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 $\mathbb{R}^{n}$, then let $B^{n}$ be a $n$-ball such that $% $, this closed set is the manifold with boundary since $% $, meantime $x_{1}^{2} + \dots + x_{n}^{2} = 1$ 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 $n$-manifold $M$ with boundary $\partial M$, then this is a $(n-1)$-manifold with corners. Formally

Definition: $\small\boxed{\text{Let} \ \mathbb{R}_{k}^{n} = [0, \infty)^{k} \times \mathbb{R}^{n-k} \ \text{. The singular space} \ M \ \text{is a second countable Hausdorff topological space in} \ n \ \text{dimensions, modeled on} \ \mathbb{R}_{k}^{n} \text{.} }$

Let $U$ be an open subset in $\mathbb{R}_{k}^{n} = [0, \infty)^{k} \times \mathbb{R}^{n-k}$ for some $0 \leq k \leq n$, and let $\phi: U \longrightarrow M$ be a homeomorphism with a nonempty set $\phi(U)$, then the pair $(U, \phi)$ is a $n$-dimensional chart on $M$.

Until here, we know what it is a manifold with corners and how we can define a $n$-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.

Sheaves

In order to define sheaves, first we need to define pre-sheaves. Let $M$ be a manifold with corners, then a presheaf consists of two sets of data:

-Sections over open sets, for each open set $U \subseteq M$ an abelian group $\Gamma(U)$.

-Restriction maps, for every inclusion $V \subseteq U$ of open sets in $M$ a group homomorpshim $p_{V}^{U}:F(U) \longrightarrow F(V)$ subjected to the conditions

for all sequences $W \subseteq V \subseteq U$ of inclusions of open sets in $M$, where $F(R)$ are called sections of $F$ over $R$ and $p_{V}^{U}$ are called restriction maps.

Now, we are able to define sheaves. A sheaf on $M$ is a presheaf of abelian groups on $M$ satisfying the following properties:

-Locally axiom, let $\{U_{i}\}_{i \in I}$ be an open cover of the open set $U$ and let $s$ be a sectionof $F$ over $U$.

-Gluing axiom, let $\{U_{i}\}_{i \in I}$ be an open cover of the open set $U$. Given sections $s_{i}$ over $U_{i}$ matching on the intersections $U_{ij} = U_{i} \cap U_{j}$, then there is a section $s$ of $F$ over $U$ satisfying $s|_{U_{i}} = s_{i}$.

What does this mean? It means that there is a locall homeomorphism structure (sheaf) $F$ into $M$ where we can define a Poisson Structure where given a ring-valued sheaf $\mathcal{O}_{M}$ on manifold with corners $M$.

Poisson Structure

A poisson structure on $(M, \mathcal{O})$, where $M$ 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.