On admissible rank one local systems

Abstract

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 $\le 2$. 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.

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

References in corpus (5)

• Formality, Alexander invariants, and a question of Serre
• Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers
• On the irreducible components of characteristic varieties
• Characteristic varieties and constructible sheaves
• On the intersection of rational transversal subtori