Hard Lefschetz Theorem and Hodge-Riemann Relations for Intersection Cohomology of Nonrational Polytopes

The Hard Lefschetz theorem for intersection cohomology of nonrational polytopes was recently proved by K. Karu [9]. This theorem implies the conjecture of R. Stanley on the unimodularity of the generalized h-vector. In this paper we strengthen Karu's theorem by introducing a canonical bilinear form (·,·)Φ on the intersection cohomology IH(Φ) of a complete fan Φ and proving the Hodge-Riemann bilinear relations for (·,·)Φ.