Mathematical Society of Japan Memoirs

Foundations of the minimal model program

Osamu Fujino

Book information

Osamu Fujino

Publication information
MSJ Memoirs, Volume 35
Tokyo, Japan: The Mathematical Society of Japan, 2017
289 pp.

Publication date: 2017
First available in Project Euclid: 31 May 2017

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Primary: 14E30: Minimal model program (Mori theory, extremal rays) 14F17: Vanishing theorems [See also 32L20]
Secondary: 14J05 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]

Copyright © 2017, The Mathematical Society of Japan

Osamu Fujino,Foundations of the minimal model program (Tokyo: The Mathematical Society of Japan, 2017)

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Around 1980, Shigefumi Mori initiated a new theory, which is now known as the minimal model program or Mori theory, for higher-dimensional algebraic varieties. This theory has developed into a powerful tool with applications to diverse questions in algebraic geometry and related fields. One of the main purposes of this book is to establish the fundamental theorems of the minimal model program, that is, various Kodaira type vanishing theorems, the cone and contraction theorem, and so on, for quasi-log schemes. The notion of quasi-log schemes was introduced by Florin Ambro and is now indispensable for the study of semi-log canonical pairs from the cohomological point of view. By the recent developments of the minimal model program, we know that the appropriate singularities to permit on the varieties at the boundaries of moduli spaces are semi-log canonical. In order to achieve this goal, we generalize Kollár's injectivity, torsion-free, and vanishing theorems for reducible varieties by using the theory of mixed Hodge structures on cohomology with compact support. We also review many important classical Kodaira type vanishing theorems in detail and explain the basic results of the minimal model program for the reader's convenience.