Title: New examples of algebras with the Kähler package and non-Archimedean analogues of mixed volumes.
Abstract:
An algebra with the Kähler package is a graded (or bigraded) algebra satisfying Poincaré duality (PD), Hard Lefschetz (HL) property, and Hodge–Riemann (HR) inequalities. The earliest examples arise from the cohomology of compact Kähler manifolds. Other examples were discovered in the theory of valuations on convex sets and in combinatorics.
In this talk, we describe another construction of algebras with the Kähler package. While it is inspired by the theory of valuations on convex sets, the field of real numbers is replaced by p-adic fields. We prove PD, (non-mixed) HL, and a special case of the (non-mixed) HR inequalities, leaving the remaining cases as conjectures.
Certain concrete elements of these algebras behave like mixed volumes involving ellipsoids in convex geometry: they satisfy analogous identities and, conjecturally, the Alexandrov–Fenchel inequality. The latter is proved in some very special cases.
No background in algebraic geometry, convexity, and p-adic numbers is assumed.