Poisson cohomology of scalar multidimensional Dubrovin–Novikov brackets
Résumé
We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin–Novikov type with $D$ independent variables. We find that the second and third cohomology groups are generically non-vanishing in $D > 1$ . Hence, in contrast with the $D = 1$ case, the deformation theory in the multivariable case is non-trivial.