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Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations

Andreas Bernig, Jan Kotrbatý, Thomas Wannerer

Abstract

The algebra of smooth translation-invariant valuations on convex bodies, introduced by S.Alesker in the early 2000s, was in part proved and in part conjectured to satisfy properties formally analogous to those of the cohomology ring of a compact Kähler manifold: Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. Our main result establishes the hard Lefschetz theorem and the Hodge-Riemann relations in full generality. As a consequence, we obtain McMullen's quadratic inequalities, which are valid for strongly isomorphic polytopes and known to fail in general, for convex bodies with smooth and strictly positively curved boundary. Our proof is based on elliptic operator theory and on perturbation theory applied to unbounded operators on a natural Hilbert space completion of the space of smooth translation-invariant valuations.

Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations

Abstract

The algebra of smooth translation-invariant valuations on convex bodies, introduced by S.Alesker in the early 2000s, was in part proved and in part conjectured to satisfy properties formally analogous to those of the cohomology ring of a compact Kähler manifold: Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. Our main result establishes the hard Lefschetz theorem and the Hodge-Riemann relations in full generality. As a consequence, we obtain McMullen's quadratic inequalities, which are valid for strongly isomorphic polytopes and known to fail in general, for convex bodies with smooth and strictly positively curved boundary. Our proof is based on elliptic operator theory and on perturbation theory applied to unbounded operators on a natural Hilbert space completion of the space of smooth translation-invariant valuations.
Paper Structure (29 sections, 50 theorems, 135 equations)

This paper contains 29 sections, 50 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Geometric inequalities
  3. Hodge theory for smooth valuations
  4. Our methods
  5. Organization of the article
  6. Background and notation
  7. Translation-invariant continuous valuations
  8. Smooth valuations
  9. The convolution of smooth valuations
  10. Poincaré duality
  11. Differential forms on the sphere bundle
  12. Complex linear algebra
  13. The linear algebra of $(p,p)$-forms
  14. Smooth valuations and $(p,p)$-forms
  15. Injectivity in the hard Lefschetz theorem
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $0\leq k\leq \frac{n}{2}$, $N\in\mathbb{N}$, $x_i\in\mathbb{R}$, and let $A_j^i,C_l\in\mathcal{K}(\mathbb{R}^n)$ be strongly isomorphic simple polytopes. If then with equality if and only if $\sum_{i=1}^N x_i V(A_1^i,\dots,A_k^i,\mathop{\mathrm{\raisebox{-0.25ex}{$\cdot$}}}\nolimits[n-k])=0$.

Theorems & Definitions (87)

  • Theorem 1.1: McMullen McMullen:SimplePolytopes
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 77 more