一类伪加权射影平面的整体向量场及其局部指标

Global vector fields and their local indices on aclass of fake weighted projective planes

给出了四个带有奇点的Gorenstein伪加权射影平面上的所有的整体切向量场,并利用推广的Poincaré-Hopf定理,证明了这些曲面上一些特征整体向量场在每一个奇点处的GSV-指标都等于1+μ,其中μ为该奇点的Milnor数.与光滑流形不同,奇异解析簇上具有孤立零点的向量场的局部指标一般不易计算.在此方面提供了一个具体实例.

We give all the global tangent vector fields on four Gorenstein fake weighted projective planes with singularities,and by making use of the generalized Poincaré-Hopf theorem we prove the GSV-indices for some characteristic global vector fields at each of the singularities on these surfaces are equal to 1+μ,whereμis the Milnor number of the singularity.Unlike the smooth manifolds case,the computation of local indices for vector fields with isolated zeros on singular analytic varieties is difficult in general.Our work provides a concrete example with respect to this issue.