Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations

Talk by Andreas Bernig, Goethe Universität Frankfurt

The so-called Kähler package consists of a Poincaré duality, a hard Lefschetz theorem and the Hodge-Riemann relations. The origin goes back to the cohomology theory of compact Kähler manifolds, but similar structures appeared in the last few years in many different areas in mathematics such as algebraic geometry, combinatorics and polytope theory and have far reaching consequences.
After explaining some of these structures, I will report on a recent project with Jan Kotrbatý (Charles University Prague) and Thomas Wannerer (Friedrich-Schiller University Jena) where we introduce a Kähler package for convex valuations. As a consequence, we find quadratic inequalities for mixed volumes that generalize the fundamental Alexandrov-Fenchel inequality.