Calabi-Yau 三流形中的孤立有理曲线研究

Calabi-Yau 三流形是数学物理和代数几何中的重要研究对象。由于Calabi-Yau三流形的拓扑类型众多，因此研究这些拓扑结构不同的Calabi-Yau三流形间的关系对于认识这类流形的结构非常重要。 本项目将利用toric variety 上的齐次坐标以及形变理论研究toric variety 中的完全交Calabi-Yau 三流形的锥形变换（conifold transition），考察这些Calabi-Yau 三流形中的孤立有理曲线。我们的主要兴趣集中在以下几个问题： 1.讨论Toric variety 中完全交Calabi-Yau 三流形中具有指定同调类的孤立有理曲线的存在性； 2.试图证明toric variety 中处于一般位置的完全交Calabi-Yau 三流形均可以通过锥形变换成为三维球面乘积的连通和； 3. 对于三维球面乘积的连通和上复结构的模空间做一些考察和讨论。

Calabi-Yau threefolds are important objects in both mathematical physics and algebraic geometry. Since there are many different topological types of Calabi-Yau threefolds, the study of some relations of different Calabi-Yau threefolds are very important. This proposal will use homogeneous coordinates on toric varieties and develop some related deformation properties to study conifold transitions of Calabi-Yau threefolds. Our main interests concentrate on the following questions: 1. The existence of isolated rational curves with given homology class in complete intersection Calabi-Yau threefolds in toric varieties; 2. Conifold transitions of complete intersection Calabi-Yau threefolds in toric varieties. 3. Moduli space of complex structures on connected sums of S3S3.