Geometric structures on manifolds and global variation problems

在流形上的几何结构、整体变分问题和模空间上，我们得到了以下研究结果。在调和映射上，研究了从Riemann球面到对称空间的所有调和映射的模空间的连通性和基本群，以及调和映射到非紧Lie群的刻划。在复微分几何和模空间上，研究了约化代数群在紧Kahler流形上的作用。特别地，在Kahler流形范畴中引入了作用上的稳定性的概念，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主要猜想的基础上，结合数论中的Vojta猜想，给出了研究这一猜想的程序，并给出了Nevanlinna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空间上瞬时子模空间的同调群。作为ALE空间上反自对偶连通模空间的推广，引入了箭簇的概念，证明了箭簇上可构造函数的Kac-Moody代数表示的存在性。关于动力系统的几何结构，引入了紧致Kahler-Liouville流形的概念，并证明了紧致Kahler-Liouville流形是环状流形。在此基础上，研究了半线性双曲型方程的奇摄动形式下橡皮筋的运动及其解的性态。在仿射流形和信息几何方面，证明了海森流形的几何与仿射微分几何、信息几何和辛几何密切相关。

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.