On admissible rank one local systems

A rank one local system L on a smooth complex algebraic variety M is 1-admissible if the dimension of the first cohomology group H1(M,L) can be computed from the cohomology algebra H∗(M,C) in degrees ⩽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 V1(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∗(M0,C), where M0 is a Zariski open subset obtained from M by deleting some hypersurfaces determined by the translated component W, see Theorem 4.3. One consequence of this result is that the local systems L where the dimension of H1(M,L) jumps along a given positive-dimensional component of the characteristic variety V1(M) have finite order, see Theorem 4.7. Using this, we show in Corollary 4.9 that dimH1(M,L)=dimH1(M,L−1) for any rank one local system L on a smooth complex algebraic variety M.