Problems on Manifolds

1.我们对代数奇点和复解析奇点进行了研究：（1）任意特征线上无穷远极近奇点的分析。(2)复曲面因子概念的一般理论表述。(3)许多新的非Kaehler歧管的构造。关于自守函数，我们从函数方程、留数和特殊值的角度研究了zeta函数、L-函数。我们可以发现流形的几何性质与各种重要的联系.我们对代数簇和复流形进行了如下研究：（1）复曲面簇的双有理几何。(2)具有正规交叉簇的不等特征非奇异提升。(3)仿射代数流形上的Ricci平坦Kaehler度量。(4)完全交簇的Torelli问题。(5)一般型正则代数曲面的构造及数值不变量.我们开展了有界域的李群理论研究以及李群、代数群、仿射李代数及其表示的代数分析研究.我们可以在算术代数几何的框架内建立二维共形场论。我们与日本国内外的数学家交换了研究信息。

1. We carried out investigations of algebraic and complex analytic singularities: (1) Analysis of infinitely very near singular points in arbitrary characteristics. (2) General theoretical formulation of the notion of toric divisors. (3) Construction of many new non-Kaehler manifolds.2. In connection with automorphic functions, we studied zeta functions, L-functions from the point of view of their functional equations, residues and special values. We could find various important connections with the geometric properties of manifolds.3. We carried out the following research on algebraic varieties and complex manifolds: (1) Birational geometry of toric varieties. (2) Unequal characteristic non-singular lifting of varieties with normal crossings. (3) Ricci-flat Kaehler metrics on affine algebraic manifolds. (4) Torelli problem for complete intersection varieties. (5) Construction and numerical invariants of regular algebraic surfaces of general type.4. We carried out Lie-group-theoretic study of bounded domains as well as algebro-analytic study of Lie groups, algebraic groups, affine Lie algebras and their representations.5. We could formulate two-dimensional conformal field theory in the framework of arithmetic algebraic geometry.6. We exchanged research information with mathematicians inside and outside japan.