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Part 1 Combinatorial Convexity 3

Ⅰ．Convex Bodies 3

1．Convex sets 3

2．Theorems of Radon and Carathéodory 8

3．Nearest point map and supporting hyperplanes 11

4．Faces and normal cones 14

5．Support function and distance function 18

6．Polar bodies 24

Ⅱ．Combinatorial theory of polytopes and polyhedral sets 29

1．The boundary complex of a polyhedral set 29

2．Polar polytopes and quotient polytopes 35

3．Special types of polytopes 40

4．Linear transforms and Gale transforms 45

5．Matrix representation of transforms 53

6．Classification of polytopes 58

Ⅲ．Polyhedral spheres 65

1．Cell complexes 65

2．Stellar operations 70

3．The Euler and the Dehn-Sommerville equations 78

4．Schlegel diagrams,n-diagrams,and polytopality of spheres 84

5．Embedding problems 88

6．Shellings 92

7．Upper bound theorem 96

Ⅳ．Minkowski sum and mixed volume 103

1．Minkowski sum 103

2．Hausdorff metric 107

3．Volume and mixed volume 115

4．Further properties of mixed volumes 120

5．Alexandrov-Fenchel's inequality 129

6．Ehrhart's theorem 135

7．Zonotopes and arrangements of hyperplanes 138

Ⅴ ．Lattice polytopes and fans 143

1．Lattice cones 143

2．Dual cones and quotient cones 148

3．Monoids 154

4．Fans 158

5．The combinatorial Picard group 167

6．Regular stellar operations 179

7．Classification problems 186

8．Fano polytopes 192

Part 2 Algebraic Geometry 199

Ⅵ．Toric varieties 199

1．Ideals and affine algebraic sets 199

2．Affine toric varieties 214

3．Toric varieties 224

4．Invariant toric subvarieties 234

5．The torus action 238

6．Toric morphisms and fibrations 242

7．Blowups and blowdowns 248

8．Resolution of singularities 252

9．Completeness and compactness 257

Ⅶ．Sheaves and projective toric varieties 259

1．Sheaves and divisors 259

2．Invertible sheaves and the Picard group 267

3．Projective toric varieties 273

4．Support functions and line bundles 281

5．Chow ring 287

6．Intersection numbers.Hodge inequality 290

7．Moment map and Morse function 296

8．Classification theorems.Toric Fano varieties 303

Ⅷ．Cohomology of toric varieties 307

1．Basic concepts 307

2．Cohomology ring of a toric variety 314

3．？ech cohomology 317

4．Cohomology of invertible sheaves 320

5．The Riemann-Roch-Hirzebruch theorem 324

Summary:A Dictionary 329

Appendix Comments,historical notes,further exercises,research problems,suggestions for further reading 331

References 343

List of Symbols 359

Index 363