variety, toric      see “Toric Fano variety”
Farkas' theorem      174
Fermat variety      98 99
Finite, complete fan      16 18 72 81 196
Finite, continued fraction      25
Finite, continued fraction, duality for      29
Flanders inequality, the      103 188
Flip conjecture, the      51
Functional equation      130 195
Fundamental group      14 152 168 170 172
Fundamental group, of toric variety      14
GAGA theorem      21 71 136
GAGA theorem, the relative      72
Galois correspondence      82 93 156 176 182 184
Gauge function for convex polytope      185
Generalized, extremal ray      107 111
Generalized, Jacobian variety      147
Generalized, moment map      94
Generating function for convex polytope      194
Globally normalized dualizing complex      124 125 130 159
Gordan's lemma      3
Gorenstein, ring      126 127 135 196
Gorenstein, space      125 126
Grauert — Riemenschneider's vanishing theorem      126 157 159
Grothendieck — Deligne's spectral sequence      133
Group algebra      5 17
Half, Inoue surface      172
Half, space      173 175
Hausdorff, distance      186
Hausdorff, topology      186
Hilbert — Samuel function      195
Hilbert, modular cusp singularity      152 153 162 169
Hilbert, modular cusp singularity, dual      172
Hilbert, modular variety      147 154
Hilbert, polynomial, for integral convex polytope      79 100
Hilbert, polynomial, for toric projective variety      100
Hironaka's theorem      37 163
Hirzebruch surface      9 16 38 42 56 59 88—90 108 137 143 146
Hirzebruch — Mumford's proportionality theorem      167
Hirzebruch's resolution of cusp singularity      25
Hodge spectral sequence, the      133
Hodge — Atiyah — Singer's index theorem      102 132 163—165
Holomorphic vector field      115 136
Homogeneous open convex cone      154 178
Hopf manifold      170
Hopf manifold, primitive      170
Hopf surface      170
Hopf surface, degeneration of      170
Hopf surface, primitive      170
Hyperbolic Inoue surface      157 171 172
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