  A rank one local system $\LL$ on a smooth complex algebraic variety $M$ is
1-admissible if the dimension of the first cohomology group $H^1(M,\LL)$ can be
computed from the cohomology algebra $H^*(M,\C)$ in degrees $\leq 2$. Under the
assumption that $M$ is 1-formal, we show that all local systems, except
finitely many, on a non-translated irreducible component $W$ of the first
characteristic variety $\V_1(M)$ are 1-admissible, see Proposition 3.1. The
same result holds for local systems on a translated component $W$, but now
$H^*(M,\C)$ should be replaced by $H^*(M_0,\C)$, where $M_0$ is a Zariski open
subset obtained from $M$ by deleting some hypersurfaces determined by the
translated component $W$, see Theorem 4.3.
