I am currently considering a family of filtered algebras over an algebraically closed field of positive characteristic with the property that the associated graded algebra is a finitely generated polynomial ring. By some very general principles, such an algebra defines a Poisson structure on the associated graded and I'd like to know some basic properties about this structure.
My first question is a reference request: can anybody suggest a good set of notes which focus on Poisson algebras in positive characteristic?
I am aware that in characteristic zero the symplectic leaves of a Poisson variety need not be algebraic. My second question is: are there any general criteria which ensure that the leaves are algebraic? Are there some nice constructions which allow you to produce examples where the leaves are not algebraic?
I don't know if there is a very good way of defining symplectic leaves in positive characteristic, however I believe that the correct substitute is the definition of a symplectic core, as defined by Brown and Gordon in their paper "Poisson orders, symplectic reflection algebras and representation theory". Let $\mathbb{k}$ be an algebraically closed field and $A$ a finitely generated Poisson algebra over $\mathbb{k}$. Suppose that $\mathfrak{m}$ is a maximal ideal of $A$ and define the Poisson core $\mathcal{C}(\mathbb{m})$ to be the largest Poisson ideal contained in $\mathfrak{m}$ (such an ideal exists since the sum of two Poisson ideals is again Poisson). Now we can define the symplectic core of $\mathfrak{m}$ to be the collection of all maximal ideals which have Poisson core equal to $\mathcal{C}(\mathfrak{m})$. This is the smallest Poisson subvariety containing $\mathfrak{m}$. It is more or less clear that, when the symplectic leaves are defined, a leaf is algebraic if and only if it equals the symplectic core of any one of its points (see Brown and Gordon, Lemma 3.5). Equivalently, the leaves are algebraic if and only if the symplectic cores are symplectic varieties. This last condition can be phrased in positive characteristic, which leads to my next question: does anybody know any examples in characteristic $p > 0$ where the symplectic cores are not symplectic? Is it possible that the existence of non-algebraic leaves is a strictly characteristic zero thing?
Edit: I have recently realised that the symplectic cores as described above are inapplicable in positive characteristic. Suppose $A$ is an affine Poisson algebra in characteristic $p > 0$ and let $\mathfrak{m} = (x_1,...,x_n)$ be a maximal ideal. Then $(x_1^p,...,x_n^p) \subseteq \mathcal{P}(\mathfrak{m})$ and so the cores of maximal ideals are point sets. I am currently working on adapting this definition to the positive characteristic case.
