期刊
COMPOSITIO MATHEMATICA
卷 155, 期 10, 页码 1924-1958出版社
CAMBRIDGE UNIV PRESS
DOI: 10.1112/S0010437X1900753X
关键词
Floer cohomology; Fukaya category; holomorphic Lagrangian; hyperkahler; local; Lojasiewicz inequality; pseudoholomorphic map; real analytic; special Lagrangian
类别
资金
- ERC [337560]
- ISF [1747/13]
- Russian Academic Excellence Project '5-100'
- CNPq [313608/2017-2]
- European Research Council (ERC) [337560] Funding Source: European Research Council (ERC)
Let (M; I; J; K; g) be a hyperkahler 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 fi nite-energy (xJ + yK)-holomorphic curve with boundary in a collection of I-holomorphic Lagrangians must be constant. By an argument based on the Lojasiewicz 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(infinity) algebra of a holomorphic Lagrangian is formal. Our result also explains why the special Lagrangian condition holds without instanton corrections for holomorphic Lagrangians.
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