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Remarks on the Poisson structures on the plane and on other powers of volume forms. (Russian. English summary) Zbl 0668.58012
Author’s summary: “The hierarchy of forms $$f(dx)^{\alpha}$$ for a fixed $$\alpha$$ begins with series A,D,E. Their singularity hypersurfaces admit quasihomogeneous representatives $$g=0$$. We calculate the moduli space dimension for those forms having a fixed g. It is equal to the number of monomials h such that $$g(hdx)^{\alpha}$$ has weight zero. For the Poisson structures on 2 manifolds $$(\alpha =1$$, $$\dim x=2)$$ this dimension is equal to $$h^{1,0}$$ of the mixed Hodge structure and is smaller by one than the number of the irreducible components of the curve $$g=0.''$$
Reviewer: J.Andres

##### MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
##### Keywords:
Hodge structure; Poisson structures