Geometric structures on manifolds and global variation problems

流形上的几何结构和全局变分问题

• 批准号: 06302005
• 负责人: SAKANE Yusuke
• 金额: $ 8.19万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Co-operative Research (A)
• 财政年份: 1994
• 资助国家: 日本
• 起止时间: 1994 至 1995
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-06302005/
• 关键词: harmonic map; moduli space of harmonic maps; value distribution theory; Vojta conjecture; Nevanlinna theory; quiver varietie; Kahler-Liouville manifolds; Hessian manifolds; 極小曲面; 平坦な共形構造; フアノ多様体; ヤンミルズ接続; ALE空間; 中心アファイン的変形; モジュライ空間; hyperka

On geometric structures on manifolds, global variation problems and moduli spaces, we obtained following research results.1. On harmonic maps, the connectedness and fundamental group of the moduli spaces of all harmonic maps from Riemann sphere to symmetric spaces are investigated and a characterization of harmonic maps into non-compact Lie groups are also studied.2. On complex differential geometry and moduli spaces, actions of reductive algebraic groups on compact Kahler manifolds are studied. In particular, the notion of stability on the action are introduced and existence of quotient space are proved in the category of Kahler manifolds. Further, on main conjectures on value distribution theory of holomorphic curves, a program to investigate this conjecture together with Vojta conjecture in number theory are proposed, and an algebraic geometric proof for second main theorem of Nevanlinna theory are obtained.3. On gage theory including Yang-Mills fields and its application, homology groups of instantons moduli spaces on ALE spaces are studied. And as a generaization of moduli spaces of anti-self-dual connection on ALE spaces, the notion of quiver varieties are introduced and the existence of representations of Kac-Moody algebras of constructible functions on quiver varieties are proved.4. On geomertic structures of dynamical system, the notion of compact Kahler-Liouville manifolds are introduced and it is proved that compact Kahler-Liouville manifolds are toric manifolds. Further, motions of rubber bands are investigated as singular perturbation of semilinear hyperbolic equations and behavior of solutions are studied.5. On affine manifolds and information geometry, it is shown that geometry of Hessian manifolds is closely related to affine differential geometry, information geometry and symplectic geometry.

关于流形上的几何结构、整体变分问题和模空间，我们得到了以下研究结果.在调和映射上，研究了从Riemann球面到对称空间的调和映射的模空间的连通性和基本群，并给出了到非紧李群的调和映射的一个刻画.在复微分几何和模空间上，研究了约化代数群在紧致Kahler流形上的作用。特别地，在Kahler流形范畴中引入了作用稳定性的概念，并证明了商空间的存在性。进一步，在全纯曲线值分布理论的基础上，提出了将该猜想与数论中的Vojta猜想一起研究的方案，并给出了Nevanlinna理论第二主要定理的代数几何证明.在包含Yang-Mills场的规范理论及其应用的基础上，研究了ALE空间上瞬子模空间的同调群。作为ALE空间上的反自对偶联络模空间的推广，引入了可构造函数的Kac-Moody代数的可构造簇的概念，证明了可构造函数的Kac-Moody代数在可构造簇上的表示的存在性.在动力系统的几何结构方面，引入了紧Kahler-Liouville流形的概念，证明了紧Kahler-Liouville流形是环面流形。进一步，将橡皮筋的运动作为半线性双曲型方程的奇摄动问题来研究，并研究了解的性态.在仿射流形和信息几何方面，证明了Hessian流形的几何与仿射微分几何、信息几何和辛几何密切相关。

项目成果

S.Nayatani and H.Urakawa: "Morse indices of Yang-Millo connections over the unit sphere" Compositio Mathematica. 98. 177-192 (1995)

S.Nayatani 和 H.Urakawa：“单位球面上的 Yang-Millo 连接的莫尔斯指数”Compositio Mathematica。

H.Shima: "Hessian manifolds of constant Hessian sectional curvature" to appear in J.Math.Soc.Japan.

H.Shima：“Hessian 截面曲率恒定的 Hessian 流形”出现在 J.Math.Soc.Japan 中。

H.Nakajima: "Instantons on ALE spaces, quiver variteies, and Kac-Moody algebras" Duke Math.vol.76. 365-416 (1994)

H.Nakajima：“ALE 空间、箭袋变体和 Kac-Moody 代数上的瞬时”Duke Math.vol.76。

A.Fujiki: "The fixed point set of C-actions on a compact complex space" Osaka J.Math.to apbear. 32. (1995)

A.Fujiki：“紧凑复杂空间上的 C 动作的不动点集”Osaka J.Math.to apear。

H.Shima: "Hessian manifolds of constant sectional curvature" J.Math.Soc.Japan. (to appear).

H.Shima：“恒定截面曲率的 Hessian 流形”J.Math.Soc.Japan。