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Fall 2004 Relative Poincaré lemma, contractibility, quasi-homogeneity and vector fields tangent to a singular variety
W. Domitrz, S. Janeczko, M. Zhitomirskii
Illinois J. Math. 48(3): 803-835 (Fall 2004). DOI: 10.1215/ijm/1258131054

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 $\mathbb R^n$.

Citation

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W. Domitrz. S. Janeczko. M. Zhitomirskii. "Relative Poincaré lemma, contractibility, quasi-homogeneity and vector fields tangent to a singular variety." Illinois J. Math. 48 (3) 803 - 835, Fall 2004. https://doi.org/10.1215/ijm/1258131054

Information

Published: Fall 2004
First available in Project Euclid: 13 November 2009

zbMATH: 1062.58037
MathSciNet: MR2114253
Digital Object Identifier: 10.1215/ijm/1258131054

Subjects:
Primary: 32B10
Secondary: 58K50

Rights: Copyright © 2004 University of Illinois at Urbana-Champaign

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Vol.48 • No. 3 • Fall 2004
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