Singular Hodge theory for combinatorial geometries
Abstract
We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy conjecture and the nonnegativity of the coefficients of Kazhdan-Lusztig polynomials for all matroids.
Fields of Study & Subjects
Advanced Combinatorial Mathematics
Algebraic Geometry and Number Theory
Commutative Algebra and Its Applications
Publication details
- Journal / source
- Journal of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Publication date
- 1 July 2026
- Language
- EN