Birational Geometry of Algebraic Varieties (Cambridge Tracts in Mathematics)

Birational Geometry of Algebraic Varieties (Cambridge Tracts in Mathematics)
Janos Kollár,Shigefumi Mori | 1998-10-28 00:00:00 | Cambridge University Press | 264 | Mathematics
One of the major discoveries of the past two decades in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. This generalization, called the minimal model program, or Mori's program, has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond. This book provides the first comprehensive introduction to the circle of ideas developed around the program, the prerequisites being only a basic knowledge of algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.
Reviews
This book provides a neat introduction to (basically) what happened in birational geometry in the '80s. It contains a description of so-called minimal model program which deals which birational classification of complex algebraic varieties. Some level of familiarity with standard algebraic geometry (like what're written in Hartshorne's famous book) is required to be able to read this book. All the important results in MMP were established in this book except the (general) existence of flips in 3 dimension, which doesn't appear anywhere else but the original research articles. One other thing that's missing in this book is discussions of complex analytic methods, which are proved to be extremely useful in dealing with some problems (for example, Fujita conjecture, invariance of plurigenera, so on...).
In conclusion, this is THE must-read book for people who would like to know MMP. It may not be very reader-friendly, but it's a book that you must read if you want to study birational geometry.

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