TGiF-Seminar: Tropical geometry in Frankfurt (Second meeting Winter Semester 2023/24)

Andreas Bernig (Goethe-Universität Frankfurt)

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.