Homological mirror symmetry for the symmetric squares of punctured spheres
Homological mirror symmetry for the symmetric squares of punctured spheres
复制标题
穿孔球对称正方形的同调镜像对称性
DOI：
10.1016/j.aim.2023.108942
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发表时间：
2023
影响因子：
1.7
通讯作者：
Polishchuk, Alexander
中科院分区：
文献类型：
--
作者：
Lekili, Yankı;Polishchuk, Alexander
为了选择合适的Z-分次结构，我们证明了（k+ 3）-穿孔球面的对称正方形的包裹福谷范畴，即作为CP 2中（k+ 3）条一般线的补的Weinstein流形，拟等价于构造为环面Landau-Ginzburg模型边界的奇异曲面Z2，k上的相干层的导出范畴（X2，k，w2，k）.我们这样做，首先构建一个准等价双方的某些类别决议，然后本地化。我们还提供了一个一般的同调镜像对称猜想有关的所有高对称功率的穿孔球。相应的复曲面LG-模型（Xn，k，w n，k）由穿孔球面上的曲线组合构造而成，并且与奇点x1... xn + 1= v1... vk的小复曲面分辨率有关。
For an appropriate choice of a Z-grading structure, we prove that the wrapped Fukaya category of the symmetric square of a (k+ 3)-punctured sphere, ie the Weinstein manifold given as the complement of (k+ 3) generic lines in C P 2 is quasi-equivalent to the derived category of coherent sheaves on a singular surface Z 2, k constructed as the boundary of a toric Landau-Ginzburg model (X 2, k, w 2, k). We do this by first constructing a quasi-equivalence between certain categorical resolutions of both sides and then localizing. We also provide a general homological mirror symmetry conjecture concerning all the higher symmetric powers of punctured spheres. The corresponding toric LG-models (X n, k, w n, k) are constructed from the combinatorics of curves on the punctured sphere and are related to small toric resolutions of the singularity x 1… x n+ 1= v 1… v k.