Seminar of Algebra

GKZ (Gelfand, Kapranov, Zelevinsky) system is a certain holonomic system on an Affine space whose basis of solutions can be constructed in terms of hypergeometric series ($\Gamma$-series). We show that GKZ hypergeometric system has various realisations as twisted Gauss -Manin connections: Laplace type, Euler-Laplace type, and Residue-Laplace type. In particular, when GKZ system is regular holonomic, this realisation combined with Riemann-Hilbert correspondence yields a new description of its solution sheaf on non-singular locus. If time permits, we will discuss its relation to intersection theory of twisted cycles and some other related works.