Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations

TL;DR

This paper proves the hard Lefschetz theorem and Hodge-Riemann relations for the algebra of smooth translation-invariant valuations on convex bodies, extending geometric and algebraic analogies to a broad setting.

Contribution

It establishes these classical geometric properties for valuations, using elliptic operator theory, in a general context previously conjectured but not proven.

Findings

Proves the hard Lefschetz theorem for convex valuations.

Establishes Hodge-Riemann relations in this setting.

Derives McMullen's quadratic inequalities for smooth convex bodies.

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 K\"ahler manifold: Poincar\'e 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.