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**Introduction to toric varieties. The 1989 William H. Roever lectures in geometry.**
*(English)*
Zbl 0813.14039

Annals of Mathematics Studies. 131. Princeton, NJ: Princeton University Press. xi, 157 p. (1993).

From the author’s preface: “Toric varieties provide an ... elementary way to see many examples and phenomena in algebraic geometry. ..., toric varieties have provided a remarkably fertile testing ground for general theories. Toric varieties correspond to objects much like the simplicial complexes ..., and all the basic concepts on toric applications and relations in commutative algebra, geometry of polyhedra and combinatorics toric varieties became more and more interesting and fashionable.

Simply speaking, a toric variety is a normal variety that contains an algebraic torus as a dense open subset together with an action of the torus on the variety that extends the natural action of the torus on itself. Concretely, it will be constructed from a lattice and a fan of strongly convex rational polyhedral cones. The detailed explanation of this concept is contained in the first chapter. Toric varieties may have (rational) singularities. Whether it is the case or not can be described by the cone structure, only. This fact, the resolution of the singularities, the surface case and the compactness property are treated out in the second chapter. In chapter 3 the fundamental groups, Euler characteristics and the cohomology of line bundles are studied. This is continued in chapter 4 by the consideration of the tangent bundle and the Serre duality. The Betti numbers of nonsingular toric varieties are computed. The closing chapter contains results about the basis of the Chow groups (resp. cohomology groups) and the whole Chow ring in the nonsingular projective case. But last not least, it contains an application to the geometry of polytopes in terms of R. Stanley’s proof of P. McMullen’s upper bound conjecture.

After T. Oda’s book “Convex bodies and algebraic geometry” (1988; Zbl 0628.52002), the book under review is the second one giving an introduction to the theory of toric varieties. Although the author denotes it a mini-course, it has many of the characteristics of a text book. Abundantly interspersed exercises help the beginners in the subjet to learn the fundamental things. Many hints to connections with other subjects are given by the notes placed together at the end of the book.

Simply speaking, a toric variety is a normal variety that contains an algebraic torus as a dense open subset together with an action of the torus on the variety that extends the natural action of the torus on itself. Concretely, it will be constructed from a lattice and a fan of strongly convex rational polyhedral cones. The detailed explanation of this concept is contained in the first chapter. Toric varieties may have (rational) singularities. Whether it is the case or not can be described by the cone structure, only. This fact, the resolution of the singularities, the surface case and the compactness property are treated out in the second chapter. In chapter 3 the fundamental groups, Euler characteristics and the cohomology of line bundles are studied. This is continued in chapter 4 by the consideration of the tangent bundle and the Serre duality. The Betti numbers of nonsingular toric varieties are computed. The closing chapter contains results about the basis of the Chow groups (resp. cohomology groups) and the whole Chow ring in the nonsingular projective case. But last not least, it contains an application to the geometry of polytopes in terms of R. Stanley’s proof of P. McMullen’s upper bound conjecture.

After T. Oda’s book “Convex bodies and algebraic geometry” (1988; Zbl 0628.52002), the book under review is the second one giving an introduction to the theory of toric varieties. Although the author denotes it a mini-course, it has many of the characteristics of a text book. Abundantly interspersed exercises help the beginners in the subjet to learn the fundamental things. Many hints to connections with other subjects are given by the notes placed together at the end of the book.

Reviewer: K.Drechsler (Halle-Neustadt)

### MSC:

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14J17 | Singularities of surfaces or higher-dimensional varieties |

14L30 | Group actions on varieties or schemes (quotients) |

14F35 | Homotopy theory and fundamental groups in algebraic geometry |

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

51M20 | Polyhedra and polytopes; regular figures, division of spaces |