Fukaya category with supports

Abstract

We associate to each compact subset $K$ of a closed symplectic manifold $M$ an $A_{\infty }$-category $\Fuk_K$, called the Fukaya category with support on $K$. The objects are tautologically unobstructed global Lagrangians endowed with appropriate decorations. The dependence on $K$ is detected by the morphism complexes which can be computed by completed telescopes associated with acceleration data for $K$. In particular, these vanish for Lagrangians disjoint of $K$. For $K=M$ this recovers the usual Fukaya category of tautologically unobstructed objects. These categories admit restriction functors under inclusions of compact subsets. We prove a Mayer--Vietoris descent theorem for weakly involutive covers. The construction is carried out over the Novikov ring and is established whenever classical transversality methods apply.

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