This paper is about the Fukaya category of a Fano hypersurface \(X \subset \mathbf {CP}^{n}\). Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed–open string maps, weak proper Calabi–Yau structure, Abouzaid’s split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface \(X\): we construct a configuration of monotone Lagrangian spheres in \(X\), and compute the associated disc potential. The result coincides with the Hori–Vafa superpotential for the mirror of \(X\) (up to a constant shift in the Fano index 1 case). As a consequence, we give a proof of Kontsevich’s homological mirror symmetry conjecture for \(X\). We also explain how to extract non-trivial information about Gromov–Witten invariants of \(X\) from its Fukaya category.

中文翻译：

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关于射影空间中的Fano超曲面的Fukaya类

本文是关于Fano超曲面\（X \ subset \ mathbf {CP} ^ {n} \）的Fukaya类别的。因为这些辛流形是单调的，所以在Fukaya类别的定义中涉及的分析和代数都大大简化了。本文的第一部分致力于在单调情况下建立Fukaya类的主要结构：封闭的开放式字符串图，弱的适当Calabi-Yau结构，Abouzaid的分裂代判据以及弱边界共链时的类似物。包括在内。然后，我们转到超曲面\（X \）的Fukaya类别的计算：我们在\（X \）中构造单调拉格朗日球的配置，并计算相关的圆盘电位。结果与\（X \）的镜像的Hori-Vafa超势重合（在Fano指数为1的情况下恒定位移）。结果，我们证明了Kontsevich关于\（X \）的同构镜对称猜想。我们还将说明如何从Fukaya类别中提取有关\（X \）的Gromov–Witten不变量的非平凡信息。