Studies on moduli spaces from the view point of mathematical physics

代数变量和向量束的模空间不仅在代数几何中占有重要地位，而且在数学物理中也占有重要地位。从数学物理的角度研究了模空间。上野主要研究与点黎曼曲面模空间有密切关系的共形场理论。他证明了非阿贝尔共形场理论，即WZW模型可以定义在合理数域上，甚至可以定义在离散估值环上。他还与Y.Shimizu和T.Suzuki一起研究了保形块层间保护平面连接的明确描述。Maruyama提出了在代数曲面上构造稳定轴的模空间的新方法，对模空间边界的研究具有重要意义。Mukai研究了K3曲面上向量束的模空间，发现了与代数曲线模空间的有趣关系。Kawamata用纯代数方法证明了Calabi-Yau流形变形的无干扰性。Katsura研究了具有正特征的椭圆曲面的多正则系统，并证明了它们具有与特征o相似的性质。这样我们发现了模空间的许多有趣的性质，也显示了与数学物理的深刻联系。

Moduli spaces of algebraic varieties and vector bundles play am important role not only in algebraic geometry but alsomathematical physics. We studied moduli spaces from the view point of mathematical physics.Ueno studied mainly conformal field theory which has deep relationship with moduli spaces of pointed Riemann surfaces. He showed that non-abelian conformal field theory, so called the WZW modelis defined overtherational number field and even more it can be defined over a discrete valuation ring. With Y.Shimizu and T.Suzuki he also studied explicit description of protectively flat connection of the sheaf of conformal blocks.Maruyama showed a new method how to construct moduli sapce of stable sheaves on algebraic surfaces, which will play an important role to study boundaries of the moduli spaces. Mukai studied the moduli spaces of vector bundles on K3 surfaces and found interesting relationship with the moduli spaces of algebraic curves. Kawamata showed unobstractedness of deformations of Calabi-Yau manifolds by purely algebraic method. Katsura studied pluricanonical sytems of elliptic surfaces in positive characteristics and showed that they have similar properties as in characteristic O.In this way we found many interesting properties of moduli spaces and also showed deep relationship with mathematicalphysics.