Hodge theory for combinatorial geometries

TY - JOUR

T1 - Hodge theory for combinatorial geometries

AU - Adiprasito, Karim

AU - Huh, June

AU - Katz, Eric

PY - 2018/9

Y1 - 2018/9

N2 - 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.

AB - 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.

KW - hard Lefschetz theorem

KW - Hodge-Riemann relation

KW - Bergman fan

KW - matroid

U2 - 10.4007/annals.2018.188.2.1

DO - 10.4007/annals.2018.188.2.1

M3 - Journal article

SN - 0003-486X

VL - 188

SP - 381

EP - 452

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -