Research on Periods of Algebraic Varieties and Hypergeometric Functions

在项目的周期内，我们调查了以下的主题，并获得了以下的结果。本研究的主要内容包括：（一）Calabi-Yau歧管的镜像对称性几何;（二）理性椭圆曲面中高基因计数曲线;（三）疼痛方程的Lie理论方面;（三）疼痛方程的代数几何方面; GKZ超算系统及其Grobner变形。作为（i），我们已经为某些Calabi-Yau 3-folds研究Gromov-Witteninvariants，并计算了一个部分的A模型准备材料，由E-D28-D2-晶格的特征，同时他们的B模型准备材料已被计算为GKZ超算计量系列。作为一个结果，我们已经对MSC进行了数学检查。在（ii）我们在理性椭圆形表面中发现了高基因曲线的问题。我们提出全息异常均衡（HAE），即预测性应该满足的地方。根据Jacobi的三重产品公式和相对的Lefschetz分解，我们已经检查了HAE的潜在满意度（（iii）），作为Painleve equations的研究（iii）），我们的小组已经开发了Lie理论方法和代数几何学方法，确定了Painleve equations中的关系，Affine Weyl小组的对称性和理性表面的几何学。

During the period of the project, we have investigated the following subjects and obtained the following results. (i)Mirror Symmetry Conjecture for Calabi-Yau manifolds, (ii)Counting Curves of higher genus in Rational Elliptic Surfaces, (iii)Lie theoretic aspects on Painleve equations, Algebro-geometric aspects on Painleve equations, GKZ hypergeometric systems and their Grobner deformations. As for (i), we have been studying Gromov-Witteninvariants for certain Calabi-Yau 3-folds and computed a part of A-model prepotentials by means of the theta function of the EィイD28ィエD2-lattice while their B-model prepotential had already calculated by GKZ hypergometric series. As a result, we have checked MSC mathematically for those cases. Developing further, in (ii)we have investigated counting problems of curves of higher genus in rational elliptic surfaces. We propose the holomorphic anomaly equation(HAE), which the prepotential should satisfy. By using the Jacobi's triple product formula and the relative Lefschetz decomposition, we have checked the prepotential satisfies the HAE.As for the studies of Painleve equations ((iii)), our group have been developing Lie theoretic approach and algeblo-geometric approach, which clarify the relations among Painleve equations, the symmetry of Affine Weyl groups and the geometry of rational surfaces.