Studies on moduli spaces from the view point of mathematical physics

从数学物理角度研究模空间

• 批准号: 07304003
• 负责人: UENO Kenji
• 金额: $ 3.65万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1995
• 资助国家: 日本
• 起止时间: 1995 至 1996
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07304003/
• 关键词: moduli space; conformal field theory; conformalblock vector bundle; Calabi-Yaumanifold; pointed Riemannsurface; arithmeticgeometry; q-analog; 弦理論; 代数曲線; ベクトル束のモジュライ空間; リーマン面のモジュライ空間; 単純リイ代数; ゲージ対称性; KZ方程式; 閉リーマン面; 共形場ブロック; ベクトル束; 楕円曲面; 変

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.

代数簇和向量丛的模空间不仅在代数几何中，而且在数学物理中都有重要的作用。我们从数学物理的角度研究模空间，上野主要研究与点黎曼曲面模空间有着深刻关系的共形场论。他证明了非阿贝尔共形场论，即所谓的WZW模型，是在有理数域上定义的，甚至可以在离散赋值环上定义。他与清水和铃木一起研究了共形块层的保护平坦联络的显式描述，丸山提出了一种构造代数曲面上稳定层的模空间的新方法，这对研究模空间的边界有重要作用。Mukai研究了K3曲面上向量丛的模空间，并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流形变形的无障碍性。Katsura研究了具有正特征的椭圆曲面的多元典范系统，并证明它们具有与特征O类似的性质。这样我们发现了模空间的许多有趣性质，也表明了与数学物理的深刻关系。

项目成果

UENO, Kenji: "Q-strurcture of conformal field theory" Progress in Math.129. 511-531 (1995)

UENO，Kenji：“共形场论的 Q 结构”数学进展 129。

桂利行: "Multicanonical system of elliptic surfaces in small characteristic" Compositio Math.97. 119-134 (1995)

Toshiyuki Katsura：“小特征椭圆曲面的多规范系统”Compositio Math.97 (1995)。

上野健爾: "代数幾何入門" 岩波書店, 343 (1995)

Kenji Ueno：《代数几何导论》岩波书店，343（1995）

川又雄二郎: "Unobstracted deformations II" J. Algebraic Geometry. 4. 277-279 (1995)

Yujiro Kawamata：“无阻碍变形 II”J.代数几何。4. 277-279 (1995)

Kenji Ueno: Algebraic Geometry I,Foundation of Modern Mathematics. Iwanami, (1997)

Kenji Ueno：代数几何I，现代数学基础。