Hard Lefschetz Theorem and Hodge-Riemann Relations for Convex Valuations

Explore a 51-minute lecture on the algebra of smooth valuations and its connection to Kähler geometry, algebraic geometry, and combinatorics. Delve into the mixed hard Lefschetz theorem and mixed Hodge-Riemann relations, and discover how these properties translate into quadratic inequalities for mixed volumes of sufficiently smooth convex bodies. Learn about the generalization of the classical Alexandrov-Fenchel inequality and compare the findings to McMullen's work on simple polytopes and van Handel's counter-example for non-smooth convex bodies. Gain insights into the proof methodology, which employs perturbation theory of unbounded linear operators, in this joint work by Andreas Bernig, Jan Kotrbatý, and Thomas Wannerer presented at the Hausdorff Center for Mathematics.