Calabi-Yau manifolds and mirror symmetry

Batyrev的反身多面体构造和gigiental的后续工作几乎解释了迄今为止已经解决的所有镜像对称情况。然而，在代数几何中，我们遇到的Calabi-Yau空间在环面变换中不是作为完全交点给出的，一个非常有趣的问题是在这些情况下找到镜像流形，以了解镜像对称现象的普遍性。我们的计划是通过计算更多的例子来测试镜像对称的范围，特别是我们想研究皮卡德1号的例子并试图找到镜像流形。在镜像面，我们寻求构造具有小h1,2的Calabi-Yau流形，例如通过施加奇点，并在它们的模空间中搜索最大单幂偶的点。

The reflexive polytope construction of Batyrev and the subsequent work of Givental accounts for almost all cases of mirror symmetry that have been worked out so far. In algebraic geometry however, one encounters Calabi-Yau spaces that are not given as complete intersections in toric varieties and a very interesting problem is to find mirror manifolds in these cases, in order to understand how general the mirror symmetry phenomenon is. The plan is to test the range of mirror symmetry by working out more examples, in particular we want to study examples with Picard number one and try to find mirror manifolds. On the mirror side, we seek to construct Calabi-Yau manifolds with small h1,2, for example by imposing singularities, and search their moduli spaces for points of maximal unipotent monodromy.