I’ve been trying to write down the construction for the Poisson structure of the knot character varieties, and it got me thinking about the Poisson geometry of the Grothendieck resolution and how it ties with our rank 1 and higher rank microlocal support conditions.

Let’s consider two trivial cases of our knot character varieties. Consider a rank $n$ vector bundle on the disk $D$, and we’re going to put our links around the origin. Moreover, let’s fix the framing at a point in the boundary, so that the monodromy map takes values in $G = GL_n$ (instead of $G/G$).

Case 1 is the usual picture of regular singularities: around the origin, put a single strand with “rank n”. Then the character variety is just $G$, since the only invariant up to gauge transformations is the monodromy. This has a quasi-Poisson G-structure (v. http://arxiv.org/abs/math/0006168), which implies that there’s a moment map $\mu$ to $G$ (identity in this case), and an action by $G$ (conjugation in this case), such that the quotient $\mu^{-1}(C)/G$ is a Poisson manifold (in this case just a point…) But here since we fixed the framing at a point on the boundary we don’t have to divide by conjugation, so let’s define $\mathcal M_1 = G$

Case 2 is where we replace the rank $n$ strand by $n$ concentric circles, with rank $1$ each. As in the STZ paper, the character variety is the Grothendieck resolution $\tilde G$, and a particularly nice presentation of this space is the quotient $G \times_B B$, where $B$ acts on $G\times B$ by $b\cdot (g,b_0) = (gb, b^{-1} b_0 b)$. In this form the map $\tilde G \to G$ is given by $(g,b)\mapsto gbg^{-1}$. This is also the global monodromy map.

Now there is a particular (quasi-?) Poisson structure $\pi$ on $\tilde G$ (v. http://arxiv.org/abs/math/0610123), which is induced by some Poisson structures on $G$ and $B$ when we take the quotient. The question mark is because I don’t know if O believe the reference, I think that the structure they have is merely a quasi-Poisson structure in general, still have to check myself. But anyway the point is that with this structure the map $\mu:\tilde G \to G$ is a Poisson morphism, and moreover the conjugation action $G \times \tilde G \to \tilde G$ is also a Poisson morphism.

So this got me thinking, could it be that these maps are Poisson because they correspond to fusing strands together? I don’t know what are the recent developments on this blurring map mentioned on the previous post, but is there anything about this map that would imply that the map on the character varieties is Poisson? Here we are not blurring any interesting links (concentric strands only) but would something like this extend to other braids?

This would be interesting, I think, because it would give an explanation of why the Grothendieck resolution is Poisson, and would also give a way of constructing (quasi-)Poisson structures on partial resolutions (fuse only some of the strands). I did a quick search and didn’t find anything about such structures on partial resolutions in the literature.