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**Convex bodies and algebraic geometry. An introduction to the theory of toric varieties.**
*(English)*
Zbl 0628.52002

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 15. Berlin etc.: Springer-Verlag. VIII, 212 p.; DM 148.00 (1988).

The theory of toric varieties, torus embeddings, relates algebraic geometry to the geometry of convex figures in real affine spaces. Ever since the foundation of the theory was laid down at the beginning of 1970’s, tremendous progress has been made and various interesting applications have been found. This book tries to explain in unified form as many results as possible obtained so far on toric varieties. Though this book may be a high level of introduction, this is a book worth reading as enlightenments for modern mathematics.

Chapter 1. Fans and toric varieties: We define “fans” in real affine spaces and then maps among them. Accordingly, we can construct many interesting examples of complex analytic spaces and holomorphic maps. We can also reduce interesting basic questions on the birational geometry of toric varieties into those on subdivisions of fans.

Chapter 2. Integral convex polytopes and toric projective varieties: We deal with the cohomology of compact toric varieties as well as toric varieties embeddable into projective spaces.

Chapter 3. Toric varieties and holomorphic differential forms: We deal with holomorphic differential forms on toric varieties. Not only are they related to deformations and degenerations of complex analytic spaces, but they might also be of some interest in commutative algebra.

Chapter 4. Applications: We can see basic results on convex sets which we need in this book, without proof.

Chapter 1. Fans and toric varieties: We define “fans” in real affine spaces and then maps among them. Accordingly, we can construct many interesting examples of complex analytic spaces and holomorphic maps. We can also reduce interesting basic questions on the birational geometry of toric varieties into those on subdivisions of fans.

Chapter 2. Integral convex polytopes and toric projective varieties: We deal with the cohomology of compact toric varieties as well as toric varieties embeddable into projective spaces.

Chapter 3. Toric varieties and holomorphic differential forms: We deal with holomorphic differential forms on toric varieties. Not only are they related to deformations and degenerations of complex analytic spaces, but they might also be of some interest in commutative algebra.

Chapter 4. Applications: We can see basic results on convex sets which we need in this book, without proof.

Reviewer: S.Ohyanagi

### MSC:

52-02 | Research exposition (monographs, survey articles) pertaining to convex and discrete geometry |

52A05 | Convex sets without dimension restrictions (aspects of convex geometry) |

32J15 | Compact complex surfaces |

14E05 | Rational and birational maps |

14E25 | Embeddings in algebraic geometry |

14M99 | Special varieties |

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

14F25 | Classical real and complex (co)homology in algebraic geometry |

32C15 | Complex spaces |