Hodge-Riemann Relations for Polytopes: A Geometric Approach

Polytopes are the higher-dimensional versions of polygons: flat-sided shapes in any number of dimensions. For every polytope one can compute a list of numbers, the generalized h-vector, that counts faces in a weighted way. Stanley conjectured, and Karu proved, that this list is symmetric, non-negative, and unimodal. The core of the proof is a set of positivity statements called the Hodge-Riemann relations, which say that certain bilinear forms have a definite sign on specially chosen subspaces.

This paper redoes Karu's proof in geometric language. The authors start from the known fact that the relations hold for simple polytopes, where only a fixed number of edges meet at every vertex. They then measure how far a general polytope is from being simple by counting certain faces, called normally stout faces. Cutting off a minimal such face reduces this count. The proof shows that the difficult piece, the part cut off, is a product of simpler polytopes up to a continuous deformation, and that a polytope obtained by gluing two pieces along a common face inherits the relations from its pieces. This yields a double induction: first on dimension, then on the number of normally stout faces.

Nothing in the paper is a new theorem. The authors state that the strategy follows Karu and Bressler-Lunts, and the goal is accessibility. The proof still relies on an elaborate machine of intersection cohomology for fans, built in earlier papers by the same group, so the exposition is aimed at specialists rather than beginners.