Geometric study of linear differential equations in complex variables

虽然高斯超几何微分方程与自同构形式理论和微分几何有着丰富的联系，但复变常微分方程理论一直局限于解析和局部研究。本文的研究目的是将高斯关于超几何微分方程作为投影线上有三个奇异点的双折线均匀化方程的理论进行推广和专门化。在1994年和1995年，我们得到了以下结果。(1)射影子流形的等价问题。设D是一个对称空间，等价地嵌在最小维射影空间P^N中。我们考虑这样一个问题：对于子流形V * P^N，给出V的微分几何条件，使其局部投影等价于d。这个问题对于研究均匀化方程是至关重要的。当D为VI型（即超二次型）时，问题解决。(2)构型几何…更多离子空间。设X （k, n）为P^<k-1>中n个一般位置点的位形空间。这个空间是本世纪初代数几何研究的主要对象之一。由于这个空间是（k, n）型超几何微分方程定义的自然域，这又成为一个热门话题。明确了（2,n）和（3,6）型位形空间的组合拓扑性质。(3)构形空间的模块化解释。问题：对称域D和不连续群GAMMA的构形空间是否同构于商空间D/GAMMA ？关于空间X （2, n）的研究很多。我们成功地正解了X(3,6),(4)旋（co）同调的交理论问题。我们建立了扭曲同调的交点理论。我们要在扭转上同调中建立它。本文还比较了一个品种的（co）同源性与一个分支覆盖物的同源性。(5) selberg型积分的研究。研究了一类selberg型积分。(4)中所述的理论是有用的。(6)超几何地图图像。我们证明了如果(k, n) * (2, n),(3,6)，那么超几何映射的像不是局部射影等价于由Plucker映射嵌入到射影空间中的Grassmannian。这个否定的结果摧毁了许多数学家的梦想。少

Though the Gauus's hypergeometric differential equation has a fruitful relationship with the theory of automorphic forms and differential geometry, the theory of ordinary differential equations in complex variables has been restricted to analytic and local studies. The present research aims generalizations and specializations of the Gauss's theory regarding the hypergeometric differential equation as the uniformizing equation of the obifolds with three singular points on the projective line. In 1994 and 1995, we got the following results.(1) Equivalence problem for projective submanifolds. Let D be a symmetric space equivariantly embedded in the minimal dimensional projective space P^N. We consider the problem : For a submanifold V * P^N, state a differential geometrical condition for V so that it is locally projectively equivalent to D.This problem is crucial for studing uniformizing equations. When D is of type VI (i.e.a hyperquadric), the problem is solved.(2) Geometry of configurat … More ion spaces. Let X (k, n) be the configuration space of n points in general postion in P^<k-1>. This space is one of the main objects in algebraic geometry in the beginning of this century. Since this space turned out to be the natural domain of definition of the hypergeometric differential equation of type (k, n), this became a hot topic again. We clarified the combinatorial topological properties of the configuration spaces of type (2, n) and (3,6).(3) Modular interpretation of configuration spaces. Problem : Is a configuration space isomorphic to the quotient space D/GAMMA for a symmetric domain D and a discontinuous group GAMMA? There are many studies about the space X (2, n). We succeeded to solve positively the problem for X (3,6),(4) Intersection theory for twisted (co) homolgies. We established the intersection thery for twisted homolgies. We are about to establish it for twisted cohomolgies. Comparison of (co) homolgies of a variety and those of a branched covering is also studied.(5) Studies of Selberg-type integrals. A kind of Selberg-type integrals is studied. The theory stated in (4) is useful.(6) The image of the hypergeometric maps. We proved that if (k, n) * (2, n), (3,6), then the image of the hypergeometric map is not locally projective equivalent to the Grassmannian embedded in the projective space by the Plucker map. This negative result destroyed the dream of many mathematicians. Less