Problems on Manifolds

1. 我们进行了代数奇异性和复解析奇异性的研究：(1)任意特征的无限极近奇异点的分析。(2)环面因子概念的一般理论表述。(3)许多新的非凯勒流形的构造。结合自同构函数，从函数方程、残数和特殊值的角度研究了zeta函数、l函数。我们可以从流形的几何性质中找到各种重要的联系。本文对代数变量和复流形进行了如下研究：(1)环型变量的双分几何。(2)正交品种的不等特征非奇异扬升。(3)仿射代数流形上的ricci -平坦Kaehler度量。(4)完全交变的Torelli问题。(5)一般类型正则代数曲面的构造与数值不变量。我们进行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解析研究。我们可以在算术代数几何的框架内表述二维共形场论。我们与日本国内外的数学家交流了研究信息。

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.