Solomon--Terao Algebras: Top Degree, Dimension, and Homology

Abstract

We study the Solomon--Terao algebra and the Solomon--Terao complex of a central hyperplane arrangement $\mathcal A$ in $\mathbb C^\ell$. Let $η$ be a homogeneous polynomial of degree $d\geq2$ whose restriction to every flat of $\mathcal A$ has no critical point away from the origin. We prove that the Solomon--Terao algebra $\mathrm{ST}(\mathcal A,η)$ has top degree $|\mathcal A|+\ell(d-2)$ and a one-dimensional top graded component with an explicit generator, solving a conjecture of Abe, Maeno, Murai, and Numata. For $d=2$, we show that the total dimension of $\mathrm{ST}(\mathcal A,η)$ is bounded below by that of the Orlik--Solomon algebra. Both results hold without assuming that the arrangement $\mathcal A$ is tame. Finally, we prove that $\mathcal A$ is tame if and only if all positive-degree homology groups of the Solomon--Terao complex associated with $η$ vanish.