an analytic morphism ${\mathbb{C}}^n= H^1(M_{{{\mathcal A}}},{\mathbb{C}}) \overset {\exp} \longrightarrow  {\mathbb{T}}_{G}=({\mathbb{C}}^*)^n$.

The local system associated to a character $\rho \in {\mathbb{T}}_{G}=\operatorname{Hom}(G_{ab},{\mathbb{C}}^*)$ is called [*equimonodromical*]{} if it is constant on the natural basis of $G_{ab}$.

A [*system of complex weights*]{} $\alpha=(\alpha_1, \dots, \alpha_n), \; \alpha_i \in {\mathbb{C}},\; 1 \leq i \leq n$, on the hyperplanes of ${{\mathcal A}}$, or equivalently an element $\alpha_{1}e_1+ \dots \alpha_ne_n \in A^1_{{\mathbb{C}}}({{\mathcal A}})$, exponentiates to an element of the character torus, $\rho=(\exp(\alpha_1), \dots, \exp(\alpha_n))$. If $\alpha_i=2 \pi \sqrt{-1}\frac{m_i}{d},\; 1 \leq i \leq n$, for some integers $m_1, \dots, m_n$ with $gcd(m_1, \dots m_n)=1$ and $d \in {\mathbb{Z}}_{>0}$, the character is called [*rational*]{}. Rational equimonodromical characters are very important in the study of the Milnor fiber cohomology (see Section \[sect:Milnor\]).

Let $(\tilde{C}_*, \tilde{d}_*)$ denote the equivariant chain complex of the universal cover of $M$. It is a free chain complex of finitely generated left ${\mathbb{Z}}[G]$-modules. The [*twisted*]{} homology (respectively cohomology) with respect to a local system $L$ is the homology of $L\otimes_{{\mathbb{Z}}[G]} (\tilde{C}_*, \tilde{d}_*)$ (respectively the cohomology of $\operatorname{Hom}_{{\mathbb{Z}}[G]} ((\tilde{C}_*, \tilde{d}_*), L)$). For cohomology, we turn the right ${\mathbb{Z}}[G]$-module $L$ into a left ${\mathbb{Z}}[G]$-module, in the usual way. Notation: $H_*(M, L)$ (respectively $H^*(M, L)$).

For example, the trivial 