Lewis J.D. — CRM Monograph Series, vol.10: A Survey of the Hodge Conjecture :: Электронная библиотека попечительского совета мехмата МГУ

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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.

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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

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Предметный указатель

$A(\mathbb{C})$ (underlying complex torus of Abelian variety A)      299
$A^{k}(X)$       272
$A^{k}(X)$ (Chow group of cycles algebraically equivalent to zero)      173
$A^{k}(X)_{ab-jac}$ (Chow group of cycles Abel — Jacobi equivalent to zero)      265
$A^{k}(X)_{inc}$ (Chow group of cycles incident equivalent to zero)      265
$A^{k}(X)_{tor}$ (torsion subgroup)      186
$B(\ast)$ (standard Lefschetz conjecture)      258
$B_{D}(X)$ (blow-up of X along a manifold D)      21
$b_{n}$ (X) (Betti number)      193
$B_{p}(X)$ (blow-up of X at a point p)      21
$CH^{k}(X)$ (Chow group)      173
$CH^{k}(X)_{hom}$ (Chow group of cycles homologous to zero)      173
$CH^{k}(X)_{num}$ (Chow group of cycles numerically equivalent to zero)      273
$CH^{k}(X)_{\mathbb{Q}}$       272
$CH_{k}(X)$ (Chow group)      272
$ch_{y}(V)$       121
$cl^{r}(z)$ (cycle class map)      275
$cl_{k}(z)$ (cycle class map)      287
$codim_{X}Y$ (codimension)      42 87
$C^{+}(V)$ (even Clifford algebra of orthogonal space V)      303
$C^{k}(X)$ (cycle group)      61
$C^{k}(X)_{alg}$ (cycles algebraically equivalent to zero)      180
$C^{k}(X)_{hom}$ (cycles homologous to zero)      173
$C^{k}(X)_{rat}$ (cycles rationally equivalent to zero)      180
$c_{k}(V)$ (Chern class of a vector bundle V)      106
$c_{k}(X)$ (Chern class of X)      106
$c_{t}(V)$       108
$c_{t}(X)$       118
$Der_{F}$ (A,A) (F-derivations from A to A)      3O2
$d\phi_{x}$ (differential of morphism $\phi$ at point x)      3O2
$End^{0}(A)$ (endomorphism algebra of Abelian variety A)      306
$E^{k}_{X}$       30
$E^{p,q}_{X}$       30
$F^{alg}$ (algebraic closure of F)      299
$F^{p}_{a}H^{l}(X, \mathbb{Q})$ (Grothendieck’s arithmetic filtration)      87
$F^{p}_{h}H^{l}(X, \mathbb{Q})$ = $F^{p}_{\mathbb{Q}}H^{l}(X, \mathbb{Q})$ (rational Hodge filtration)      88
$f_{\ast}$ (push-forward)      172
$Gal(F^{alg}/F)$ (absolute Galois group of F)      299
$GL(r,\mathbb{C})$ (general linear group)      17
$Gr^{l}_{N} H\cdot(X, \mathbb{Q})$ (graded piece of filtration by coniveau)      272
$Gr^{W}_{l}V$ (graded piece of weight filtration)      284
$G_{ab}$ (Abelian part of reductive linear algebraic group)      300
$G_{ss}$ (semisimple part of reductive linear algebraic group)      300
$h: \mathcal{S} \rightarrow GL(V\mathbb{R})$ (morphism defining a Hodge structure)      310
$h: \mathcal{S} \rightarrow GL(W\mathbb{R})$ (morphism defining a complex structure)      305
$Hodge^{p,p} (X, Q)$ (popular version of Hodge conjecture)      88
$H^{2k-1}_{a}$ (X, $\mathbb{Q}$ )      186
$H^{2k-1}_{h}$ (X, $\mathbb{Q}$ )      188
$H^{2p}_{et}(A_{0}\bigotimes F^{alg},\mathbb{Q}_{l}(p))$ (l-adic etale cohomology of $A_{0}$ )      348
$H^{k,k}(X,\mathbb{Q})$ (rational Hodge classes)      63
$H^{k,k}(X,\mathbb{Z})$ (integral Hodge classes)      60
$H^{k}_{de R}(X,\mathbb{C})$ (de Rham cohomology)      31
$H^{n}(X,\mathbb{Q})_{v}$ (vanishing cohomology)      206
$h^{p,q}(x)$ (Hodge number)      119
$H^{p,q}(X,\mathbb{C})$ (Dolbeault cohomology)      31
$H^{p,q}_{a}$       261
$H^{q}(X,\mathcal{S})$ (Cech cohomology)      44
$h_{1}: U(1) \rightarrow GL(V\mathbb{R})$ (morphism defining a Hodge structure      310
$h_{1}: U(1) \rightarrow GL(W\mathbb{R})$ (morphism defining a complex structure)      305
$h_{n}(X)$       191
$Int_{g}$ (inner automorphism by g)      302
$J(X_{m})$ (Jacobian of $X_{m}$ )      320
$J^{k}(X)$ (intermediate Griffiths Jacobian)      171
$J^{k}_{a}(X)$ (Lieberman Jacobian)      173
$J^{k}_{a}(X)_{tor}$ (torsion subgroup)      186
$J^{k}_{h}(X)$       188
$J^{k}_{W}(X)$ (intermediate Weil Jacobian)      171
$J^{p}(A)$ ( $p^{th}$ Weil intermediate Jacobian of A)      348
$j_{\ast}R^{n-1}f_{\ast}\mathbb{C}$ (sheaf of invariant cocycles)      229
$Mer_{X}$ (field of meromorphic functions on X)      36
$M_{m}$ (a set of m-tuples of integers)      320
$M_{m}(d)$ (a subset of $M_{m}$ )      320
$N^{p}H^{i}(X,\mathbb{Q})=F^{p}_{a}H^{i}(X,\mathbb{Q})$ (filtration by coniveau)      272
$N_{k}H_{i}(X,\mathbb{Q})$ (filtration by niveau)      285
$PGL(\mathbf{P}^{n})$ (projective general linear group)      24
$Pic^{0}(X)$ (Picard variety)      46
$Prim^{k}(M)$ (primitive cohomology)      166
$Prim^{p,q}(M)$ (primitive cohomology)      166
$Prim_{n}(M)$ (primitive homology)      232
$p^{\alpha}| N$ ( $p^{\alpha}$ exactly divides N)      341
$ReS_{K/F}V$ (Weil restriction of scalars functor)      299
$R^{i}f_{\ast}\mathbb{C}$ (Leray cohomology sheaf)      228
$T(G)_{e}$ (tangent space to G at identity e)      302
$T(X)_{\mathbb{C}}$ (complexified tangent bundle)      29
$Tr_{K/\mathbb{Q}}$ (trace from K to $\mathbb{Q}$ )      309
$T^{*}(X)_{\mathbb{C}}$ (complexified cotangent bundle)      29
$T^{0,1}(x)$       30
$T^{1,0}(x)$ (holomorphic cotangent bundle)      18
$T^{p}(X,V)$ , $T^{p}(X)$       121
$T_{0,1}(X)$       29
$T_{1,0}(X)$ (holomorphic tangent bundle)      18
$T_{y}(X)$       121
$T_{y}(X,V)$       121
$U(1,\mathbb{R})$ (unitary group of complex numbers of absolute value 1)      301
$V(\mu)$ (zero set of an ideal $\mu$ )      8
$V=H^{1}(A,\mathbb{Q})$       311
$V^{0,1}_{\mathbb{C},\sigma}$ (Hodge decomposition of $V_{C,\sigma}$ )      322
$V^{1,0}_{\mathbb{C},\sigma}$ (Hodge decomposition of $V_{C,\sigma}$ )      322
$V^{\vee}=Horn(V,F)$       300
$V_{F}$ (algebraic variety or vector space over field F)      299
$V_{k}=V_{F}\times_{spec F}spec K$ (base change from F to K)      299
$V_{sing}$ (singular set)      61
$V_{\mathbb{C},\sigma}$ (summand of Vc where K acts via $\sigma$ )      322
$V_{\mathbb{C}}=V\bigotimes_{\mathbb{Q}}\mathbb{C}$       310
$V_{\mathbb{R}}=V\bigotimes_{\mathbb{C}}\mathbb{C}$       310
$W=H_{1}(A,\mathbb{Q})$       311
$W_{l}$ (weight filtration)      241
$X_{1}(N)(\mathbb{C})$ (modular curve)      330
$X_{m}$ (Fermat curve of degree m)      320
$Z_{\ast}$       182
$\alpha*\beta$ (a convolution on $\mathfrak{A}_{m}^{r}\times\mathfrak{A}_{m}^{s}$ )      319
$\ast$ (star operator)      139
$\bigwedge^{*}V$ (exterior algebra of V)      304
$\bigwedge^{+}V$ (even part of exterior algebra of V)      3O4
$\bigwedge^{-}V$ (odd part of exterior algebra of V)      3O4
$\bigwedge^{2n}_{K}V$ (2n exterior power of V as K-vector space)      321
$\bigwedge^{2}V$ (exterior square of V)      304
$\bigwedge^{p,q}T^{\ast}(X)$       30
$\cdot$ (a mod m scalar multiplication)      319
$\chi(M)$ (Euler chararteristic)      193
$\chi(X,V)$       119
$\chi^{p}(X)$       120
$\chi^{p}(X,V)$       120
$\chi_{y}(X)$       121
$\chi_{y}(X,V)$       120
$\Delta$ (Laplacian)      140
$\delta: C^{q}(\mathcal{U},\mathcal{S}) \rightarrow C^{q+1}(\mathcal{U},\mathcal{S})$ (Cech coboundary operator)      44
$\delta=d^{\ast}$ (adjoint)      140
$\Delta_{\overline{\partial}}$ (Laplacian)      142
$\Delta_{\partial}$ (Laplacian)      160
$\Gamma(\mathcal{S})$ (global sections of the sheaf $\mathcal{S}$ )      37
$\Gamma_{1}(N)$ (congruence subgroup of level N)      33O
$\kappa(X)$ (Kodalra dimension)      269
$\kappa_{N,M}$       248
$\kappa_{N}$       248
$\langle,\rangle$ (natural pairing between V and $V^{\vee}$ )      300
$\mathbb{C}$ (complex numbers)      1
$\mathbb{C}(X)$ (function field of X)      8
$\mathbb{H}$ (Hamiltonian quaternion algebra)      304
$\mathbb{N}$ (natural numbers)      53
$\mathbb{Q}$ (rational numbers)      37
$\mathbb{Q}(m)$ (one-dimensional rational Hodge structure of type (-m,-m))      311
$\mathbb{R}$ (real numbers)      1
$\mathbb{S}:=Res_{\mathbb{C}/\mathbb{R}}G_{m/\mathbb{C}}$       300
$\mathbb{Z}$ (integers)      2
$\mathbf{P}^{n,\ast}$ (dual projective space)      23
$\mathbf{P}^{n}$ (complex projective space)      3
$\mathcal{D}p(A)$ ( $\mathbb{Q}$ -linear span of intersections of divisors on A)      311
$\mathcal{D}p(A)$ ( $\mathbb{Q}$ -linear span of p-fold intersections of divisors on A)      311
$\mathcal{F}(d)$       50

$\mathcal{G}$ (Galois group of $\mathbb{Q}^{alg}$ over $\mathbb{Q}$ )      318
$\mathcal{H}(A)$ (ring of all Hodge classes on A)      311
$\mathcal{H}^{k}(X)$ (harmonic space)      144
$\mathcal{H}^{p,q}(X)$ (harmonic space)      144
$\mathcal{H}^{p}(A)$ (codimension p Hodge classes on A)      311
$\mathcal{M}^{\ast}_{X}$       45
$\mathcal{M}_{X}$       45
$\mathcal{O}$       45
$\mathcal{O}$ (ring of integers)      308
$\mathcal{O}$ order in quaternion algebra)      309
$\mathcal{O}_{X}$       45
$\mathcal{O}_{X}(-D)$       50
$\mathcal{O}_{X}(d)$       50
$\mathcal{O}_{X}^{\ast}(D)$       50
$\mathcal{S}^{(N)}(X)$ (symmetric product of X)      247
$\mathcal{V}(A)$ (a sum of $\mathcal{W}(A)’s$ )      323
$\mathcal{W}(A)$ (Weil cycles on A)      321
$\mathcal{W}_{K}(A)$ (Weil classes with respect to K)      322
$\mathds{G}_{m} := GL(1)$       300
$\mathfrak{A}_{m}^{n}$ (a set of (n+2)-tuples mod m)      319
$\mathfrak{B}_{m}^{n}$ (a subset of $\mathfrak{A}_{m}^{n}$ )      319
$\mathfrak{gl}(V):=End(V)$ as Lie algebra      301
$\mathfrak{g} = \mathfrak{t} + \mathfrak{p}$ (Cartan decomposition of Lie algebra $\mathfrak{g}$ )      303
$\mathfrak{g}_{l}$ (l-adic Lie algebra)      349
$\mathfrak{hg}(V)$ (Lie algebra of Hg(V))      312
$\mathfrak{H}$ (complex upper half-plane)      330
$\mathfrak{sl}(V)$ (Lie algebra of SL(V))      303
$\mathfrak{so}(n)$ (Lie algebra of SO(n))      303
$\mathfrak{sp}(2n)$ (Lie algebra of Sp(2n))      303
$\mathfrak{S}_{m}$ (a quotient space of $\mathfrak{A}^{1}_{m}$ )      320
$\Omega^{p}_{X}$ (sheaf of germs of holomorphic p-form)      30
$\Omega^{p}_{Y}(logX)$       131
$\Omega_{X}$ (Fano variety of lines in X)      201
$\Omega_{X}(k)$ (Fano variety of k-planes on X)      218
$\overline{a}$ (integer congruent to a (mod m)      319
$\overline{\mathbb{Q}}$ (algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$ )      283
$\overline{\mathbb{Q}}(\mathbb{R})$ (algebraic closure of $\mathbb{Q}$ in $\mathbb{R}$ )      213
$\overline{\partial}$       31
$\overline{\partial}^{\ast}$ (adjoint)      142
$\partial$       31
$\partial^{\ast}$ (adjoint)      160
$\phi(m)$ (Euler phi function)      320
$\phi^{\ast}$ (pullback of a morphism)      302
$\Phi_{k}$ (Abel — Jacobi map)      173
$\Phi_{\ast}$ (cylinder homomorphism)      204
$\rho^{\vee}$ (dual representation)      300
$\rho_{m}(X)$ (Picard number)      207
$\sim alg$ (algebraic equivalent to)      180
$\sim rat$ (rational equivalent to)      173
$\theta$ (connection matrix)      103
$\Theta$ (curvature matrix)      104
$\varepsilon^{k}_{X}$       30
$\varepsilon^{p,q}_{X}$       30
$\varnothing$ (empty set)      8
$\zeta_{m}=e^{2\pi i/m}$ (primitive $m^{th}$ root of unity)      320
$^{t}Z$       182
= supp z (support of z)      294
$|\alpha|$ (a norm on $\mathfrak{A}_{m}^{n}$ )      319
( )’ (canonical involution on quaternion algebra)      304
(A,K) (Abelian variety of Weil type)      308
A*(X) (Chow group of cycles algebraically equivalent to zero)      181
Abel — Jacobi map      5 75 176 292
Abel — Jacobi map, analyticity      75 185
Abel — Jacobi map, functoriality properties      183
Abelian variety      12 93 259 298 299 307 314
Abelian variety, Albert classification      307
Abelian variety, algebraic family      336
Abelian variety, degenerate      339 340
Abelian variety, elliptic curve      307 329 330 342
Abelian variety, endomorphism algebra      306 348
Abelian variety, endomorphism ring      336
Abelian variety, four-dimensional      319
Abelian variety, fourfold      308 323 333 335 343 348
Abelian variety, general      310 342 343 345
Abelian variety, index of degeneracy      336
Abelian variety, level structure      336 348
Abelian variety, morphism      299
Abelian variety, nondegenerate      338 339
Abelian variety, of CM-type      309 327 329 337 339 340 349
Abelian variety, of Fermat type      319—321 334
Abelian variety, of Hodge type, family      349
Abelian variety, of prime dimension      308 326 339 345
Abelian variety, of Ribet type      308
Abelian variety, of RM-type      329
Abelian variety, of type H      330
Abelian variety, of type III      324 334
Abelian variety, of Weil type      308 319 321—323 333 334 341 349
Abelian variety, polarization      306 335 336 348 349
Abelian variety, principal polarization      14 175 343
Abelian variety, simple      306 324 326 332 333 335 336 338 344
Abelian variety, stably degenerate      335
Abelian variety, stably nondegenerate      330—332 335 342 345 348
Abelian variety, threefold      342 343
Abelian variety, with multiplication by an imaginary quadratic field      307
Abelian variety, with quaternionic multiplication      309 343
Abelian variety, with real multiplication      308
Abel’s theorem      5 83
Ad (adjoint representation of G on $T(G)_{e}$ )      3O2
ad (adjoint representation of Lie(G) on itself)      302
Adequate equivalence      194
Adjoint representation      302
Adjunction formula      43 357
Adjunction formula for complete intersections      43
Adjunction formula for hypersurfaces      43
Affine variety      8
Alb(X) (Albanese variety)      171
Albanese map      178
Albanese variety      171
Albert classification      307 308
Algebraic cocycle      58 189
Algebraic equivalence      84 180 194
Algebraic equivalence vs. homological equivalence      84
Algebraic group      299 301 316
Algebraic group, affine      299
Algebraic group, linear      299 300
Algebraic group, real reductive      303
Algebraic group, reductive      300 301 313
Algebraic group, semisimple      300
Algebraic torus      300 315 328 337
Almost complex structure      30
Alternating group      340
Analytic hypersurface      42
Analytic map      1 182 183
Analytic subset      11
Analytic subvariety      48
Arithmetic filtration      87 272
B(X) (standard Lefschetz conjecture)      259
Base locus      71 205
Bertini’s theorem      52 203 220
Betti number      193
Bezout’s Theorem      123 203 254
Bianchi identity      116
Bilinear form skew-symmetric      301 303 325
Bilinear form symmetric      303
Bilinear form, alternating      324 327
Bimeromorphic map      200
Birational map      200
Birationally invariance of GHC(1,n,-)      200
Bloch conjecture      254
Bloch — Beilinson filtration      273 281
Blow-up      21
Blow-up along a manifold      27
Blow-up of a point      21
Bounded symmetric domain      314 349
Branching rule      336
C(V) (Clifford algebra of orthogonal space V)      303
c(V) (total Chern class)      106
c-closed      248
c-closed dimension      248
Canonical bundle      41

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