Introduction to Toric Varieties (Annals of Mathematics Studies v. 131)
By: William Fulton (author)Paperback
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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. 1Definitions and examples1.1Introduction31.2Convex polyhedral cones81.3Affine toric varieties151.4Fans and toric varieties201.5Toric varieties from polytopes23Ch. 2Singularities and compactness2.1Local properties of toric varieties282.2Surfaces; quotient singularities312.3One-parameter subgroups; limit points362.4Compactness and properness392.5Nonsingular surfaces422.6Resolution of singularities45Ch. 3Orbits, topology, and line bundles3.1Orbits513.2Fundamental groups and Euler characteristics563.3Divisors603.4Line bundles633.5Cohomology of line bundles73Ch. 4Moment maps and the tangent bundle4.1The manifold with singular corners784.2Moment map814.3Differentials and the tangent bundle854.4Serre duality874.5Betti numbers91Ch. 5Intersection theory5.1Chow groups965.2Cohomology of nonsingular toric varieties1015.3Riemann-Roch theorem1085.4Mixed volumes1145.5Bezout theorem1215.6Stanley's theorem124Notes131References149Index of Notation151Index155
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- ID: 9780691000497
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