Singular Hodge theory for combinatorial geometries
Combinatorics
2023-04-11 v4 Algebraic Geometry
Abstract
We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e 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.
Cite
@article{arxiv.2010.06088,
title = {Singular Hodge theory for combinatorial geometries},
author = {Tom Braden and June Huh and Jacob P. Matherne and Nicholas Proudfoot and Botong Wang},
journal= {arXiv preprint arXiv:2010.06088},
year = {2023}
}
Comments
106 pages; v4: Minor corrections and improvements to the exposition