Geodesic
Hodge theory for combinatorial geometries
Karim Adiprasito1 ; June Huh2 ; Eric Katz3
1 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
2 Institute for Advanced Study, Princeton, NJ, USA and Korea Institute for Advanced Study, Seoul, Korea
3 Department of Mathematics, The Ohio State University, Columbus, Ohio, USA
Annals of mathematics, Tome 188 (2018) no. 2, pp. 381-452

Voir la notice de l'article provenant de la source Annals of Mathematics website

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 $\mathrm {M}$. 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.

DOI : 10.4007/annals.2018.188.2.1

Karim Adiprasito 1 ; June Huh 2 ; Eric Katz 3

1 Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
2 Institute for Advanced Study, Princeton, NJ, USA and Korea Institute for Advanced Study, Seoul, Korea
3 Department of Mathematics, The Ohio State University, Columbus, Ohio, USA 
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Karim Adiprasito; June Huh; Eric Katz. Hodge theory for combinatorial geometries. Annals of mathematics, Tome 188 (2018) no. 2, pp. 381-452. doi: 10.4007/annals.2018.188.2.1

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