Examples of Poisson schemes

Question

A Poisson manifold is a real manifold $M$ along with a Lie bracket $[\cdot,\cdot]$ on $C^\infty(M)$ which is a derivation in each variable. Poisson manifolds are interesting for a few reasons, among them:

1. You can define the notion of an integrable system structure on a Poisson manifold, which allows them to be applied to solving problems in physics with enough symmetry.
2. Every symplectic manifold is automatically Poisson
3. Any Poisson manifold has a foliation by symplectic leaves.

(Reference for all of this: anything on Poisson manifolds, in particular, Wikipedia.)

Now, I've seen people seriously (for instance, in Vanhaecke's book) extend this notion to affine varieties over $\mathbb{C}$, where being Poisson means that the structure sheaf is a sheaf of Poisson Algebras, specifically, a Poisson algebra is an associative algebra along with a Lie bracket that is a derivation in each variable.

Now, what I'm interested in is how far this can be generalized and still have something where there are interesting (new!) examples. For instance, is "Poisson Scheme $X$ over $S$" a real object of interest? Specifically, I'm wondering if there are any examples where $S$ is not the spectrum of a field of characteristic zero, say $S$ is a finite field, or something positive dimensional, or nonreduced, etc, and if there are examples of this form, what makes them interesting? For instance, one reason that Poisson manifolds are interesting is that they are applicable to physics and, in fact in many cases to problems related to the geometry of the moduli space of vector bundles on a Riemann surface.

Answer 1

Of course, thinking of Poisson things in a relative way isn't going to give anything new, since a family of Poisson things is itself Poisson.

But to answer whether people think about these things in an abstract way, the answer is yes. There's now a huge and absolutely beautiful theory of symplectic singularities; here's a survey by Kaledin: Geometry and topology of symplectic resolutions If you read his papers, you'll see lots of modern algebraic geometry; theorems like the local existence of tilting generators depend on reduction to characteristic $p$.

Answer 2

One way in which Poisson schemes show up is as Poisson structures on singular varieties---although this is rather less scheme-theoretic-ish than what you have in mind, I think.

A concrete example: take $\mathbb C^2$ with is usual Poisson structure (the one coming from its usual symplectic form) and let $G\subseteq\mathrm{Sp}(2,\mathbb C)=\mathrm{SL}(2,\mathbb C)$ be a finite subgroup. Clearly the action of $G$ respects the Poisson structure, so the du Val/MacKay quotients $\mathbb C^2/G$ are naturally Poisson varieties with an isolated singularity. These objects can be studied in a very Poisson-theoretic way.

Answer 3

This "example" may seem like rather weak sauce, but I think you could take any smooth quasi-projective family $X \to S$ for $S$ any scheme, since you can pull back a fiberwise symplectic form from projective space. Also, you can do anything to such a gadget that preserves Poisson-ness (although interesting explicit constructions are escaping me - maybe take a total space of a family to make something foliated?). I seem to remember structures like this used in mirror symmetry mod p (from some Arizona Winter School lectures by Candelas back in 2004 or so), but that was using algebraic symplectic structures, instead of Poisson structures in general.

Answer 4

There is also another Kaledin's paper that you may wish to look at, Normalization of a Poisson algebra is Poisson.