Higher Dimensional Algebraic Varieties

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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.