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Let $(M,I,J,K,g)$ be a hyperk\"ahler 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 curvewith boundary in a collection of $I$-holomorphic Lagrangians must be constant.By an argument based on the Lojasiewicz inequality, this result holds no matterhow the Lagrangians intersect each other. It follows that one can chooseperturbations such that the holomorphic polygons of the associated Fukayacategory lie in an arbitrarily small neighborhood of the Lagrangians. That is,the Fukaya category is local. We show that holomorphic Lagrangians aretautologically unobstructed in the sense of Fukaya-Oh-Ohta-Ono. Moreover, theFukaya $A_\infty$ algebra of a holomorphic Lagrangian is formal. Our resultalso explains why the special Lagrangian condition holds without instantoncorrections for holomorphic Lagrangians.

Locality in the Fukaya category of a hyperk\"ahler manifold