高维奇点的前沿问题研究

In this project, we plan to explore the connections among several different volumes of singularities and describe the singularities which have the same volumes, in particular, the singularities having zero volume. We will also study the semicontinuity of volumes for deformations of singularities and answer the question of whether the volume constant deformations of singularities are equisingular. Moreover, we shall prove the Fulger volumes are topological invariants for some singularities. Furthermore we will show that the volume of a singularity, which has positive volume, is bounded below by a constant which depends only on the dimension of the singularity. We shall prove that there are no negative derivations for normal isolated weighted homogeneous singularities. We shall classify the 3-dimensional isolated quotient singularities and calculate their crepant resolutions and Cox rings, and we use them to obtain information on the geometric structure of singularities.

在本项目中，我们计划发掘奇点几种不同体积之间的大小关系并且描述不同体积相等所对应的奇点，特别是体积等于零的奇点；研究这些体积在形变下的半连续性特征，解决奇点形变中保持体积不变是否一定是等奇异形变这个问题； 证明Fulger定义的体积对某些奇点是拓扑不变量；证明对体积不等于零的奇点, 它的体积有某个只依赖于奇点的维数下界。证明正规孤立加权齐次奇点没有负导子。分类三维孤立Gorenstein商奇点，计算这些所分类奇点的Crepant解消的Cox环, 并用其获得奇点的几何信息。

本项目除了完成了大部分预定目标之外，还得到了很多原计划之外的研究成果。我们证明了零维和正维数正规孤立加权齐次奇点没有负导子。特别是证明了广义Halperin猜想、广义Wahl猜想。 Brieskorn 建立了简单奇点和单李代数的联系，一个自然的基本问题是如何建立奇点和可解李代数的联系。这启发了我们就奇点发现并引入了三种不同系列的新导子李代数：k-th Yau algebra, local Hessian derivation Lie algebra, derivation Lie algebras of Nash blow up algebras。这些发现是重要的，可得到奇点的很多新的不变量。我们猜想这些新李代数也没有负导子，并就部分奇点验证了这些猜想是对的，获得了一系列成果。 毫无疑问这些发现对高维可解李代数的分类研究会起到重要作用。另外我们还分类了三维孤立Gorenstein商奇点。本项目执行期间共发表了18篇论文，其中大部分论文发表在J. Diff. Geom(2篇)、Trans. A.M.S.、Math. Z.(3篇)、CAG、MRL、Alge. Reprent. Theo.、Geo. Dedicata、J. Alebra(2篇)、Asian J. Math.、等国际知名杂志。