Locality in the Fukaya category of a hyperkähler manifold

Abstract

Let$(M,I,J,K,g)$be a hyperkähler manifold. Then the complex manifold$(M,I)$is holomorphic symplectic. We prove that for all real$x,y$, with$x^{2}+y^{2}=1$except countably many, any finite-energy$(xJ+yK)$-holomorphic curve with boundary in a collection of$I$-holomorphic Lagrangians must be constant. By an argument based on the Łojasiewicz inequality, this result holds no matter how the Lagrangians intersect each other. It follows that one can choose perturbations such that the holomorphic polygons of the associated Fukaya category lie in an arbitrarily small neighborhood of the Lagrangians. That is, the Fukaya category is local. We show that holomorphic Lagrangians are tautologically unobstructed. Moreover, the Fukaya$A_{\infty }$algebra of a holomorphic Lagrangian is formal. Our result also explains why the special Lagrangian condition holds without instanton corrections for holomorphic Lagrangians.