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 H(1) (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 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. One consequence of this result is that the local systems L where the dimension of H(1) (M, L) jumps along a given positive-dimensional component of the characteristic variety V(1) (M) have finite order, see Theorem 4.7. Using this, we show in Corollary 4.9 that dim H(1) (M, L) = dim H(1) (M, L(-1)) for any rank one local system L on a smooth complex algebraic variety M. (C) 2008 Elsevier Inc. All rights reserved.