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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