Calabi-Yau manifolds and mirror symmetry

Batyrev的自反式多面体结构和Givental随后的工作几乎解释了迄今为止已解决的所有镜像对称的情况。然而，在代数几何中，人们遇到了Calabi-Yau空间，这些空间不是给出的环面簇的完全交集，在这种情况下，一个非常有趣的问题是寻找镜像流形，以便了解镜像对称现象有多普遍。我们的计划是通过设计更多的例子来测试镜像对称性的范围，特别是我们想研究Picard数为1的例子，并试图找到镜像流形。在镜像方面，我们试图构造具有小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极大么次元的点。

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.