HODGE-RIEMANN RELATIONS FOR POLYTOPES A GEOMETRIC APPROACH

Abstract

The key to the Hard Lefschetz Theorem for combinatorial intersection cohomology of polytopes is to prove the Hodge-Riemann bilinear relations. In these notes, we strive to present an easily accessible proof. The strategy essentially follows the original approach of [Ka], applying induction à la [BreLu2], but our guiding principle here is to emphasize the geometry behind the algebraic arguments by consequently stressing polytopes rather than fans endowed with a strictly convex conewise linear function. It is our belief that this approach makes the exposition more transparent since polytopes are more appealing to our geometric intuition than convex functions on a fan. 1