Relative Poincare Lemma, Contractibility, Quasi-homogeneity and Vector Fields Tangent to a Singular Variety

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.