Hodge theory of GKZ-systems and mirror symmetry

I will explain a close relationship between GKZ hypergeometric systems after Gelfand, Kapranov and Zelevinsky, Gauss-Manin systems of families of Laurent polynomials and the intersection cohomology of hyperplane sections in a toric variety. As an application, I will show how to construct non-affine Landau-Ginzburg models which are mirror partners for complete intersections in toric varieties with a numerical effective anti-canonical divisor.