Kokyuroku Bessatsu Vol. B24
Checkout Your Cart Price. Description Details Customer Reviews Higher-dimensional algebraic geometry studies the classification theory of algebraic varieties.
This very active area of research is still developing, but an amazing quantity of knowledge has accumulated over the past twenty years. The author s goal is to provide an easily accessible introduction to the subject. The book covers preparatory and standard definitions and results, moves on to discuss various aspects of the geometry of smooth projective varieties with many rational curves, and finishes in taking the first steps towards Mori s minimal model program of classification of algebraic varieties by proving the cone and contraction theorems.
The book is well organized and the author has kept the number of concepts that are used but not proved to a minimum to provide a mostly self-contained introduction to graduate students and researchers.
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Geometry of Higher Dimensional Algebraic Varieties (Oberwolfach Seminars)
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Visitors and Participants. In recent years we have seen great breakthroughs in the classification theory of higher dimensional compact algebraic varieties and complex manifolds. The seminal results are the proofs of finite generations of canonical rings of algebraic varieties by Caucher Birkar — Paolo Cascini — Christopher D.
These results have profound influence on many areas of mathematics — including the study of higher dimensional dynamics and number theoretical dynamics. The interactions of algebraic geometry and the study of these dynamics is exactly the main theme of this program.
Here we recall the main conjectures in birational geometry which are still open.
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Abundance conjecture : For every minimal variety X, some pluri-canonical system of X is base point free. The consideration of three typical fibrations: Iitaka fibration, Albanese map and Maximal rationally connected fibrations reduces the classification of algebraic varieties to the following three building blocks:. Now let us turn our attention to the study of holomorphic dynamics.