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 x2 + y2 = 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∞ algebra of a holomorphic Lagrangian is formal. Our result also explains why the special Lagrangian condition holds without instanton corrections for holomorphic Lagrangians.
Bibliographical noteFunding Information:
J.S. was partially supported by ERC starting grant 337560 and ISF Grant 1747/13. M.V. was supported by the Russian Academic Excellence Project ‘5-100’ and CNPq - Process 313608/2017-2. This journal is ©c Foundation Compositio Mathematica 2019.
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- Floer cohomology
- Fukaya category
- holomorphic Lagrangian
- pseudoholomorphic map
- real analytic
- special Lagrangian
- Łojasiewicz inequality