The derived equivalence conjecture for the compactified Fukaya category

Let MM be an exact symplectic manifold embedded in a closed symplectic manifold XX as in Assumption. Let F(MX)gen\mathcal F(M \subset X)_{gen} be the generic fibre of the AA_\infty-deformation of the Fukaya category, and let Λt\Lambda_t be the Novikov ring. Derived equivalence conjecture. There is a canonical equivalence of triangulated categories

Dπ(F(MX)genQ[t1][[t]]Λt)Dπ(F(X)).D^\pi\left(\mathcal F(M \subset X)_{gen} \otimes_{\mathbb{Q}[t^{-1}][[t]]} \Lambda_t\right) \cong D^\pi(\mathcal F(X)).

This predicts that, after passing to derived categories and extending scalars, the generic fibre built from Lagrangians lying in MM recovers the Fukaya category of the closed manifold XX; the source indicates that the deformation construction itself has not yet been fully carried out.

Sources & referencesView supporting material

Primary source

Paul Seidel, “Fukaya categories and deformations”, arXiv:math/0206155 (2002).

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