超曲面的上同调表示与镜像对称
Representations on the cohomology of hypersurfaces and mirror symmetry
摘要：
研究了作用于投射环面簇的非退化不变超曲面的上同调的有限群的表示。我们推导出一个明确的描述时，复曲面品种在最坏的商奇点的表示。作为应用，我们猜想了一个表示论版本的Batyrev和Borisov的镜像对称之间的Calabi-Yau超曲面对，并证明了它当超曲面都是光滑的或有维数至多3.一个有趣的结果是存在的卡拉比-丘orbifolds对的霍奇钻石是镜像，相对于通常的霍奇结构奇异上同调。
英文摘要：
We study the representation of a finite group acting on the cohomology of a non-degenerate, invariant hypersurface of a projective toric variety. We deduce an explicit description of the representation when the toric variety has at worst quotient singularities. As an application, we conjecture a representation-theoretic version of Batyrev and Borisov's mirror symmetry between pairs of Calabi-Yau hypersurfaces, and prove it when the hypersurfaces are both smooth or have dimension at most 3. An interesting consequence is the existence of pairs of Calabi-Yau orbifolds whose Hodge diamonds are mirror, with respect to the usual Hodge structure on singular cohomology.
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