Relative Poincaré lemma, contractibility, quasi-homogeneity and vector fields tangent to a singular variety
Relative Poincaré lemma, contractibility, quasi-homogeneity and vector fields tangent to a singular variety
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相对庞加莱引理、可收缩性、拟同质性和与奇异簇相切的向量场
DOI：
10.1215/ijm/1258131054
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
M. Zhitomirskii
中科院分区：
文献类型：
--
作者：
W. Domitrz;S. Janeczko;M. Zhitomirskii
我们研究了标题中给出的奇异簇N⊂Rn的芽的性质与与N相切的向量场代数之间的相互作用。Poincare引理的性质意味着，在n的任意点上消失的任何闭微分(p+1)形式都是p形式的微分，它也在N的任意点上消失。特别地，我们证明了经典的拟齐性不是Poincare引理性质的必要条件；它可以被关于Rn的光滑子流形或光滑子流形链的拟齐性所代替。证明了N是拟齐次的当且仅当存在一个与N相切且具有正本征值的向量场V，V(0)=0。我们还把这个定理推广到关于Rn的光滑子流形的拟齐性。
We study the interplay between the properties of the germ of a singular variety N ⊂ Rn given in the title and the algebra of vector fields tangent to N . The Poincare lemma property means that any closed differential (p+1)-form vanishing at any point ofN is a differential of a p-form which also vanishes at any point of N . In particular, we show that the classical quasi-homogeneity is not a necessary condition for the Poincare lemma property; it can be replaced by quasi-homogeneity with respect to a smooth submanifold of Rn or a chain of smooth submanifolds. We prove that N is quasi-homogeneous if and only if there exists a vector field V, V (0) = 0, which is tangent to N and has positive eigenvalues. We also generalize this theorem to quasi-homogeneity with respect to a smooth submanifold of Rn.