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 inductionà la [BreLu 2 ], 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.

The decomposition theorem, perverse sheaves and the topology of algebraic maps

“…The vector spaces V p i are subspaces of H 4 (f −1 (p i )), and contribute to the zero perversity term p H 0 (f * Q X C 4 [4]). In order to determine their dimension, we compute the stalk…”

“…not necessarily rational, polytope. Different proofs, each one shedding new light on interesting combinatorial phenomena, have then been given by Bressler-Lunts in [27] and by Barthel-Brasselet-Fieseler-Kaup in [4]. Another example of application of methods of intersection cohomology to the combinatorics of polytopes is the solution, due to T. Braden and R. MacPherson of a conjecture of G. Kalai concerning the behavior of the g-polynomial of a face with respect to the g-polynomial of the whole polytope.…”

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“…Up to renumbering, this is the filtration abutment of the perverse Leray spectral sequence met in the crash course §1.5, and it can be defined and described geometrically regardless of the decomposition theorem (7); see §2. 4. We abbreviate mixed Hodge structures as mHs.…”

“…(7) (Generalized Lefschetz decomposition and Hodge-Riemann bilinear relations) Let i, j ∈ Z and consider the perverse cohomology groups of (4). Define P −j…”

“…In §1. 4, we state the decomposition theorem in terms of intersection cohomology without any reference to perverse sheaves. The known proofs, however, use in an essential way the theory of perverse sheaves, which, in turn, is deeply rooted in the formalism of sheaves and derived categories.…”