Advanced Combinatorial Mathematics
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Peer reviewed
Abstract
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.
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@article{Braden2026Singular,
title = {Singular Hodge theory for combinatorial geometries},
author = {Tom Braden and June Huh and Jacob P. Matherne and Nicholas Proudfoot and Botong Wang},
journal = {Journal of the American Mathematical Society},
year = {2026},
doi = {10.1090/jams/1083},
url = {https://doi.org/10.1090/jams/1083}
}FAQ
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