the algebraic torus $(\Bbbk^*)^n \cong \operatorname{Hom}(H_1(M), \Bbbk^*) \cong \operatorname{Hom}(\pi_1(M), \Bbbk^*)$. The resonance varieties ${{\mathcal R}}_{k}^{q}(M, \Bbbk)$ are homogeneous subvarieties in $\Bbbk^n \cong H^1(M, \Bbbk)$.

The resonance varieties in the case when $M$ is a complex algebraic variety are closely related to the classical results in algebraic geometry, starting with Castelnuovo-De Franchis Lemma and culminating with the isotropic subspace theorems due to Catanese and Bauer, see [@D0] for details and complete references.

We restrict our attention to the case $M=M_{{{\mathcal A}}}$, for an arbitrary arrangement ${{\mathcal A}}$, and $\Bbbk={\mathbb{C}}$, using a simplified notation (for example, ${{\mathcal R}}_{k}^{q}(M):={{\mathcal R}}_{k}^{q}(M, {\mathbb{C}})$).

By construction, the resonance varieties are combinatorial invariants of an arrangement, since $H^*(M,\Bbbk)$ depends only on the intersection lattice, see Theorem \[thm:OS\].

[ *For characteristic varieties, combinatorial determination is a major open problem.*]{} The next theorem provides a partial answer in this direction.

\[germs\] The exponential map ${\mathbb{C}}^n \\ \overset{\exp}\longrightarrow ({\mathbb{C}}^*)^n$ induces an isomorphism of analytic germs, $$\label{germs1}
({{\mathcal R}}_{k}^{q}(M_{{{\mathcal A}}}),0) \overset{\exp}\longrightarrow ({{\mathcal V}}^q_k(M_{{{\mathcal A}}}),1)\, ,$$ for all $q$ and $k$.

\[tangentcone\] The resonance variety ${{\mathcal R}}_{k}^{q}(M_{{{\mathcal A}}})$ coincides with the tangent cone of the characteristic variety ${{\mathcal V}}^q_k(M_{{{\mathcal A}}})$ at the point $1$, $TC_1 ({{\mathcal V}}^q_k(M_{{{\mathcal A}}}))$.

In [@Lib2], Libgober proves 