Osamu Fujino - Böcker

Contains the proceedings of the 6th Mathematical Society of Japan Seasonal Institute Development of Moduli Theory, which was held as the Seasonal Institute 2013, with support from the Mathematical Society of Japan and the 2013 Research Project of the Research Institute of Mathematical Science, Kyoto University. This volume consists of five survey articles and eight research articles.

Since the appearance of extremal rays and the minimal model program around 1980, we have seen the tremendous development of Algebraic Geometry. With this in mind, the conference "Minimal Models and Extremal Rays" was held at RIMS Kyoto University in June 2011. The purpose was to review the past, examine the present and enjoy discussing the future. This volume is the outcome of this conference and consists of one survey article on the mathematical work of Professor Shigefumi Mori, who turned sixty in 2011, and thirteen research papers given by the authors: L Ein-S Ishii-M Mustata, J Kollár, S Kovács-K Schwede, D Greb-S Kebekus-T Peternell, H Kawanoue-K Matsuki, A-S Kaloghiros-A Kuronya-V Lazic, J-M Hwang-K Oguiso, M Lehn-Y Namikawa-Ch Sorger-D van Straten, O Fujino, J Chen, C Hacon-C Xu, S Mukai, Y Prokhorov-M Reid.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

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.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets