Translated tori in the characteristic varieties of complex hyperplane arrangements

Abstract

We give examples of complex hyperplane arrangements $\text{A}$ for which the top characteristic variety, $\text{V}{}_1\text{(}\text{A}\text{)}$, contains positive-dimensional irreducible components that do not pass through the origin of the algebraic torus $\text{(}\text{C}{}^{\mathrm{\ast }}\text{)}{}^{\mathrm{<mtext>\mid </mtext><mtext>A</mtext><mtext>\mid </mtext>}}$. These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhibit a pair of arrangements for which the resonance varieties of the Orlik–Solomon algebra are (abstractly) isomorphic, yet whose characteristic varieties are not isomorphic. The difference comes from translated components, which are not detected by the tangent cone at the origin.