On admissible rank one local systems

TL;DR

This paper investigates the conditions under which rank one local systems on complex algebraic varieties are 1-admissible, linking their cohomology to the algebraic structure of the variety, especially under 1-formality assumptions.

Contribution

It establishes that most local systems on certain components of the characteristic variety are 1-admissible, extending results to translated components with a modified cohomology algebra.

Findings

Most local systems on non-translated components are 1-admissible.

For translated components, 1-admissibility holds when considering a Zariski open subset.

The results depend on the 1-formality of the variety.

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 $\leq 2$. 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.