Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds
Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds
复制标题
Fano环面流形中拉格朗日环面纤维的Floer上同调和盘瞬子
DOI：
10.4310/ajm.2006.v10.n4.a10
复制
发表时间：
2003
影响因子：
0.6
通讯作者：
Y. Oh
中科院分区：
文献类型：
--
作者：
Cheol;Y. Oh
本文首先给出了任意紧环面流形的拉格朗日环面纤维上的全纯圆盘（圆盘瞬子）的一个显式描述，并证明了它们的Fredholm正则性.利用这一点，我们计算了Fano复曲面流形的Lagrangian环面纤维的Fooaya-Oh-Ohta-Ono（FOOO）阻塞链和Floer上同调。特别是针对形式参数T^{2\pi} = e^{-1}$，我们的计算验证了FOOO的阻塞（co）链对应于镜像对称对应下的Landau-Ginzburg超势的民间传说，也验证了K. Hori关于Fano环面流形的Lagrange环面纤维的Floer上同调。后者指出，所有纤维的弗洛尔上同调（对于参数值$T^{2\pi} = e^{-1}$）都消失了，除了在有限数量处，即复曲面流形的欧拉特征线，动量多面体中的基点是朗道-金斯堡镜像到复曲面流形的超势的临界点。在后一种情况下，我们还证明了相应纤维的Floer上同调同构于其奇异上同调。
我们还介绍了拉格朗日子流形的Floer上同调的一个限制版本，这是一个先验更灵活的定义一般，我们称之为{\it adapted Floer上同调}。然后，我们证明了任何非奇异环面纤维的Fano环面流形的适应Floer上同调是定义良好的，不变的Hamilton合痕和同构的Bott-Morse Floer上同调的纤维。
In this paper, we first provide an explicit description of {\it all} holomorphic discs (``disc instantons'') attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute Fukaya-Oh-Ohta-Ono's (FOOO's) obstruction (co)chains and the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. In particular specializing to the formal parameter $T^{2\pi} = e^{-1}$, our computation verifies the folklore that FOOO's obstruction (co)chains correspond to the Landau-Ginzburg superpotentials under the mirror symmetry correspondence, and also proves the prediction made by K. Hori about the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. The latter states that the Floer cohomology (for the parameter value $T^{2\pi} = e^{-1}$) of all the fibers vanish except at a finite number, the Euler characteristic of the toric manifold, of base points in the momentum polytope that are critical points of the superpotential of the Landau-Ginzburg mirror to the toric manifold. In the latter cases, we also prove that the Floer cohomology of the corresponding fiber is isomorphic to its singular cohomology.
We also introduce a restricted version of the Floer cohomology of Lagrangian submanifolds, which is a priori more flexible to define in general, and which we call the {\it adapted Floer cohomology}. We then prove that the adapted Floer cohomology of any non-singular torus fiber of Fano toric manifolds is well-defined, invariant under the Hamiltonian isotopy and isomorphic to the Bott-Morse Floer cohomology of the fiber.