Poisson algebras of polynomial growth

Consider the sequence c n (V) of codimensions of a variety V of Poisson algebras. We show that the growth of every variety V of Poisson algebras over an arbitrary field is either bounded by a polynomial or at least exponential. Furthermore, if the growth of V is polynomial then there is a polynomial R(x) with rational coefficients such that c n (V) = R(n) for all sufficiently large n. We present lower and upper bounds for the polynomials R(x) of an arbitrary fixed degree. We also show that the varieties of Poisson algebras of polynomial growth are finitely based in characteristic zero.