New developments and interaction between Algebraic Geometry and Integrable Systems

代数几何与可积系统的新发展及其相互作用

• 批准号: 19104002
• 负责人: SAITO Masa-hiko
• 金额: $ 63.48万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (S)
• 财政年份: 2007
• 资助国家: 日本
• 起止时间: 2007 至 2011
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-19104002/
• 关键词: 可積分系; パンルヴェ方程式; 相空間; モジュライ空間; 量子コホモロジーとミラー対称性; 極小モデル理論; 高次元代数多様体; モノドロミー保存変形; 複素力学系; ミラー対称性

We gave an algebro-geometric construction of the moduli spaces of stable parabolic connections over curves with unramified singularities, and showed the fundamental property of the Riemann-Hilbert correspondences. These results showed the geometric Painleve property of the nonlinear isomonodromic differential equations and established the geometry of isomonodromic deformations of connections, which enables us to investigate the phase space of differential equations deeply such as Okamoto's space of initial conditions for classical Painleve equations. Together with the progress in the field of higher dimensional birational geometry and the geometry related to mirror symmetry, these results reveal deep relations between algebraic geometry and integrable systems.

给出了非分歧奇点曲线上稳定抛物联络模空间的代数几何构造，并证明了Riemann-Hilbert对应的基本性质.这些结果揭示了非线性等单道微分方程的几何Painleve性质，建立了联络的等单道变形几何，使我们能够深入研究微分方程的相空间，如经典Painleve方程的Okamoto初值空间.这些结果与高维双有理几何和与镜像对称相关的几何领域的进展一起，揭示了代数几何与可积系统之间的深刻联系。

项目成果

Asymptotic behavior of the hyperbolic Schwarz map at irregular singular points

不规则奇点处双曲 Schwarz 映射的渐近行为

• 发表时间: 2010
• 期刊: Funkcialaj Ekvacioj (in press)
• 作者: T. Koike;T. Sasaki;Masaaki Yoshida
• 通讯作者: Masaaki Yoshida

Poisson deformations of affine symplectic varieties

仿射辛簇的泊松变形

• DOI: 10.1215/00127094-2010-066
• 发表时间: 2011
• 期刊: Duke Math. J.
• 作者: Namikawa;Y
• 通讯作者: Y

Exact Lagrangian submanifolds in simply-connected cotangent bundles

• DOI: 10.1007/s00222-007-0092-8
• 发表时间: 2007-01
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: K. Fukaya;P. Seidel;I. Smith

On the eigenvalue problem of the Coulomb potential and exact WKB analysis

库仑势特征值问题及精确WKB分析

• 发表时间: 2008
• 作者: T.;Aoki;T. Kawai;T. Koike
• 通讯作者: T. Koike

Lagrangian fibrations on the moduli spaces of singular connections of curves

曲线奇异连接模空间上的拉格朗日纤维

• 发表时间: 2011
• 作者: F.Loray;M.-H. Saito and C.Simpson;Masa-Hiko Saito;Masa-Hiko Saito;Masa-Hiko Saito
• 通讯作者: Masa-Hiko Saito