Research on Periods of Algebraic Varieties and Hypergeometric Functions

代数簇和超几何函数的周期研究

• 批准号: 09440015
• 负责人: SAITO Masa-hiko
• 金额: $ 8.26万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440015/
• 关键词: Mirror Symmetry; GKZ Hypergeometric System; Calabi-Yau manifolds; Gromov-Witten invariants; Rational Elliptic Surfaces; Painleve equations; Spaces of Initial Values; Moduli spaces of Vector Bundles; Mirror 対称性; GKZ 超幾何系; カラビ・ヤウ多様体; グロモフ・ウイッテン不変

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.

在项目期间，我们对以下主题进行了调查，并获得了以下结果。（i）Calabi-Yau流形的镜像对称猜想，（ii）有理椭圆曲面中高亏格的计数曲线，（iii）Painleve方程的Lie理论问题，Painleve方程的代数几何问题，GKZ超几何系统及其Grobner变形。对于（i），我们研究了某些Calabi-Yau三重格的Gromov-Witteninvariants，并利用E_（10）D_（28）D_（2）格的θ函数计算了一部分A型预势，而它们的B型预势已经用GKZ超几何级数计算过了。因此，我们已经在数学上检查了这些情况下的MSC。进一步发展，在（ii）中，我们研究了有理椭圆曲面中高亏格曲线的计数问题。提出了前势应满足的全纯反常方程。利用Jacobi的三重积公式和相应的Lefschetz分解，我们检验了前势满足HAE.对于Painleve方程的研究（（iii）），我们的研究小组发展了Lie理论方法和代数几何方法，阐明了Painleve方程、仿射Weyl群的对称性和有理曲面几何之间的关系.

项目成果

M. -H. Saito: "Prepotentials of Yukawa couplings of certain Calabi-Ysu 3-folds and Mirror Symmetry"to appear in the Proceedings of Nato Advance Study Institute, CRM Summer School, Banff. (2000)

M.-H。

S. Hosono, Saito & Takahashi: "Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface"Adv. Theor. Math. Phys.. 3. 177-208 (1999)

S.细野斋藤

M.Noumi and Y.Yamada: "Affine Weyl groups, discrete dynamical systems and Painleve equations"Commun. Math. Phys.. 199. 281-295 (1998)

M.Noumi 和 Y.Yamada：“仿射 Weyl 群、离散动力系统和 Painleve 方程”Commun。

Takano,K.: "Defining manifolds for Painleve equations"Toward the exact WKB analysis of differential equations, linear and nonlinear. 261-269 (2000)

Takano, K.：“定义 Painleve 方程的流形”对线性和非线性微分方程进行精确的 WKB 分析。

Takano, K.: "Defining manifolds for Painleve equations""Toward the exact WKB analysis of differential equations, linear and nonlinear", (2000). 261-133 (2000)

Takano, K.：“定义 Painleve 方程的流形”“对线性和非线性微分方程进行精确的 WKB 分析”，(2000)。