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Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties (Ergebnisse Der Mathematik Und Ihrer Grenzgebiete 3 Folge)

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This book is an advanced overview of the theory of toric varieties written for individuals with a strong background in algebraic geometry, topology, and algebra. It is very formal and not for a beginning course.

The author moves right into the necessary convex geometry in the first section of Chapter 1 and then defines a toric variety in the next section. He does not hesitate to use diagrams to illustrate the examples, which is good given the level of abstraction he employs in the book. The fundamental group of a toric variety is given an explicit characterization, but the proof is omitted unfortunately. This is followed by a discussion of when a toric variety is compact and nonsingular, with detailed proofs given. The Hironaka resolution of singularities theorem is discussed for toric varieties, the proof being a lot simpler of course in this case. A concrete realization of singularity resolution using continued fractions is given in the next section. The chapter ends with a very detailed and superb discussion of the birational geometry of toric varieties.

The next chapter is very involved and deals with Cartier divisors on toric varieties and toric projective varieties. The latter are related to convex polytopes by means of moment maps. In particular, integral convex polytopes have many connections with toric projective varieties, and these are outlined in detail in this chapter. A toric version of Mori's theorem is also outlined. Toric varieties offer a nice, intuitive picture of Mori's program for rational curves on projective varieties.

Chapter 3 deals with differential forms on toric varieties. The author employs the sheaf of germs of holomorphic vector fields with logarithmic zeroes and the sheaf of germs of p-forms with logarithmic poles to study holomorphic differential forms over toric varieties. In addition, Ishida complexes are used to study complexes of coherent sheaves on toric varieties. A very interesting discussion on the automorphism groups of toric varieties is given in terms of Cremona groups.

The last chapter discusses applications, such as Mumford toroidal embeddings, quotients of toric varieties, semisimple algebraic groups, and Newton polyhedra. Unfortunately, the author does not expound on these, but refers the reader to the literature. Instead, the author explains how to construct complex manifolds in dimension two by taking the quotient of an open set of a toric variety with respect to an action of a discrete map. Some interesting examples of compact quotientsof toric varieties are given, including complex tori, Hopf surfaces, and Inoue surfaces. The latter, for the case of parabolic Inoue surfaces, use elliptic curves in their constructions, interestingly.

The book does contain a review of convex geometry for a reader not well-versed in this area. There is a lot in this book, even though it is short. The price is very high so only for individuals seriously interested in this topic.