Relative Poincaré lemma, contractibility, quasi-homogeneity and vector fields tangent to a singular variety

Abstract

We study the interplay between the properties of the germ of a singular variety $N\subset \mathbb R^n$ 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 of $N$ 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 $\mathbb R^n$ 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 $\mathbb R^n$.