Integral representations of GKZ hypergeometric functions: Gauss-Manin connection, intersection theory, and quadratic relations

2019/05/10 Fri 16:30 - 18:00

GKZ(Gelfand, Kapranov, Zelevinsky) system is a holonomic
system which describes classical hypergeometric systems in a unified
manner. In this talk, we realize GKZ system as a Gauss-Manin connection
where we treat Euler integral and Laplace integral at the same time.
Focusing on regular holonomic case, we give a method of reinterpreting
the combinatorics of regular triangulations to the construction of the
basis of twisted cycles at "toric infinity". This naturally gives rise
to an orthogonal decomposition of the twisted homology group with
respect to the intersection pairing. As an application, we give a
general quadratic relation of GKZ hypergeometric functions associated to
a unimodular triangulation. We also discuss an algorithm of computing
cohomology intersection numbers based on a joint work with Nobuki
Takayama. The techniques above produce several new quadratic relations
of hypergeometric functions of several variables.