复代数簇的极小切簇和目标刚性

The theory of varieties of minimal rational tangents (VMRT for short) is a powerful tool to study Fano geometry. Target rigidity property is a basic geometric property of algebraic varieties. We shall study TRP property of algebraic varieties by applying VMRT theory. Then we can apply these results to the study of other related fields. For instance, the study of rigidity of morphisms. Concrete research problems include: giving an algebraic proof of TRP of rational surfaces, giving a VMRT method to prove generalized Remmert-Ven problem, and study how does TRP property behave under blowing up of a subvariety of codimension greater than 0. For TRP property of uniruled varieties, a lot of problems are still open. We wish that, through our study, we could understand Fano geometry better, especially for geometry of uniruled varieties. We also wish that we can bring some new approaches and fresh viewpoints to the fields.

极小切簇理论(VMRT)是研究Fano几何的有力工具。目标刚性(TRP)是代数簇的一个基本几何性质。我们希望通过研究直纹簇的VMRT来研究它们的TRP性质，并且把这些结果应用到其它领域的研究，比如态射的刚性研究中。具体的研究问题包括有理曲面TRP性质的代数证明，广义的Remmert-Ven问题的VMRT证明还有代数簇胀开余维数大于零的子簇对TRP性质的影响等。对于直纹簇的TRP性质，还有许多未解决的问题。我们希望通过这些研究加深对Fano几何，特别是对直纹簇几何的理解，并且希望能够给相关的领域带来新的研究手段和不同的研究视角。

原计划研究直纹簇上的有理曲线和相关的问题，比如目标刚性等问题。我们的研究计划按原计划执行、基本达到了研究目标。我们系统的研究了环簇的极小切粗和目标刚性问题。事实上，与研究工作计划相比，我们考虑了更广泛的内容，所得的结果也更丰富。具体的结果包括高维典范丛公式的模部分半丰沛性猜想及其应用Fujino-Gongyo猜想, 正特征上的Iitaka猜想。