关于代数簇的Frobenius分裂的几个问题

在研究正特征的代数几何时，Frobenius分裂是被广泛考察的一个重要性质。具有Frobenius分裂性质的代数簇十分稀少。已知的具备这一性质的代数簇有toric簇、旗簇、舒伯特簇等。给定一个Frobenius分裂的代数簇和其上的一个向量丛，那么此向量丛是否也是Frobenius分裂的，对这一问题所知甚少。Kumar，Lauritzen和Thomsen证明了旗簇的余切丛是Frobenius分裂的。申请人证明了toric簇的余切丛是Frobenius分裂的。本项目拟研究A型部分旗簇和二次超曲面的余切丛是否是p-1次Frobenius分裂的，以及极限球簇是否是Frobenius分裂的。

Frobenius splitting is a widely investigated property in the study of algebraic geometry in positive characteristics. It rarely happens for a given algebraic variety to be Frobenius split. Those enjoying this property include toric varieties, flag varieties, Schubert varieties, etc. For a given Frobenius split variety and a vector bundle on it, much less is known whether this vector bundle is Frobenius split or not. Kumar, Lauritzen and Thomsen prove that the cotangent bundle of a flag variety is Frobenius split. The applicant proves the cotangent bundle of a toric variety is Frobenius split. It is the aim of this project to study whether the cotangent bundles of a partial flag varieties of type A and quadrics are (p-1)-th power Frobenius split or not and whether the horospherical varieties are Frobenius split or not.