Geometric study of linear differential equations in complex variables

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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