Introduction to Toric Varieties. (AM-131), Volume 131

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories. The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are quoted without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.

Ch. 1 Definitions and examples
1.1 Introduction 3

1.2 Convex polyhedral cones 8

1.3 Affine toric varieties 15

1.4 Fans and toric varieties 20

1.5 Toric varieties from polytopes 23

Ch. 2 Singularities and compactness

2.1 Local properties of toric varieties 28

2.2 Surfaces; quotient singularities 31

2.3 One-parameter subgroups; limit points 36

2.4 Compactness and properness 39

2.5 Nonsingular surfaces 42

2.6 Resolution of singularities 45

Ch. 3 Orbits, topology, and line bundles

3.1 Orbits 51

3.2 Fundamental groups and Euler characteristics 56

3.3 Divisors 60

3.4 Line bundles 63

3.5 Cohomology of line bundles 73

Ch. 4 Moment maps and the tangent bundle

4.1 The manifold with singular corners 78

4.2 Moment map 81

4.3 Differentials and the tangent bundle 85

4.4 Serre duality 87

4.5 Betti numbers 91

Ch. 5 Intersection theory

5.1 Chow groups 96

5.2 Cohomology of nonsingular toric varieties 101

5.3 Riemann-Roch theorem 108

5.4 Mixed volumes 114

5.5 Bezout theorem 121

5.6 Stanley's theorem 124

Notes 131

References 149

Index of Notation 151

Index 155