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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.

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4 Answers

There are a few comments about this in the notes from Kontsevich's talk at the Arbeitstagung.

I also attended the Seidel talk that people are referring to. I should have my notes somewhere; if I succeed in finding them and if they contain anything not already in Sheel Ganatra's notes, I'll scan them and post them.

Edit: Kontsevich spoke about this last week in Miami. Check out these notes from the talk. Find the abstract here.

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thanks for the link, those notes are very cool –  David Ben-Zvi Jan 10 '10 at 16:26
Good spot, Kevin! –  Tim Perutz Jan 10 '10 at 16:33
If only I could vote you up again... –  S. Carnahan Jan 27 '10 at 3:23
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On this topic I've only seen Ganatra's notes from Paul's Talbot talk (see Scott's answer). An aspect of this, that a Fukaya category can behave in a sheafy way, is part of Nadler's proof that "microlocal branes are constructible sheaves".

The most relevant reference I can think of for the Fukaya category of the crepant resolution of $\mathbb{C}^2/\Gamma$ is Abouzaid's paper about Floer theory in plumbings.

Bad form as it may be, I'll ask a reciprocal question in this answer box. Suppose you knew that the Fukaya category of this crepant resolution was the constructible derived category on the exceptional fibre (was that what you had in mind?). What could one deduce? Would there then be a quiver presentation? A reasonable description of Hochschild cohomology?

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I asked Paul a couple months ago, and he said that nothing has been written, but there may be notes floating around from some talks Kontsevich gave in France last March. There are a few examples given in the last talk at Talbot on the bottom of this page. I was unable to work out a precise mathematical statement from the information given, but I'm not a symplectic geometer.

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Not sure if this is helpful to you, but Bing came upon this paper by Ruan which I'm thinking you likely have already seen and "get."

The Fukaya category of symplectic neighborhood of a non-Hausdorff manifold

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