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Donaldson K., Kronheimer P.B. — Geometry of Four-Manifolds
Donaldson K., Kronheimer P.B. — Geometry of Four-Manifolds

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Название: Geometry of Four-Manifolds

Авторы: Donaldson K., Kronheimer P.B.

Аннотация:

The last ten years have seen rapid advances in the understanding of differentiable four-manifolds, not least of which has been the discovery of new 'exotic' manifolds. These results have had far-reaching consequences in geometry, topology, and mathematical physics, and have proven to be a mainspring of current mathematical research. This book provides a lucid and accessible account of the modern study of the geometry of four-manifolds. Consequently, it will be required reading for all those mathematicians and theoretical physicists whose research touches on this topic. The authors present both a thorough treatment of the main lines of these developments in four-manifold topology — notably the definition of new invariants of four-manifolds — and also a wide-ranging treatment of relevant topics from geometry and global analysis. All of the main theorems about Yang-Mills instantons on four-manifolds are proven in detail. On the geometric side, the book contains a new proof of the classification of instantons on the four-sphere, together with an extensive discussion of the differential geometry of holomorphic vector bundles. At the end of the book the different strands of the theory are brought together in the proofs of results which settle long-standing problems in four-manifolds topology and which are close to the frontiers of current research. Co-author Donaldson is the 1994 co-recipient of the prestigious Crafoord Prize.


Язык: en

Рубрика: Математика/Алгебра/Алгебраическая геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1990

Количество страниц: 440

Добавлена в каталог: 12.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$CP^2$ (complex projective plane)      3 124 127 237 398 400
$E_s \oplus E_s$ (form)      317
$F^{(r)}$ (lower bundle)      318
$g_E, g_r$ (adjoint bundle)      32
$L^p$ theory      291 425
$p_g$, (geometric genus)      12 376
$R^4$, exotic      28
$R_p$ (complex surface)      14 401
$S_4$, (complex surface)      12
$V_{\Sigma}$ (codimension-2 submanifold)      193
$\delta$ (elliptic operator)      55
$\delta$(operator)      43
$\delta$(operator), integrable      45
$\hat{A}$ class      185
$\mathscr{A}$ (space of all connections)      33
$\mathscr{A}^{||.||}$ (space of connections)      209
$\mathscr{G}$ (gauge group)      34
$\mathscr{L}_{\Sigma}$ (line bundle)      189 271
$\mathscr{P}$ dibar      53
$\mu$ (map on homology)      177 271
$\nu$ (four-dimensional class)      180 361
$\nu(\Sigma)$ (tubular neighbourhood)      192 321
$\zeta$-function      254
ADHM (Atiyah, Drinfeld, Hitchin, Martin)      96
ADHM (Atiyah, Drinfeld, Hitchin, Martin) correspondence      98
ADHM (Atiyah, Drinfeld, Hitchin, Martin) data      97 103 117
ADHM (Atiyah, Drinfeld, Hitchin, Martin) equations      97 250
Adjoint, formal      37 422
Agmon, S. and Nirenberg, L.      151
Almost-complex structure      11
Anomaly      206 257
Anti-self-dual (ASD), condition      38
Anti-self-dual (ASD), equations      39 47
Anti-self-dual (ASD), form      7 152
Asymptotic, behaviour on $R^4$      103
Asymptotic, behaviour on $R^4$, form of heat kernel      255
Atiyah — Singer index theorem      184 265 271
Automorphism, of homology      363
Base-point fibralion      174 180
Belinson spectral sequence      124
Bianchi identity      37 71
Blow-up      13 377
Bootstrap      57 61 166
Boundary, manifold with      16
Boundary, manifold with, value problem      68 192
Branched cover      14 375 400
Bundle of Banach spaces      146
Bundle, holomorphic      43
Bundle, principal      31
Bundle, semi-stable      209 230
Bundle, stable      209 211
Bundle, universal      397 413
Cauchy kernel      51 258
Cauchy — Riemann, equations      51
Centre of mass, of instanton      115
Characteristic class      6 39 42
Characteristic element      5
Chern character      184
Chern class      11
Chern — Simons invariant      164 178
Chern — Weil theory      39 42 164 195
Classifying space      174
Clifford multiplication      77
Cobordism      18 143
Cohomology bundle      82
Cohomology, de Rham      2
Cohomology, de Rham, of $\mathscr{B}$      176 178
Compactification of moduli space      156 230 273 322 412
Compactness      54
Compactness, of structure group      64
Completion, of moduli space      117
Complex analytic space      241
Complex structure, compatible with connection      45 75
Complex surface      10 208
Complex, deformation      138 237
Complex, deformation, Dolbeault      105
Complex, deformation, Koszul      79 391
Cone      149
Configurations of points      391 406
Conformal invariance      39 41 96 100
Conformal invariance of conformal Laplacian      101
Conformal invariance of Dirac operator      102
Connected sum      4 26 284 303 363 371
Connected sum of ASD connections      290 303
Connected sum of connections      286
Connected sum, bundle      283
Connected sum, multiple      305
Connection      31
Connection, (anti-)self-dual (ASD)      38 40 75
Connection, compatible with complex structure      45
Connection, concentrated      172 198 323 326
Connection, convergence of      59 160
Connection, cutting off      168
Connection, flat      36
Connection, form      34
Connection, framed      173
Connection, Hermitian Yang — Mills      215
Connection, ideal      157
Connection, Levi-Civita      31 37
Connection, partial      44
Connection, product      33
Connection, reducible      131 149 186
Connection, regular ASD      147
Convergence, of connections      59 160 308
Convergence, of connections, strong      309 369
Convergence, of connections, weak      157 308 367 368
Coulomb gauge      55 57 167
Coulomb gauge, local      58
Coulomb gauge, relative      131
Coulomb gauge, relative, symmetry of      56
Covariant constant section      33
Curvature      36
Curvature, type (1, 1)      46 75
Cutting-off, connections      168
Deformation complex      138 237
Deformation, (uni-)versal      238
Degree      208 378 386
Delta function, in energy density      117
Derivative, covariant      32
Derivative, exterior      35
Determinant line bundle      187 188 208
Determinant line bundle, curvature      252 255
Determinant line bundle, extending      263 273
Difference operator      267
Differential form      see form
Dimension, of moduli space      138 298
Dimension, of singular set      357 384
Dimension, virtual      138 263 264 383
Dimension-counting      351
Dirac operator      77 185 336
Dirac operator and Dolbeault complex      80
Dirac operator, coupled      77 263
Dolbeault complex      43 79
Double complex      88 104
Double plane      14 375 400
Doubling construction      67
Electromagnetism      38
Elliptic operator      54 421
Energy density, of instanton      117
Euler class      6 342 370
Euler — Lagrange equations      41
Even type      4
Excision property      264
Excision property for families      271
Exotic, $R^4s$      28
Ext group      394
Extended equations      372
Extension, of bundles      388
Family of connections      174
Family of equations      142
Family, flamed      175
Family, universal      175
Fintashel, R. and Stern, R.      318
Form, on $\mathscr{B}$      197
Formal adjoint      37 422
Forms, self-dual, anti-self-dual      7 152
Fourier transform for holomorphic bundles      85
Fourier transform, for ASD connections      75 84 123
Fourier transform, inversion      93
Frame      32
Fredholm, alternative      55 57 66 84 167 211 422
Fredholm, alternative map      136 145 265
Fredholm, alternative operator      423
Freed, D. and Uhlenbeck, K.K.      146
Freedman, M.H. and Taylor, L.      24
Frobenius theorem      49
Fundamental group      1
Fundamental group, representation of      49
G1 (gluing parameters)      286 294 324
Gandle      18 161
Garmonic form      9 12 141
Gauge group      34 129
Gauge group, complex      210 225
Gauge, choice of      34 49
Gauge, Coulomb      55 57 167
Gauge, fixing      49 53 295
Gauge, Lorentz      55
Gauge, radial      54 150
General position      141
Generic metric      149
Geometric genus      12 376
Gieseker's projective embedding      379
Global Ext      394
Gluing      283
Gradient flow      217 233
Green's operator      101
h-cobordism      21 28 345
h-cobordism theorem      18
Hasse — Minkowski classification      5
Heat equation      217 221
Heat kernel, asymptotic form      255
Hermitian Yang — Mills condition      215
Hilbert criterion      249 381
Hirzebruch signature theorem      6
Hodge metric      208 378
Hodge theorem      9 12 264
Holomorphic bundle, construction      388
Holomorphic bundle, stable      347; see also Bundle
Holonomy      49 177
Homology      1
Homology 3-sphere      17
Homotopy type      15 174
Hyperkaehler structure      125
Hyperplane class      12
Hypersurface      12
Ideal instanton      123
Ideal sheaf      391
Implicit function theorem      57 65 420
Index formula      137 264 298
Index theorem      184 271
Index, of critical point      19
Index, of family      183 338
Index, of operator      182
Instanton      96 126 323
Instanton, charge one      116
Instanton, ideal      123
Integrability, of flat connection      48
Integrability, of holomorphic structure      45 50
Integral operators      51 101 425
Integration along fibre      195
Intersection form      2 18
Intersection form, definite      25 317 326
Intersection form, even      4 25 326
Intersection form, indefinite      5
Intersection form, non-standard part      319
Invariant      26
Invariant calculation      410
Invariant integer      343
Invariant polynomial      341 349 352
Invariant theory      249
Inverse function theorem      419
Irrational surface      13
Isomorphism class, of SO(3) bundle      41
Isoperimetric inequality      424
Isotropy group      132
Jacobian torus      50 83 182
k (characteristic number)      42
K-theory      182
K3 surface      14 341 345 377 417
K3 surface, not connected sum      346
Kaehler identities      212
Kaehler manifold      80
Kodaira embedding theorem      378
Koszul complex      79 391
Lattice, 4-dimensional      83
Lattice, dual      83
Lattice, Leech      5 319
Leibnitz rule      32 35
Line bundle, canonical      80
Line bundle, complex      2
Line bundle, determinant      187 188 208 252 255 263
Line bundle, positive      378
Link of lower strata      333
Link of reducible connection      186
Link, framed      16
Link, homology class of      335
Local centre and scale      323 340
Manifold with boundary      16
Manifold, topological      27
Maximum principle      221
Mehta, V.B. and Ramanathan, A.      384
Method of continuity      63 71 167 261 297
Metric on instanton moduli space      125
Metric, generic      148
Milnor, J.      15
Moduli space      118 134
Moduli space as algebraic variety      384
Moduli space as complex analytic space      241
Moduli space for $R_4$      405
Moduli space, compactified      156 273 322 412
Moduli space, dimension      138 263 298
Moduli space, empty      366
Moduli space, examples      126 139
Moduli space, local model      137 149 237 324
Moduli space, multiplicity      408
Moduli space, one-dimensional      318
Moduli space, regular      147 149
Moduli space, universal      149 344
Moment map (momentum map)      244
Moment map (momentum map) and ADHM construction      125 250
Monad      81
Monad and spectra      397
Monopoles      124
Morse function      19 143
Mukai, S.      85
multi/set (symmetric product)      127 391 157
Neck      263
Neck, shrinking      309 365
Neumann problem      192
Odd type      4
Orbit space of connections      129
Orbit space, is Hausdorff      130
Orientation, of moduli space      203 281
Ouillen, D.G.      208 244
Parallel transport      33
Patching argument      158
Path space      179
Perturbations      155 373
Physics, mathematical      38
Picard group      402
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