MathOverflow Asked by rori on February 23, 2021
Kontsevich cosheaf conjecture roughly states that wrapped Fukaya category can be recovered from local information on the Lagrangian skeleton. What are some reasons why would one believe it? I believe it looks most reasonable in the microlocal approach to Fukaya category but maybe there is some intuition for other models of Fukaya category as well.
On the suggestion of DamienC, I'm converting my comments into an answer. I didn't do this before because I don't really know the answer to the deeper question of why the microlocal sheaf category is the correct thing to give the Fukaya category (and I mostly just say things which are either obvious or wild speculation).
Some naive reasons why it's reasonable to expect local information on the skeleton to know everything:
It's true for cotangent bundles and represents a nice categorical enhancement of many historical results about cotangent bundles, starting with Viterbo/Abbondandolo-Schwarz results that symplectic homology of a cotangent bundle is the homology of the loopspace of the zero section.
Liouville flow retracts a Weinstein manifold onto its skeleton, so everything about symplectic topology of the completion should be determined by the germ of the manifold along the skeleton.
More speculatively:
In fact, Kontsevich explains the motivation for the conjecture in his own words in his paper "Symplectic geometry of homological algebra": https://www.ihes.fr/~maxim/TEXTS/Symplectic_AT2009.pdf
Edit^2: The following comments are misguided but I'll leave them here to give context to John Pardon's clarifying (and very illuminating) comments below.
Edit: I should add that the conjecture isn't going to be true if you allow arbitrary skeleta, only skeleta with mild (arboreal) singularities. In a talk by Daniel Alvarez-Gavela the other week, he mentioned the following example to see why this is true.
Take a Legendrian knot in the sphere (boundary of the ball) and attach a Weinstein handle along it. For a suitable choices of Liouville vector field, the skeleton of the resulting handlebody is just the core of the handle union the cone on the Legendrian. Topologically this is just homeomorphic to a sphere (but it is very singular at the cone point). The skeleton therefore doesn't depend on the knot, but the Fukaya category of the handlebody certainly does. You need to "arborealise" the singularity before you get a skeleton to which the Kontsevich conjecture can possibly apply.
Answered by Jonny Evans on February 23, 2021
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