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

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66576The 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 $(\cdot ,\cdot)_Φ$ on the intersection cohomology $IH(Φ)$ of a complete fan $Φ$ and proving the Hodge-Riemann bilinear relations for $(\cdot ,\cdot)_Φ$.Final version to appear in Indiana University Mathematics JournalAlgebraic GeometryCombinatoricsHard Lefschetz theorem and Hodge-Riemann relations for intersection cohomology of nonrational polytopestext