Hodge theory for combinatorial geometries
Hodge theory for combinatorial geometries
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DOI：
10.4007/annals.2018.188.2.1
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发表时间：
2018-09-01
影响因子：
4.9
通讯作者：
Katz, Eric
中科院分区：
文献类型：
--
作者：
Adiprasito, Karim;Huh, June;Katz, Eric
本文证明了与任意拟阵M相关联的交换环的硬Lefschetz定理和Hodge-Riemann关系。我们利用Hodge-Riemann关系解决了Heron、Rota和Welsh的一个猜想，该猜想假定M的特征多项式的系数的对数为零。我们进一步得出了拟阵的独立复形的f-向量构成对数凹序列的结论，证明了Mason和Welsh关于一般拟阵的一个猜想。
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.