An analogue of the Hodge-Riemann relations for simple convex polytopes

Abstract

Contents Introduction § 1. Combinatorics of simple polytopes 1.1. Simple polytopes 1.2. The Dehn-Sommerville equations 1.3. Stanley's theorem 1.4. Kähler manifolds 1.5. The Hodge-Riemann form on a Kähler manifold 1.6. The Lefschetz decomposition 1.7. Integrally simple polytopes § 2. The volume polynomial and the polytope algebra 2.1. The volume polynomial 2.2. The polarization of a homogeneous polynomial 2.3. A differential operator connected with a polytope 2.4. The volume polynomial of a face 2.5. Construction of an algebra for a given polynomial 2.6. The polytope algebra § 3. Flips 3.1. Flips 3.2. A combinatorial description of flips 3.3. The transformation of the h-vector under a flip 3.4. The action of a flip on faces 3.5. Transition polytopes § 4. Flips and the volume polynomial 4.1. Variation of the volume polynomial 4.2. The volume polynomial of a simplex 4.3. The change of the polytope algebra under a flip § 5. An analogue of the Hodge-Riemann form 5.1. An analogue of the Hodge-Riemann form 5.2. Outline of the proof 5.3. An analogue of the Lefschetz decomposition 5.4. Some corollaries 5.5. The hard Lefschetz theorem 5.6. The signature of the Hodge-Riemann form 5.7. The mixed Hodge-Riemann relationsBibliography