IUMJ

Title: Hard Lefschetz theorem and Hodge-Riemann relations for intersection cohomology of nonrational polytopes

Authors: Paul Bressler and Valery A. Lunts

Issue: Volume 54 (2005), Issue 1, 263-308

Abstract:

The Hard Lefschetz theorem for intersection cohomology of nonrational polytopes was recently proved by K. Karu [8]. 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 $(\cdot,\cdot)_{\Phi}$.