General hypergeometric functions and geometry of the space of arrangements of points with infinitesimal neighborhoods

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The general hypergeometric functions(GHF) and the structure of the twsted cohomology group. The conjugacy classes of the centralizers of regular elements of GL(N) are determined by partitions of N. GHF is a multi-valued function on the Grassmannian manifold Gr(n+1, N) defined as a Radon transform of a character of the universal covering group of the centralizer. For an integer q > 0, consider a partition (q, 1,...,1) of N. To clarify the structure of the solution space of general hypergeometric system, we computed the rank and a basis of the associated de Rham cohomology group. When GHF is given by n dimensional integral, we found that the k-th cohomology group vanishes for k different from n, and the rank of the n-th cohomology group is (N-2)!/n!(N-n-2)!. We gave a basis for this group explicitly using Schur functions.Schlesinger system and its generalizations. We started the research of giving this generalizations from the point of view of twistor theory. When one consider the genera … More lized anti-self dual Yang-Mills equation(GASDYM) on the Grassmannian manifold Gr(2, N), its solution corresponds to a holomorphic vector bundle on the twistor space PN-1 via the Ward correspondence which is trivial when restricted to twistor lines. Let H be a maximal abelian subgroup of GL(N) as in 1) and consider its natural action on the twistor space PN-1. Moreover we assume that the action of H can be lifted to the holomorphic vector bundle corresponding to a solution to the GASYM equation. Then this action determines a flat connection on the bundle and when restricted to twistor lines, this flat connection describes a monodromy preserving deformation of ODEs. We gave the explicit form of the flat connection and by this explicit expression we made clear the analogy to the definition of GHF. We derived in a unified way the general Schlesinger systems from this point of view as the differential equations on Gr(2,N) which corresponds to the Painleve equations(including the degenerated ones). We also made clear that the Weyl group associated with H describes a group of symmetry of the general Schlesinger system. By this, we can give the group theoretic understanding for the fact that the number of parameters in the Painleve equations deceases after the degeneration. We could also construct the process of degeneration (confluence) for the general Schlesinger systems. Less