Hodge theory for combinatorial projective bundles

In algebraic geometry, certain spaces like projective bundles have cohomology rings that obey strong structural rules called the Hard Lefschetz theorem and Hodge-Riemann relations. These rules help control how classes multiply and intersect. The authors define abstract algebraic rings using combinatorial data that copy the multiplication tables of those geometric cohomology rings. They then prove the same structural rules hold for these combinatorial versions. Because the rings come from matroid data rather than actual geometry, the proof gives new instances where the standard conjecture of Hodge type is known to hold. It also produces results about special classes attached to matroids that behave like tautological classes in geometry.