Higher Dimensional Algebraic Varieties

这个项目的中心目的是研究高维代数簇，特别是从外射线的观点。我们的目的是为了研究代数变种的各个方面，通过支持20名参加MSRI(92/93)和RGI暑期研究所93的代数几何年的日本数学家。该项目还部分支持了Fulton，Looijenga和Matsukki。Kollar，Miyaoka和Mori一直在联合研究代数曲线在代数簇上的变形。Kollar在他最近关于具有大代数基本群的代数簇的研究中，将形变方法与基本群相结合来构造Shafarevich映射。对于三维极小模型理论，Matsuki与S.Keel和J.McKernan共同将Kawamata的重要丰度定理推广到对数情形。Kawamata证明了半稳定极小模型理论的正性特征。Mori证明了Reid关于任意三维反转压缩的一般大象猜想。还研究了特殊的变种：Oguiso研究了Calabi-Yau 3-折叠的纤维空间结构，Mukai on Gorenstein Fano 3-折叠和Konno研究了一般类型曲面的特殊正则映射。Cho和Miyaoka正在研究射影空间的刻画。一些来自其他方面的工作：Kyoji Saito将TeichMuller空间表示为定义在整数环上的实半代数仿射格式。Usui构造了Hodge结构的分类空间的算术商的某些部分紧化，而Masahiko Saito利用Hodge结构的正则扩张研究了农隆模型。

The central purpose of this project was to study higher dimensional algebraic varieties, specially from the viewpoint of external rays. We aimed at the study of the various aspects of algebraic varieties by supporting 20 Japanese mathematicians participating in the algebraic geometry year at MSRI (92/93) and RGI summer institute 93. The project also partially supported Fulton, Looijenga and Matsuki.Kollar, Miyaoka and Mori have been jointly studying the deformation of algebraic curves on an algebraic variety. Kollar has combined the deformation method with fundamental groups to construct the Shafarevich maps in his recent study of algebraic varieties with big algebraic fundamental groups. They also generalized Kawamata's boundedness theorem of Q-Fano 3-folds to arbitrary Picard number case.As for 3-dimensional minimal model theory, Matsuki generalized Kawamata's important abundance theorem to the log case jointly with S. Keel and J. McKernan. Kawamata proved the semi-stable minimal model theory in positive characteristics. Mori proved Reid's general elephant conjecture for arbitrary 3-dimensional flipping contractions.Special varieties are also studied : Oguiso studied on fiber space structures of Calabi-Yau 3-folds, Mukai on Gorenstein Fano 3-folds and Konno on special canonical mappings of surfaces of general type. Cho and Miyaoka are working on the characterization of projective spaces.A few from other aspects : Kyoji Saito has expressed Teichmuller spaces as real semi-algebraic affine schemes defined over the ring of integers. Usui constructed certain partial compactifications of the arithmetic quotients of the classifying spaces of Hodge structures, and Masahiko Saito studied Neron models using canonical extensions of Hodge structures.