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