# Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces

## TL;DR

This paper investigates the cohomology of local systems on complements of arrangements of smooth algebraic hypersurfaces, demonstrating vanishing results and duality properties in specific geometric contexts.

## Contribution

It establishes vanishing theorems for local system cohomology on certain quasi-projective varieties with hypersurface arrangements, extending duality concepts to these spaces.

## Key findings

- Cohomology of local systems vanishes under specified conditions
- Complements of linear, toric, and elliptic arrangements are duality spaces
- Provides criteria for Stein manifold structures in hypersurface arrangements

## Abstract

We consider smooth, complex quasi-projective varieties $U$ which admit a compactification with a boundary which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on $U$ vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.