Hodge Theory for Combinatorial Geometries
Combinatorics
2018-05-02 v2 Algebraic Geometry
Abstract
We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a conjecture of Heron, Rota, and Welsh that postulates the log-concavity of the coefficients of the characteristic polynomial of M. We furthermore conclude that the f-vector of the independence complex of a matroid forms a log-concave sequence, proving a conjecture of Mason and Welsh for general matroids.
Cite
@article{arxiv.1511.02888,
title = {Hodge Theory for Combinatorial Geometries},
author = {Karim Adiprasito and June Huh and Eric Katz},
journal= {arXiv preprint arXiv:1511.02888},
year = {2018}
}
Comments
63 pages. Minor revision