Birational geometry of higher dimensional varieties

提出的研究涉及高维复杂代数几何的主题（即研究几个复杂变量的多项式方程组的解）。如果两个变体（即解的不可约集）具有同构开子集，则它们是生的。这个项目的主要焦点是在复杂的射影变异的自然几何问题，特别是在最小模型程序相关的问题。二十世纪初，意大利学派理解了曲面的出生地分类。如果一个曲面没有被有理曲线覆盖，那么自然会选择一个双曲面（称为最小模型），它有许多有用的性质。最小模型程序旨在将这些结果推广到更高的维度。该程序在3维中完成（由Mori， Kawamata, Kollar, Reid， Shokurov等人完成），并且已知可用于各种一般类型（由Birkar， Cascini, Hacon, McKernan， Siu等人完成）。这个项目希望回答在这种情况下自然产生的一些剩余的问题和猜想。

The proposed research deals with topics in Higher Dimensional Complex Algebraic Geometry (i.e. the study of the solutions of sets of polynomial equations in several complex variables). Two varieties (i.e. irreducible sets of solutions) are birational if they have isomorphic open subsets.The main focus of this project is on natural questions inthe birational geometry of complex projective varietiesand in particular on problems related to the Minimal Model Program.The birational classification of surfaces was understood by theItalian school at the beginning of the twentieth century.If a surface is not covered by rational curves then there is a naturalchoice of a birational surface (known as the minimal model)which has many useful properties. The Minimal Model Program aims to generalize these results to higher dimensions. This program is complete in dimension 3 (by work of Mori, Kawamata, Kollar, Reid, Shokurov and others) and is known to work for varieties of general type (by work of Birkar, Cascini, Hacon, McKernan, Siu and others). This project hopes to answer some of the remaining questions and conjectures that naturally arise in this context.