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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
Icosahedral triangulation of sphere      65 191
Ideal in convex cone      177 185
Ideal in convex cone, support function for      27 155 185
INDEX      132 162 165
Index, logarithmic      166
Index, theorem, Hodge — Atiyah — Singer's      102 132 163—165
Induction theorem of Brueckner — Eberhard, the      190
Inequality, Alexandrov — Fenchel's      102 187
Inequality, Bonnesen's      103 188
Inequality, Bruenn — Minkowski's      103 187
Inequality, Flanders'      103 188
Inequality, Miyaoka's      98
Inequality, Teissier's      102 103
Inequality, the isoperimetric      79 187 188
Infinitesimal deformation of Tsuchihashi cusp singularity      169
Inoue surface, half      see “Half Inoue surface”
Inoue surface, hyperbolic      see “Hyperbolic Inoue surface”
Inoue surface, parabolic      see “Parabolic Inoue surface”
Inoue — Hirzebruch surface      171 172
Inoue — Kato manifold      172
Inradius      103 104 188
Integral convex polytope      79 82 88 100
Integral convex polytope, absolutely simple      83 94 96 97
Integral convex polytope, Hilbert polynomial for      79 100
Integral convex polytope, lattice point in      77 100
Interior      100 117
Intersection, matrix      151
Intersection, number      52 78 81 102 105 187 189
Invariant, Cartier divisor      68 72 73
Invariant, Cartier divisor, associated to support function      69 81
Invariant, divisor      68 134
Invariant, for Tsuchihashi cusp singularity      162 163 165 167
Invertible sheaf, ample      83 93 130
Invertible sheaf, associated to Cartier divisor      69
Invertible sheaf, generated by global sections      76 79 81 103
Invertible sheaf, on toric variety      72
Invertible sheaf, vanishing theorem for      77
Invertible sheaf, very ample      82 83
Ishida's complex, of $\mathbb{Z}$-modules      119 122 124 126 127
Ishida's complex, of coherent sheaves      100 121 129 130 164
Ishida's construction theorem      125
Ishida's criteria      126 127
Isoperimetric inequality, the      79 187 188
Jurkiewicz — Danilov's theorem      102 108 127 134 135 196 197
Kummer holomorphic map      30 97
Lattice point in integral convex polytope      77 100
Laurent monomial      4 98 157
Lebesgue measure      29 33 50 100 104 157 178 186
Lefschetz's hyperplane section theorem      135
Leray spectral sequence, the      75
Lie algebra      115 136 137
Limit of one-parameter subgroup      10 16 96
Line bundle, ample      196
Link      53 127
Local duality theorem, the      129
Locally star closed subset of fan      119
Logarithmic, arithmetic genus      166
Logarithmic, Chern class      165 167
Logarithmic, derivative      71
Logarithmic, index      166
Logarithmic, pole      116
Logarithmic, zero      116 118 131
Macaulay's theorem      195
Manifold with corners      13 16 94 149 156 170 171
Manifold with corners, orbit in      13
Map of fans      19
Matsusaka's theorem      105
Matsushima's theorem      140
McMullen's conjecture      194
Minimal, model problem, the      51
Minimal, with respect to birational morphisms      37
Minimal, with respect to blowing-ups      37
Minimal, with respect to equivariant blowing-ups      59 142
Minkowski sum      79 180 183
Mixed volume      79 186 188
Miyaoka's inequality      98
Moment map      88 94 140
Moment map, generalized      94
Monodromy condition      56 168
Mori's theorem      106 111
Morphology for convex polytope      127 182 189 194
Motzkin's upper bound conjecture      194
Multiplicity      29 33
Murre — Matsumura — Oort — Artin's criterion      136
N-weight      52 55 144 168
N-weighted triangulation, admissible      55 59
Nagata's theorem      17
Nakai criterion      86
Nakai criterion, the toric      86 89
Nash blowing-up      37
Neron — Severi group      102 105
Newton polyhedron      148 185
Nondegenerate open convex cone      154
Nonnegative real locus of toric variety      13 95
Nonnegative real locus of toric variety, orbit in      13
Nonnegative scalar multiple of convex set      180 183
Nonprojective variety      51 65 84
Nonsingular, fan      15
Nonsingular, rational convex polyhedral cone      15 123
Nonsingular, toric variety      15
Nonsingularity of, closure of orbit in toric variety      12
Nonsingularity of, equivariant blowing-up      37
Normalization morphism      21
Normally of, closure of orbit in toric variety      12
Normally of, equivariant blowing-up      37
Numerical, effectiveness      106 107 111
Numerical, equivalence      105 108
O-sequence      195 196
One-parameter subgroup      4 10 138
One-parameter subgroup, limit of      10 16 96
One-parameter subgroup, unipotent      138 140
Open convex cone      153—155 178
Open convex cone, dual      155 178
Open convex cone, duality for      178
Open convex cone, homogeneous      154 178
Open convex cone, nondegenerate      154
Open convex cone, self-dual      154
Orbit, dimension of      10
Orbit, in manifold with corners      13
Orbit, in nonnegative real locus of toric variety      13
Orbit, in toric variety      10
Orbit, in toric variety, closure of      12 24 68 94 96
Orbit, in toric variety, nonsingularity of closure of      12
Orbit, in toric variety, normality of closure of      12
Order map for toric variety      13 149 156 171
Parabolic Inoue surface      171
Parabolic Inoue surface, degeneration of      171
Partially ordered set of faces      176 181
Period for complex torus      169
Periodic continued fraction      52 148
Perron — Frobenius theorem, the      172
Picard, group      67 74
Picard, number      59 65 74 89 105
Pick's formula      101
Pluri-genus      158
Poincare, duality theorem, the      102 197
Poincare, lemma, the      129
Poincare, residue map, the      120 164 166
Poincare, series      130 196 197
Polar convex, polyhedral set      27 181
Polar convex, polytope      88 93 102 182 184
Polar convex, set      156 181
Polarity      156 182 186
Polyhedral cone, convex      see “Convex polyhedral cone”
Polytope, convex      see “Convex polytope”
Positively homogeneous function      27 70 156 183 185
Primitive, element      24 49 67 152
Primitive, Hopf manifold      170
Primitive, Hopf surface      170
Projection of convex polyhedral cone      177
Projective, line      8 24 40 41 44 56 59 88 90 92 137 143 145
Projective, line, bundle      9 38 42 59 90 92 108 138 146
Projective, plane      9 16 38 42 56 59 88 90 92 98 108 137 143 145
Projective, plane, bundle      59 90 146
Projective, space      96
Projective, space, bundle      58
Projective, three-space      59 90 144
Projective, toric variety      17 65 see
Projective, variety, toric      see “Toric projective variety”
Properness criterion, the      21
Proportionality theorem, Hirzebruch — Mumford's      167
Pseudo-ample cone      105 111
Purely, elliptic singularity      158
Purely, periodic continued fraction      148 150 151
Quadratic irrational number      148
Quadratic irrational number, reduced      149 150
Quasi-Buchsbaum ring      159
Quasi-projective open set      10
Quotient singularity      102 107
Rational, convex polyhedral cone      2
Rational, partial polyhedral (r.p.p.) decomposition      2
Rational, point of sphere      52
Rational, singularity      23 125 157 158
Real, locus of toric variety      95
Real, quadratic field      149
Reduced quadratic irrational number      149 150
Regular, continued fraction      25
Regular, sequence      135 196
Reid's theorem      107—114
Relative, boundary      175
Relative, GAGA theorem, the      72
Relative, interior      174
Resolution of, singularity, equivariant      23 101 125 159
Resolution of, singularity, simultaneous      31
Resolution of, Tsuchihashi cusp singularity      159 160
Riemann — Roch theorem, the      78 165 167
Rigidity lemm, the      141
Root system      137 138 140 142—146
Roots, saturated set of      142
Saturated, additive subsemigroup      3
Saturated, set of roots      142
Segre embedding      97
Self-dual open convex cone      154
Self-intersection number      24 44 53 80 102 151 152 165
Semigroup algebra      5 17 126
Semistable reduction theorem, the      147
Separation theorem, the      174
Serre — Grothendieck duality theorem, the      101 125 129 131 197
Shed      50
Siegel, domain      154 155
Siegel, upper half plane      147 155
Signature      102 132
Simple convex polytope      94 97 102 182 190
simplex      96 181 182
Simplicial, cone      3 32 123 173 175
Simplicial, convex polytope      88 182 190 194
Simplicial, fan      69 87 129 135
Simultaneous resolution of singularities      31
Splitting of vertex      190 191
Stable singularity      128
Stanley — Reisner ring      127 135 196
Stanley's theorem      196 197
Star, closed subset of fan      119
Star, of vertex      53
Star, subdivision of fan      23 38 40 49 54—56 88 89
Stein factorization      20
Steinitz's theorem      189
Strictly upper convex support function      82 93 94 184 185
Strong Lefschetz theorem, the      102 135 197
Strongly convex cone      1 2 7 122 153 178 185
Subcomplex of fan      119
Subdivision of fan      23 38 46 101 159 168
Subdivision of fan, star      23 38 40 49 54—56 88 89
Sumihiro's theorem      10
Support function, equivariant line bundle associated to      68
Support function, for compact convex set      182
Support function, for convex polytope      77 184
Support function, for convex set      180
Support function, for ideal in convex cone      27 155 185
Support function, invariant Cartier divisor associated to      69 81
Support function, linear with respect to fan      59 66 67 72 196
Support function, strictly upper convex      82 93 94 184 185
Support of fan      2 16 66
Supporting hyperplane      183
Surface, area      187
Surface, of class $VII_0$      170—172
Symmetric domain      154 162
Symmetric domain, arithmetic quotient of      147 153 154 167 169
Symmetric domain, compact dual of      167
T-complex for toric divisor      162 172
Tangent, complex      128
Tangent, sheaf      87
Tautological line bundle      58
Teissier's, inequalities      102 103
Teissier's, problem      103 188 189
Terminal lemma      34 83 97
Terminal lemma, of White — Frumkin      36 48
Terminal, cone      36 46
Terminal, fan      46
Tetrahedral triangulation of sphere      190
Todd polynomial      163
Toric, affine variety      4 11 18
Toric, Chow's lemma      17 85
Toric, del Pezoo surface, classification of      88 89
Toric, del Pezzo surface      88 92
Toric, divisor      161 165 167
Toric, divisor, T-complex for      162 172
Toric, Fano threefold      51 88
Toric, Fano threefold, classification of      90
Toric, Fano variety      88
Toric, Fano variety, centrally symmetric      88 148
Toric, Fano variety, Einstein — Kaehler metric on      88 95 140
Toric, Kodaira vanishing theorem, the      101
Toric, Nakai criterion, the      86 89
Toric, projective variety      13 84 85 93 94 182 189 196
Toric, projective variety, Hilbert polynomial for      100
Toric, variety      7 9
Toric, variety, as algebraic variety      5 8 72 126 133
Toric, variety, as ambient space      97—99
Toric, variety, as scheme      8
Toric, variety, automorphsim group of      136 140 143 144
Toric, variety, classification of      42 55 59
Toric, variety, compact      16 17
Toric, variety, dimension of      7
Toric, variety, fundamental group of      14
Toric, variety, invertible sheaf on      72
Toric, variety, nonnegative real locus of      13 95
Toric, variety, nonsingular      15
Toric, variety, orbit in      see “Orbit in toric variety”
Toric, variety, orbit in nonnegative real locus of      13
Toric, variety, order map for      13 149 156 171
Toric, variety, real locus of      95
Toroidal embedding      115 147
Torus, algebraic      see “Algebraic torus compact“
Torus, compact      see “Compact torus”
Torus, complex      see “Complex torus”
Torus, embedding      7 see
Totally, positive element      150 155
Totally, real algebraic number field      154 155
Triangulation of, compact real manifold      161 168
Triangulation of, sphere      52 127 135 190—193 196
Triangulation of, sphere, icosahedral      65 191
Triangulation of, sphere, tetrahedral      190
Truncation of, dualizing complex      159
Truncation of, edge or vertex      190
Tsuchihashi cusp singularity      52 153 157 158 160 168 185
Tsuchihashi cusp singularity, dual      157
Tsuchihashi cusp singularity, infinitesimal deformation of      169
Tsuchihashi cusp singularity, invariant for      162 163 165 167
Tsuchihashi cusp singularity, resolution of      159 160
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