At some point in this past year, some Fukaya people I know got very
excited about the Fukaya categories of symplectic manifolds with "Lagrangian skeletons." As I understand it, a
Lagrangian skeleton is a union of Lagrangian submanifolds which a
symplectic manifold retracts to. One good example would be the
zero-section of a cotangent bundle, but there are others; for example,
the exceptional fiber of the crepant resolution of $\mathbb
C^2/\Gamma$ for $\Gamma$ a finite subgroup of $SL(2,\mathbb C)$. From the rumors I've heard, apparently there's some connection between the geometry of the skeleton and the Fukaya category of the symplectic manifold; this is understood well in the case of a cotangent bundle from work of Nadler and Nadler-Zaslow
I'm very interested in the Fukaya categories of some manifolds like
this, but the only thing I've actually seen written on the subject is
Paul Seidel's moderately famous picture of Kontsevich carpet-bombing
his research program
which may be amusing, but isn't very
mathematically rigorous. Google searching hasn't turned up much, so I
was wondering if any of you have anything to suggest.
Best Answer
I noticed this question has been bumped up to the front page, and the most recent answer is about 8 years old: the subject has moved on since then, and more has been written. Here is my understanding of some of the recent developments.
Caveat: This whole area has undergone rapid progress in the last few years, and I am not working on this question, so what I say is probably not up-to-date (even as I write, never mind in the future).
Kontsevich's idea was that you can relate the Fukaya category of a Weinstein manifold to the microlocal sheaf category of its skeleton. As far as I know, a proof of this conjecture is work in progress by Ganatra-Pardon-Shende (one preliminary part of which is already available).
I think the idea of their proof is roughly the following:
You prove that the Fukaya category has a co-sheaf property, which means it can be computed locally first on some subsets and then the answer can be glued together using homotopy colimits. For this, you need suitable functors relating the Fukaya category of a subset to the Fukaya category of the whole manifold. This is complicated by the fact that your "subsets" might not be very nicely embedded in the whole manifold: for example, if your ambient manifold is $T^*M$ then you want to allow subsets like $T^*M'$ where $M'\subset M$ is a codimension zero submanifold with boundary (that's because you ultimately want to work locally on the skeleton). That is problematic because the Liouville vector field for $T^*M'$ and the Liouville vector field for $TM$ don't match up nicely. The first GPS paper constructs these categories and functors for "Liouville sectors" (a suitably broad class of inclusions, related to Sylvan's notion of stops and partially wrapped Floer homology). I think the proof of the co-sheaf property is still ongoing work?
Now you compute the local pieces of the Fukaya category and show that they agree with the microlocal sheaf categories; since both have co-sheaf gluing, you get the same global answers.
The second part relies on some local computations of Fukaya categories. Nadler has introduced the notion of "arboreal skeleton" which is a skeleton with certain "generic" singularities. For example, trivalent graphs in dimension 1; trivalent graph times interval or cone on 1-skeleton of a tetrahedron in dimension 2; etc. He computes the microlocal sheaf category for these; I'm not sure if the corresponding partially-wrapped Fukaya categories have been calculated in all cases yet. Finally, you want to show that any Weinstein manifold has an arboreal skeleton: Starkston has some results in this direction which may represent the state of the art.
Leaving this aside for a moment, there are also special cases where the Konstevich conjecture/local-to-global results for the Fukaya category has been established independently of this general program. These include (but again, I'm probably missing some):