三维加权终端奇点篮与高维射影簇的地理学分类

This project aims to study the geographical problem concerning the correspondence between the set of formal baskets of 3-dimensional terminal singularities and that of projective 3-folds. It also studies all kinds of inequalities of Noether type and Miyaoka-Yau type for 3-folds of general type and the constructing of 3-folds with large slope. Besides we shall study the Noether type of inequality for higher dimensional varieties of general type and the birational geometry of projective varieties with very large canonical volume. Possibly some other related conjectures and open problems will be considered as well.

本项目主要研究形式化的加权终端奇点篮集合与三维射影簇集合之间的对应关系的地理学问题；研究一般型三维簇的各类 Noether 型不等式、Miyaoka-Yau 型不等式和高斜率的实例构造方法；研究四维以上的一般型簇的 Noether 型不等式；研究典范体积很大的射影簇的典范稳定双有理指数和双有理几何分类；研究代数几何中的其他相关问题和猜想。

本项目主要研究高维代数簇的双有理几何和典范分类等前沿基本问题。本项目发现并证明了三维代数簇的Noether不等式，对小不变量的三维簇作出了深入的地理学分类，证明了一般型三维簇的四典范双有理性的刻画猜想，证明了大体积的四、五维簇的典范递归猜想, 对弱Fano三维簇给出了有效分类。受本项目资助完成论文11篇，其中7篇论文发表或被接受发表于 Duke Math J, Ann. Inst. Fourier (Grenoble), Comm Anal Geom, Proc. ICM 2018等国际一流出版物。