On the homological mirror symmetry conjecture for pairs of pants
On the homological mirror symmetry conjecture for pairs of pants
复制标题
关于裤子的同调镜像对称猜想
DOI：
--
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Nick Sheridan
中科院分区：
文献类型：
--
作者：
Nick Sheridan
n维裤子被定义为CP^n中n+2个一般超平面的补。我们构造了裤子中的浸入拉格朗日球，并计算了它在福谷范畴中的自同态A_{infty}代数.在上同调的层次上，它是一个有n+2个生成元的外代数。它不是形式的，我们计算某些更高的产品，以确定它的准同构。这使我们能够为同调镜像对称猜想提供一些证据：这条裤子被证明是Landau-Ginzburg模型（C^{n+2}，W）的镜像，其中W = z_1. z_{n+2}。证明了拉格朗日量的自同态A_{infty}代数与镜像原点的结构层的自同态dg代数是拟同构的。这意味着对裤子的有限覆盖，特别是对某些仿射费马超曲面，有类似的结果。
The n-dimensional pair of pants is defined to be the complement of n+2 generic hyperplanes in CP^n. We construct an immersed Lagrangian sphere in the pair of pants and compute its endomorphism A_{infty} algebra in the Fukaya category. On the level of cohomology, it is an exterior algebra with n+2 generators. It is not formal, and we compute certain higher products in order to determine it up to quasi-isomorphism. This allows us to give some evidence for the homological mirror symmetry conjecture: the pair of pants is conjectured to be mirror to the Landau-Ginzburg model (C^{n+2},W), where W = z_1 ... z_{n+2}. We show that the endomorphism A_{infty} algebra of our Lagrangian is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror. This implies similar results for finite covers of the pair of pants, in particular for certain affine Fermat hypersurfaces.