Chern flow and Chern moment algebras

Abstract

We construct realizable-volume models over every field for the factorial normalizations of homogeneous Lascoux, Lascoux-atom, and positive Grothendieck packets, including their minimal homogenizations and layers. In particular, the construction realizes the factorial normalizations of all Schubert and key polynomials and of the minimal sign-corrected homogeneous Grothendieck polynomials. The normalized polynomials are Lorentzian, and the ordinary supports are the lattice points of integral generalized polymatroids. On a Bott--Samelson tower, row and co-row filtrations assemble the local factors into globally generated bundles; a creation-state graph absorbs the remaining kernel factors by Chern flow. We also construct intrinsic algebras of joint Chern moments. Positive inverse-Chern presentations give these algebras Hard Lefschetz and Hodge--Riemann relations, and supply source-level Hodge completions of the packets. For globally generated tropical toric bundles in the sense of Kaveh--Manon, finite generating witnesses and matroid duality provide the presentations required by Larson--Partida's theorem, without representability. These constructions yield joint Chern-number inequalities, nonvanishing polymatroids, and equality criteria.