Birational geometry of algebraic varieties

所提出的研究涉及高维代数几何的主题。主要研究射影簇的双有理几何中的自然问题，如极小模型规划的相关问题。PI将研究非一般类型品种的最小模型的存在性，丰度和翻转的终止，正特征中3倍翻转的存在性，Fano品种的几何形状（特别是对数Fanos的有界性），正规簇和乘子理想层的奇异性，代数簇的复正则映射和不规则簇的双有理几何。曲面的双有理分类在世纪初被意大利学派所理解。如果曲面没有被有理曲线覆盖，那么自然会选择具有许多有用性质的双有理曲面（称为最小模型）。最小模型程序旨在将这些结果推广到更高的维度。这个程序在3维中是完整的（由Mori，Kawamata，Kollar，Reid，Shokurov和其他人的工作），并且已知适用于一般类型的复杂投射变体（由Birkar，Cascini，Hacon，McKernan，Siu和其他人的工作）。这个项目希望回答一些在这种情况下自然出现的遗留问题和疑问。

The proposed research deals with topics in Higher dimensional Algebraic Geometry. It is mainly focused on natural questions in the birational geometry of projective varieties such as the study of questions related to the minimal model program. The PI will investigate the existence of minimal models for varieties not of general type, abundance and the termination of flips, the existence of flips for 3-folds in positive characteristic, the geometry of Fano varieties (in particular the boundedness of log-Fanos), the singularities of normal varieties and multiplier ideal sheaves, the pluricanonical maps of algebraic varieties and the birational geometry of irregular varieties.The birational classification of surfaces was understood by the Italian school at the beginning of the twentieth century. If a surface is not covered by rational curves then there is a natural choice 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 complex projective 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.