Studies on moduli spaces from the view point of mathematical physics

代数簇和向量丛的模空间不仅在代数几何中有重要的作用，而且在数学物理中也有重要的作用。我们从数学物理的角度对模空间进行了研究。Ueno主要研究了与点黎曼曲面的模空间有很深关系的共形场理论。他证明了非阿贝尔保形场论，即所谓的WZW模型，是定义在过热数域上的，甚至可以定义在离散赋值环上。他还与Y.Shimizu和T.Suzuki研究了保形块层的保护性平坦联络的显式刻画。Maruyama给出了一种构造代数曲面上稳定层的模空间的新方法，这将对研究模空间的边界起到重要作用。Mukai研究了K3曲面上向量丛的模空间，发现了与代数曲线的模空间之间有趣的关系。Kawamata用纯代数方法证明了Calabi-Yau流形的变形是不明显的。Katsura研究了具有正特征的椭圆曲面的多正则系，证明了它们具有与特征O相似的性质，从而发现了模空间的许多有趣的性质，并与数学物理有了很深的联系。

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.