Hodge theory for local systems and cohomological support loci

In this article, we pursue two main objectives. The first is to show that the fundamental results of Green-Lazarsfeld (1987, 1991) on generic vanishing theorems, and works of Budur-Wang (2015, 2020) on cohomology jumping loci, can be established within a unified framework based on suitable versions of the $\partial\bar{\partial}$-lemma. Our second-and primary-goal is to develop the technical tools required for this approach, namely an $L^2$-Hodge theory for the cohomology of rank-one local systems on quasi-compact K\"ahler manifolds. Further developments concerning higher-rank local systems, as well as several geometric applications, will be presented in a companion paper and are briefly outlined in the introduction.