Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations

Abstract: The hard Lefschetz theorem and the Hodge-Riemann relations have their origin in the cohomology theory of compact Kähler manifolds. In recent years it has become clear that similar results hold in many different settings, in particular in algebraic geometry and combinatorics (work by Adiprasito, Huh and others). In a recent joint work with Jan Kotrbatý and Thomas Wannerer, we prove the hard Lefschetz theorem and Hodge-Riemann relations for valuations on convex bodies. These results can be translated into an array of quadratic inequalities for mixed volumes of smooth convex bodies, giving a smooth analogue of the quadratic inequalities in McMullen's polytope algebra. Surprinsingly, these inequalities fail for general convex bodies. Our proof uses elliptic operators and perturbation theory of unbounded operators.

algebraic geometrycombinatorics

Audience: researchers in the topic

Tropical Geometry in Frankfurt/Zoom TGiF/Z

Series comments: Description: An afternoon seminar series on tropical geometry, known as the TGiZ ("Tropical Geometry in Zoom") or the TGiF ("Tropical Geometry in Frankfurt") seminar.

Please send an email to one of the organizers at the latest one day before a session if you wish to receive the Zoom link and password.

Videos of some past talks are available on YouTube here, while some slides can be found here.