论文标题
$ \ mathbb {c} p^n $的同源镜对称性及其产品通过摩尔斯
Homological mirror symmetry of $\mathbb{C}P^n$ and their products via Morse homotopy
论文作者
论文摘要
当复杂的歧管是光滑的紧凑型孢子歧管时，我们提出了一种理解同源镜像对称性的方法。到目前为止，在许多示例中，将衍生的类别$ d^b（coh（x））$在复式歧管上$ x $上的相干滑轮$与相应的兰道吉茨堡电位的Milnor纤维的Fukaya-Seidel类别进行了比较。相反，我们将双圆环纤维化$π：m \ to $ x $中的圆环隔板的补充b $，其中$ \ bar {b} $是折叠式折线$ x $的双重元素。该设置中同源镜子对称性的自然表述是定义$ fuk（\ bar {m}）$ fukaya类别的变体，并显示等价$ d^b（coh（x））\ simeq d^b（fuk（fuk（\ bar {m}））$。作为中级步骤，我们在$ p：= \ bar {b} $上构建了$ mo（p）$（p）$（p）$，作为对Kontsevich-Soibelman提出的加权福卡亚-OH类别的自然概括。然后，我们显示一个完整的子类别$ mo _ {\ Mathcal {e}}（p）$ of $ mo（p）$ of $ mo（p）$ oferates $ d^b（coh（x））$ x $是一个复杂的投射空间及其产品。
We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many example, the derived category $D^b(coh(X))$ of coherent sheaves on a toric manifold $X$ is compared with the Fukaya-Seidel category of the Milnor fiber of the corresponding Landau-Ginzburg potential. We instead consider the dual torus fibration $π:M \to B$ of the complement of the toric divisors in $X$, where $\bar{B}$ is the dual polytope of the toric manifold $X$. A natural formulation of homological mirror symmetry in this set-up is to define $Fuk(\bar{M})$ a variant of the Fukaya category and show the equivalence $D^b(coh(X)) \simeq D^b(Fuk(\bar{M}))$. As an intermediate step, we construct the category $Mo(P)$ of weighted Morse homotopy on $P:=\bar{B}$ as a natural generalization of the weighted Fukaya-Oh category proposed by Kontsevich-Soibelman. We then show a full subcategory $Mo_{\mathcal{E}}(P)$ of $Mo(P)$ generates $D^b(coh(X))$ for the cases $X$ is a complex projective space and their products.
