In my last post I wrote about Poisson Structure on manifolds with singularities. Sorokina’s research (See her published) motivated me to do research in this topic and thanks to her support I sent a paper to arXiv about how can we describe a Poisson Structure on manifolds with corners. As you can see by the title, is really important to define What it is a manifold with corners, but all this work was defined by Mr. Joyce on his papers On manifolds with corners and A generalization of manifolds with corners, so let’s just recall some definitions.
Manifolds with corners
What it is the intuition behind manifolds with corners? First, we need to remember what does a manifold with boundary is. Suppose we are on , then let be a -ball such that , this closed set is the manifold with boundary since , meantime is the unit hyper-sphere. This is indeed a manifold, since every second countable Hausdorff space is locally homeomorphic to Euclidean space. Now, we need to generalize the previous concept, if we take a -manifold with boundary , then this is a -manifold with corners. Formally
Let be an open subset in for some , and let be a homeomorphism with a nonempty set , then the pair is a -dimensional chart on .
Until here, we know what it is a manifold with corners and how we can define a -dimensional chart on it. In order to define a Poisson structure on a manifold with corners, we need to know how we can describe a sheaf structure on it.
In order to define sheaves, first we need to define pre-sheaves. Let be a manifold with corners, then a presheaf consists of two sets of data:
-Sections over open sets, for each open set an abelian group .
-Restriction maps, for every inclusion of open sets in a group homomorpshim subjected to the conditions
for all sequences of inclusions of open sets in , where are called sections of over and are called restriction maps.
Now, we are able to define sheaves. A sheaf on is a presheaf of abelian groups on satisfying the following properties:
-Locally axiom, let be an open cover of the open set and let be a sectionof over .
-Gluing axiom, let be an open cover of the open set . Given sections over matching on the intersections , then there is a section of over satisfying .
What does this mean? It means that there is a locall homeomorphism structure (sheaf) into where we can define a Poisson Structure where given a ring-valued sheaf on manifold with corners .
A poisson structure on , where is a manifold with corners, is a sheaf morphism
that is a derivation (satisfies the Leibniz rule) in each argument and also satisfies the Jacobi identity (See my previous post for details). The proof is short, but since we need to recall some lemmas, please take your time to read it.