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.