On the Fukaya category of a Fano hypersurface in projective space
On the Fukaya category of a Fano hypersurface in projective space
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射影空间中 Fano 超曲面的 Fukaya 范畴
DOI：
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发表时间：
2013
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通讯作者：
Nick Sheridan
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文献类型：
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作者：
Nick Sheridan
本文是关于Fano超曲面X∧CPn$X \子集\mathbf {CP}^{n}$的深谷范畴。由于这些辛流形是单调的，在定义深谷范畴时所涉及的分析和代数都大大简化了。本文第一部分建立了单调情况下Fukaya范畴的主要结构：闭开弦映射、弱固有Calabi-Yau结构、Abouzaid的分裂生成判据，以及它们在包含弱边界共链时的类似性质。然后我们转向超曲面X$X$的深谷范畴的计算：我们在X$X$中构造了单调拉格朗日球的一个构形，并计算了相关的盘势。该结果与X$X$镜面的堀里-瓦法超势相吻合（在Fano指数1的情况下达到恒定位移）。因此，我们给出了X$X$的Kontsevich的同调镜像对称猜想的证明。我们还解释了如何从X$X$的Fukaya范畴中提取关于X$X$的Gromov-Witten不变量的非平凡信息。
This paper is about the Fukaya category of a Fano hypersurface X⊂CPn$X \subset \mathbf {CP}^{n}$. 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$X$: we construct a configuration of monotone Lagrangian spheres in X$X$, and compute the associated disc potential. The result coincides with the Hori–Vafa superpotential for the mirror of X$X$ (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$X$. We also explain how to extract non-trivial information about Gromov–Witten invariants of X$X$ from its Fukaya category.