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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

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems