Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus
Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus
复制标题
克利福德环面的全纯盘、自旋结构和弗洛尔上同调
DOI：
10.1155/s1073792804132716
复制
发表时间：
2003
影响因子：
1
通讯作者：
Cheol
中科院分区：
文献类型：
--
作者：
Cheol
我们使用所有可能的自旋结构计算 ℙ n 中 Clifford 环面的 Bott-Morse Floer 上同调。已知每个自旋结构决定全纯盘模空间的方向，我们根据克利福德环面自旋结构的变化来分析方向的变化。此外，我们通过建立此类盘的马斯洛夫指数公式，对边界位于 Clifford 环面上的所有全纯盘进行分类。结果，我们证明在奇数维度中，存在两个自旋结构，它们给出 Clifford 环面的非零弗洛尔上同调，而在偶数维度中，只有一个这样的自旋结构。当弗洛尔上同调不为零时，它与环面的奇异上同调同构（以诺维科夫环作为其系数）。作为推论，我们证明当相交是横向时，克利福德环面的任何哈密顿变形至少在 2 n 个不同的交点处与其相交。我们还计算了带有扁线束的 Clifford 环面的 Floer 上同调，并使用镜像对称计算验证了 Hori 的预测。
We compute the Bott-Morse Floer cohomology of the Clifford torus in ℙ n with all possible spin structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also, we classify all holomorphic discs with boundary lying on the Clifford torus by establishing a Maslov index formula for such discs. As a result, we show that in odd dimensions, there exist two spin structures which give nonvanishing Floer cohomology of the Clifford torus, and in even dimensions, there is only one such spin structure. When the Floer cohomology is nonvanishing, it is isomorphic to the singular cohomology of the torus (with a Novikov ring as its coefficients). As a corollary, we prove that any Hamiltonian deformation of the Clifford torus intersects with it at least at 2 n distinct intersection points when the intersection is transversal. We also compute the Floer cohomology of the Clifford torus with flat line bundles on it and verify the prediction made by Hori using a mirror symmetry calculation.