2025/11/10 by Cao, Junyan, Deng, Ya, Hacon, Christopher D. +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds
paper · doi:10.48550/arxiv.2511.06773
openalex publication_date 2025/11/10 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28
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 ∂∂-lemma. Our second-and primary-goal is to develop the technical tools required for this approach, namely an L2-Hodge theory for the cohomology of rank-one local systems on quasi-compact Kähler 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.