Introduction 1
A Short History：Les débuts de la théorie des faisceaux by Christian Houzel 7
Ⅰ．Homological algebra 23
Summary 23
1.1．Categories and functors 23
1.2．Abelian categories 26
1.3．Categories of complexes 30
1.4．Mapping cones 34
1.5．Triangulated categories 38
1.6．Localization of categories 41
1.7．Derived categories 45
1.8．Derived functors 50
1.9．Double complexes 54
1.10．Bifunctors 56
1.11．Ind-objects and pro-objects 61
1.12．The Mittag-Leffler condition 64
Exercises to Chapter Ⅰ 69
Notes 81
Ⅱ．Sheaves 83
Summary 83
2.1．Presheaves 83
2.2．Sheaves 85
2.3．Operations on sheaves 90
2.4．Injective,flabby and flat sheaves 98
2.5．Sheaves on locally compact spaces 102
2.6．Cohomology of sheaves 109
2.7．Some vanishing theorems 116
2.8．Cohomology of coverings 123
2.9．Examples of sheaves on real and complex manifolds 125
Exercises to Chapter Ⅱ 131
Notes 138
Ⅲ．Poincaré-Verdier duality and Fourier-Sato transformation 139
Summary 139
3.1．Poincaré-Verdier duality 140
3.2．Vanishing theorems on manifolds 149
3.3．Orientation and duality 151
3.4．Cohomologically constructible sheaves 158
3.5．γ-topology 161
3.6．Kernels 164
3.7．Fourier-Sato transformation 167
Exercises to Chapter Ⅲ 178
Notes 184
Ⅳ．Specialization and microlocalization 185
Summary 185
4.1．Normal deformation and normal cones 185
4.2．Specialization 190
4.3．Microlocalization 198
4.4．The functor μhom 201
Exercises to Chapter Ⅳ 214
Notes 215
Ⅴ．Micro-support of sheaves 217
Summary 217
5.1．Equivalent definitions of the micro-support 218
5.2．Propagation 222
5.3．Examples：micro-supports associated with locally closed subscts 226
5.4．Functorial properties of the micro-support 229
5.5．Micro-support of conic sheaves 241
Exereises to Chapter Ⅴ 245
Notes 247
Ⅵ．Micro-support and microlocalization 249
Summary 249
6.1．The category Db（X；Ω） 250
6.2．Normal cones in cotangent bundles 258
6.3．Direct images 263
6.4．Microlocalization 268
6.5．Involutivity and propagation 271
6.6．Sheaves in a neighborhood of an involutive manifold 274
6.7．Microlocalization and inverse images 275
Exercises to Chapter Ⅵ 279
Notes 281
Ⅶ．Contact transformations and pure sheaves 283
Summary 283
7.1．Microlocal kernels 284
7.2．Contact transfornations for sheaves 289
7.3．Microlocal composition of kernels 293
7.4．Integral transformations for sheaves associated with submanifolds 298
7.5．Pure sheaves 309
Exercises to Chapter Ⅶ 318
Notes 318
Ⅷ．Constructible sheaves 320
Summary 320
8.1．Constructible sheaves on a simplicial complex 321
8.2．Subanalytic sets 327
8.3．Subanalytic isotropic sets and μ-stratifications 328
8.4．R-constructible sheaves 338
8.5．C-constructible sheaves 344
8.6．Nearby-cycle functor and vanishing-cycle functor 350
Exercises to Chapter Ⅷ 356
Notes 358
Ⅸ．Characteristic cycles 360
Summary 360
9.1．Index formula 361
9.2．Subanalytic chains and subanalytic cycles 366
9.3．Lagrangian cycles 373
9.4．Characteristic cycles 377
9.5．Microlocal index formulas 384
9.6．Lefschetz fixed point formula 389
9.7．Constructible functions and Lagrangian cycles 398
Exercises to Chapter Ⅸ 406
Notes 409
Ⅹ．Perverse sheaves 411
Summary 411
10.1．t-structures 411
10.2．Perverse sheaves on real manifolds 419
10.3．Perverse sheaves on complex manifolds 426
Exercises to Chapter Ⅹ 438
Notes 440
Ⅺ．Applications to O-modules and D-modules 441
Summary 441
11.1．The sheaf Ox 442
11.2．Dx-modules 445
11.3．Holomorphic solutions of Dx-modules 453
11.4．Microlocal study of Ox 459
11.5．Microfunctions 466
Exercises to Chapter Ⅺ 471
Notes 474
Appendix：Symplectic geometry 477
Summary 477
A.1．Symplectic vector spaces 477
A.2．Homogeneous symplectic manifolds 481
A.3．Inertia index 486
Exercises to the Appendix 493
Notes 495
Bibliography 496
List of notations and conventions 502
Index 509