On admissible rank one local systems

Details On admissible rank one local systems On admissible rank one local systems

2007-07-31

arxiv.org/abs/0707.4646v2

Journal of Algebra 321(2009), 3145-3157.

Mathematics

Algebraic Geometry

The second version contains a couple of new results, namely Theorem 4.7 and Corollary 4.9

Scientific paper

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.

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