Hodge theory for combinatorial geometries

By Karim A. Adiprasito, June Huh, Eric Katz. Annals of Mathematics, 2015

325 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/07554075afa976bbae7da3959d632b7b3fc8b64e

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Citations in the corpus

Simplicial generation of Chow rings of matroids (2019, background): “ed permutohedra, and show that the volume polynomial is strongly log-concave in its positive orthant. As a consequence, we obtain a more direct proof of the portion of the Hodge theory of matroids in [AHK18] that implies the Rota-Heron-Welsh conjecture on the log-concavity of the coeﬃcients of the c”
Superforms, tropical cohomology, and Poincaré duality (2015, methodology): “ality for general tropical manifolds is then established from the local situation via standard methods. In recent work, Adiprasito, Huh and, Katz consider an intersection ring associated to a matroid [AHK15]. For a matroid M, the graded ring A∗(M) is shown to satisfy many striking properties in line”
Combinatorial Lefschetz theorems beyond positivity (2018, methodology): “braic shifting, that is unnecessary for the proof but reduces the splitting to a more pleasant combinatorial one (though we lose sight of the Poincaré pairing). Additionally, much of the reasoning of [AHK18] can be simpliﬁed using this technique, as the proof of the Hodge-Riemann relations for matro”
Lagrangian geometry of matroids (2020, methodology): “e Bergman fans of matroids [IKMZ19]. Adiprasito, Huh, and Katz showed that the Chow ring of the Bergman fan of M satisﬁes Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations [AHK18]. Furthermore, they interpreted the entries of the f-vector of the reduced broken circuit com”
A semi-small decomposition of the Chow ring of a matroid (2020, methodology): “The present description was used in [1], where the Chow ring of M was denoted A(M).”
Tautological classes of matroids (2021, methodology): “The authors of [1, 4] combined this Hodge theory of matroids with formulas like Theorem 2.”
COMBINATORIAL APPLICATIONS OF THE HODGE–RIEMANN RELATIONS (2017, background): “ are ‚ jPF x F; ‚ jRF x F: The two elements do not depend on the choice of j, and they are limits of elements of the form ‘ cfor a strictly submodular function c. A bijective counting argument in [2] shows that e ipMqdegpid iqfor every i; where e ipMqis the sequence of integers satisfying the iden”