We prove Kontsevich's homological mirror symmetry conjecture for certain mirror pairs arising from Batyrev-Borisov's `dual reflexive Gorenstein cones' construction. In particular we prove HMS for all Greene-Plesser mirror pairs (i.e., Calabi-Yau hypersurfaces in quotients of weighted projective spaces). We also prove it for certain mirror Calabi-Yau complete intersections arising from Borisov's construction via dual nef partitions, and also for certain Calabi-Yau complete intersections which do not have a Calabi-Yau mirror, but instead are mirror to a Calabi-Yau subcategory of the derived category of a higher-dimensional Fano variety. The latter case encompasses Kuznetsov's `K3 category of a cubic fourfold', which is mirror to an honest K3 surface; and also the analogous category for a quotient of a cubic sevenfold by an order-3 symmetry, which is mirror to a rigid Calabi-Yau threefold.

中文翻译：

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广义格林-普莱瑟镜的同调镜面对称

我们证明了由 Batyrev-Borisov 的“双自反 Gorenstein 锥”构造产生的某些镜像对的 Kontsevich 的同调镜像对称猜想。特别地，我们证明了所有 Greene-Plesser 镜像对（即加权投影空间商中的 Calabi-Yau 超曲面）的 HMS。我们还证明了由鲍里索夫通过双 nef 分区构造产生的某些镜像卡拉比 - 丘完全交叉点，以及某些没有卡拉比 - 丘镜像，而是镜像到卡拉比 - 丘子类别的卡拉比 - 丘完全交叉点高维 Fano 变体的派生范畴。后一种情况包含 Kuznetsov 的“三次四重 K3 范畴”，它是诚实 K3 曲面的镜像；