Hodge theory for combinatorial geometries

Karim Adiprasito, June Huh, Eric Katz

Research output: Contribution to journalArticlepeer-review

94 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)381-452
Number of pages72
JournalAnnals of Mathematics
Issue number2
StatePublished - 2018

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Bergman fan
  • Hard Lefschetz theorem
  • Hodge-Riemann relation
  • Matroid


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