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 [Ka]. 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 $(\cdot ,\cdot {)}_{\mathrm{\Phi }}$ on the intersection cohomology $IH(\mathrm{\Phi })$ of a complete fan $\mathrm{\Phi }$ and proving the Hodge-Riemann bilinear relations for $(\cdot ,\cdot {)}_{\mathrm{\Phi }}$.