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The theory of toric varieties (also called torus embeddings) describes a fascinating interplay between algebraic geometry and the geometry of convex figures in real affine spaces. This book is a unified up-to-date survey of the various results and interesting applications found since toric varieties were introduced in the early 1970's. It is an updated and corrected English edition of the author's book in Japanese published by Kinokuniya, Tokyo in 1985. Toric varieties are here treated as complex analytic spaces. Without assuming much prior knowledge of algebraic geometry, the author shows how elementary convex figures give rise to interesting complex analytic spaces. Easily visualized convex geometry is then used to describe algebraic geometry for these spaces, such as line bundles, projectivity, automorphism groups, birational transformations, differential forms and Mori's theory. Hence this book might serve as an accessible introduction to current algebraic geometry. Conversely, the algebraic geometry of toric varieties gives new insight into continued fractions as well as their higher-dimensional analogues, the isoperimetric problem and other questions on convex bodies. Relevant results on convex geometry are collected together in the appendix.
List of contents
1. Fans and Toric Varieties.- 1.1 Strongly Convex Rational Polyhedral Cones and Fans.- 1.2 Toric Varieties.- 1.3 Orbit Decomposition, Manifolds with Corners and the Fundamental Group.- 1.4 Nonsingularity and Compactness.- 1.5 Equivariant Holomorphic Maps.- 1.6 Low Dimensional Toric Singularities and Finite Continued Fractions.- 1.7 Birational Geometry of Toric Varieties.- 2. Integral Convex Polytopes and Toric Projective Varieties.- 2.1 Equivariant Line Bundles, Invariant Cartier Divisors and Support Functions.- 2.2 Cohomology of Compact Toric Varieties.- 2.3 Equivariant Holomorphic Maps to Projective Spaces.- 2.4 Toric Projective Varieties.- 2.5 Mori's Theory and Toric Projective Varieties.- 3. Toric Varieties and Holomorphic Differential Forms.- 3.1 Differential Forms with Logarithmic Poles.- 3.2 Ishida's Complexes.- 3.3 Compact Toric Varieties and Holomorphic Differential Forms.- 3.4 Automorphism Groups of Toric Varieties and the Cremona Groups.- 4. Applications.- 4.1 Periodic Continued Fractions and Two-Dimensional Toric Varieties..- 4.2 Cusp Singularities.- 4.3 Compact Quotients of Toric Varieties.- Appendix. Geometry of Convex Sets.- A.1 Convex Polyhedral Cones.- A.2 Convex Polyhedra.- A.3 Support Functions.- A.4 The Mixed Volume of Compact Convex Sets.- A.5 Morphology for Convex Polytopes.- References.
