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Lewis J.D. — CRM Monograph Series, vol.10: A Survey of the Hodge Conjecture
Lewis J.D. — CRM Monograph Series, vol.10: A Survey of the Hodge Conjecture



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Название: CRM Monograph Series, vol.10: A Survey of the Hodge Conjecture

Автор: Lewis J.D.

Аннотация:

This book provides an introduction to a topic of central interest in transcendental algebraic geometry: the Hodge conjecture. Consisting of 15 lectures plus addenda and appendices, the volume is based on a series of lectures delivered by Professor Lewis at the Centre de Recherches Mathematiques (CRM). The book is a self-contained presentation, completely devoted to the Hodge conjecture and related topics. It includes many examples, and most results are completely proven or sketched. The motivation behind many of the results and background material is provided. This comprehensive approach to the book gives it a ``user-friendly'' style. Readers need not search elsewhere for various results. The book is suitable for use as a text for a topics course in algebraic geometry; includes an appendix by B. Brent Gordon.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 368

Добавлена в каталог: 30.10.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Jacobian, generalized intermediate      236
Jacobian, higher      348
Jacobian, intermediate      343
Jacobian, intermediate, Griffiths      171
Jacobian, intermediate, Weil      171
Jacobian, Lieberman      173
Kaehler form      32
Kaehler manifold      11 29 32
Kaehler manifold, test      34
Kaehler metric      11 32
Kodaira dimension      266
Kodaira — Serre duality      60
Koszul complex      130
Kuenneth formula      161
L(G) (space of left-invariant derivations on G)      302
Lefschetz (1,1) theorem      58 79
Lefschetz (1,1) theorem, analog for Stein manifolds      67
Lefschetz decomposition      166 275
Lefschetz decomposition theorem      165
Lefschetz group      316 326 328 330 344 347
Lefschetz hyperplane section      71
Lefschetz pairing      167
Lefschetz pencil      71 227
Lefschetz theorem, strong (hard)      165 167 227 268 283 290
Lefschetz theorem, strong (hard), for Chow groups      275
Lefschetz theorem, weak      166 167
Lefschetz theorem, weak, for Chow groups      239
Lefschetz theorem, weak, for hypersurfaces      270
Lefschetz vanishing cocycle      206
Lefschetz vanishing cone      74 228
Lefschetz vanishing cycle      72 74 206 228
Leray cohomology sheaf      113 228
Leray spectral sequence      113 228 231
Level      265
Level of a Chow group      272 277 288
Level of a Hodge structure      265 272 345
Lf(A) (Lefschetz group of Abelian variety A)      316
Lie algebra      301 304 315
Lie algebra of an algebraic group      302
Lie algebra, bracket      301
Lie algebra, reductive      344
Lie algebra, semisimple part      344
Lie group      314
Lie group, compact      3
Lie group, complex analytic      3
Lie group, connected      3
Lie(G) (Lie algebra of algebraic group G)      302
Line bundle      20
Line bundle, (weakly) negative      22
Line bundle, (weakly) positive      22
Line bundle, ample      23
Line bundle, associated to a divisor      49
Line bundle, Chern class      33 46
Line bundle, metric on      32
Line bundle, positive      23 33
Line bundle, tautological      21 112
Line bundle, very ample      23
Linear algebraic group      298
Linear equivalence      49 83 84
Linear system      23 83
Lines, Fano variety      201
Local invariant cycle property      229
Local monodromy transformation      73 206 229 241
Local ring      48
Local trivialization      17
Lsogenous      316
Lsogeny      306 316 320
Lsogeny, degree of      306
Lsogeny, dual      306
Manifold      1—13
Manifold, flag      91
Manifold, Grassmannian      12
Manifold, Hermitian      24
Manifold, Hodge      12
Manifold, Hopf      13 34
Manifold, Kaehler      11 29 32
Manifold, Moishezon      11 15
Manifold, projective algebraic      11
Manifold, Riemannian      24
Manifold, Stein      8 9 66
Maximal order      305
Meromorphic form      50 76 129
Meromorphic form, Poincare residue      50 76 131
Meromorphic function field      11 36
Meromorphic map      200
Metric connection      106 116
Metric on a complex vector bundle      24
MHS (mixed Hodge structure)      241
Mixed Hodge structure      241 284 292
Mixed Hodge structure, extension      292
Moishezon manifold      11 15
Monodromy group      206
Monodromy theorem      241
Monodromy transformation      see also “Local monodromy transformation”
MT(A) (Mumford — Tate group of Abelian variety A)      312
mt(V) (Lie algebra of MT(V))      312
MT(V) (Mumford — Tate group of rational Hodge structure V)      312
Mumford — Serre conjecture      349
Mumford — Tate group      312 313—317 328 332 333 337
Mumford — Tate group, special      312
Mumford’s CM fourfold      333
Mumford’s example      333
Mumford’s theorem      252
N(Y/X) (normal bundle of Y in X)      42
Node      9
Noether — Lefschetz theorem      207
Non-singular point      8
Non-singular variety      10
Normal bundle      42
Normal function      76 236
Normal function, homology class      77 237
Normal function, horizontal      236
Normal function, main theorem      237 240
Numerical equivalence      273 274
Ordinary double point      71 80
Orientation, standard      34
Parseval’s Inequality      147
Period      5 75 178
Pg(X) (geometric genus)      41
Picard group      45
Picard group, continuous part      46
Picard variety      46
Picard — Lefschetz formula      73
Picard — Lefschetz transformation      see also “Local monodromy transformation”
Pie(X) (Picard group)      45
Pluecker embedding      13
Pluricanonical differential form      254
Pohlmann’s criterion      337
Pohlmann’s theorem      338
Poincare divisor      84
Poincare duality      60
Poincare lemma      31
Poincare’s complete reducibility theorem      183 250 305 348
Poincare’s existence theorem      80 238
Polarization      12 14 97 313 314 316 321 324 349
Polarization, principal polarization      14
Positive involution      306
Presheaf      34
Presheaf, associated sheaf      35
Presheaf, complete      35
Primitive cohomology      123 128 134 166 193 211
Primitive homology      74 228
Principal torus      174
Privileged extension      235
Projection formula      111 189 257 286 290
Projective algebraic manifold      11
Projective bundle      111
Projective space      7
Projective space, canonical bundle      41
Projective space, cohomology      8
Projective space, Picard group      63
Projective variety      10
Proper transform      190 243
Prym variety, generalized      343
q(X) (irregularity of X)      83
QM-type      309
Quadratic transform      see also “Blow-up”
Quartic threefold, Hodge conjecture      209
Quartic threefold, uniruledness      98 213
Quasi-decomposable      320
Quaternion algebra      304 330
Quaternion algebra, canonical involution      304
Quaternion algebra, definite      304 324 325
Quaternion algebra, Hamiltonian      304 307 345
Quaternion algebra, indefinite      304 325
Quaternion algebra, maximal order      305
Quaternion algebra, order      305
Quaternion algebra, partially indefinite      336
Quaternion algebra, reduced norm      304
Quaternion algebra, reduced trace      304
Quaternionic multiplication      309 330 343
Rank of a vector bundle      17
rank(K,S) (rank of CM-type)      337
Rational algebraic cohomology class      63
Rational curve      3
Rational equivalence      49 83 194
Rational form      129
Rational function field      8
Rational map      200
rdim A (reduced dimension of A)      331
Real multiplication      308 329
Real multiplication, generalized      329
Reduced dimension      331
Regular map      182
Regularity theorem      151 154
rel dim(A) (relative dimension of A)      333
Relative dimension      333 335
Rellich lemma      148 151
Representation      300 313
Representation, contragredient      300
Representation, dual      300
Representation, G-equivariant      300
Representation, G-linear      300
Representation, half-spin      304 315
Representation, irreducible      300
Representation, odd      341
Representation, reductive      313
Representation, semisimple      313
Representation, spin      303 304 315
Representation, standard      300 315 346
Representation, subrepresentation      300
Representation, symplectic      315
Res (residue)      127
Residue      76 127
Residue map, generalized      86
Restriction of scalars functor      299 300
Riemann form      305 306 309 314 327 349
Riemann form, equivalent      306
Riemann — Roch theorem      see also “Hirzebruch — Riemann — Roch theorem”
Riemannian manifold      24
Riemannian metric      24
Rigidity      336
RM-type      329
Rosati involution      307 323
Sheaf      35
Sheaf of invariant cocycles      229
Sheaf, coherent      9 51
Sheaf, cohomology      37
Sheaf, exact sequence      37
Sheaf, fine      37
Sheaf, invertible      51
Sheaf, locally free      51
Sheaf, resolution      38
Sheaf, torsionless      38
Siegel half-space      310 335 343
Similtude norm      301
Simple Abelian fourfold      323
Simple abelian variety      326
Singular homology      60
Singular point      9 (see also “Double point”)
Skyscraper sheaf      37 110 127
SL(n) (special linear group of size n)      301
Smooth base locus      228
Smooth cubic      125
Smooth hypersurface      92
Smooth point      see also “Non-singular point”
SO(n) (special orthogonal group in n variables)      301
SO(V,S) (special orthogonal group on V with respect to symmetric form S)      301
Sobolev lemma      147 151
Sobolev Norm      147
Sobolev s-norm      146 151
Sobolev s-space      146
Sp(2n) (symplectic group in 2n variables)      301
Sp(V,E) (symplectic group on V with respect to symplectic form E)      301
Special orthogonal group      332
Special unitary group      301
Sporadic cycle      338
Standard conjectures, Grothendieck’s      349
Standard Lefschetz conjecture      259
Standard Lefschetz conjecture for Abelian variety      259
Standard Lefschetz conjecture for complete intersection      259
Standard Lefschetz conjecture for flag manifold      259
Standard orientation      34
Standard representation      318 324 332 346 347
Star operator      139 141 145 349
Stein manifold      9 66
Stein manifold, Theorem A      66
Stein manifold, Theorem B      67
Stoke’s theorem for analytic varieties      61
SU(n):=SU(n,0) (special unitary group on n variables)      301
SU(p,q) (special unitary group of signature p, q)      301
SU(V,H) (special unitary group on V with respect to Hermitian form H)      301
Sub-Hodge structure rational      346 347
Sub-Hodge structure rational, irreducible      347
supp (support)      37
Symmetric group      340
Symmetric product      75 247
Symplectic group      301 332 345
Symplectic group, semisimple      301
Symplectic representation      323 324 332 335 336
Symplectic similitude group      301
Symplectic space      303
T(X) (real tangent bundle)      18
T*(X) (real cotangent bundle)      18
Tangent bundle      18 29
Tangent space      8 302
Tate conjecture      348 349
td(V), td(X) (Todd class)      119
Todd class      119
Torsion algebraic cocycle      68
Torsion class      9 60
Total transform      189 243
Totally indefinite division quaternion algebra      326
Totally real algebraic number field      307
Totally real field      323 325—327
Transition functions      2 17
Triangular polymer without double bond      337
Triangular polymer without Hexatram      337
Triangular polymer without short cycle      337
Type H      330
Type of Abelian variety      307
U(k,n) (universal bundle)      18
U(n) (unitary group)      13 301
U(p,q) (unitary group of signature p, q)      301
U(V,H) (unitary group on V with respect to Hermitian form H)      301
Uniformization      9
Unimodular      175
Unirational variety      92 94 200
Unirational variety, Hodge conjecture      92 200
Uniruled variety      92 214
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