Real Algebraic and Analytic Geometry
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77. W. Domitrz, S. Janeczko, M. Zhitomirskii:
Relative Poincare Lemma, Contractibility, Quasi-Homogeneity and Vector Fields Tangent to a Singular Variety.

e-mail: , ,

Submission: 2004, January 12.

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

Mathematics Subject Classification (2000): 32B10, 14F40, 58K50.

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