Generic vanishing and characterizations of semi-abelian varieties

In this project I propose a revolutionary approach to study the geometry of algebraic varieties, with special focus on non-projective ones. Some of the research questions asked here appear already in the work of Iitaka and Kawamata from 45-years ago. However, till some my recent papers, evolutions of this theory had suffered the limit of the available techniqes.

Hence the need for new tools.Building on existing results on cohomology of local systems, I aim to extend Green-Lazarsfeld generic vanishing theory to non-compact algebraic varieties.

There are two main objectives, one concering surfaces - which evolves from my recent publications with Mendes Lopes and Pardini - and one, to be considered more long term, which focuses on higher dimensional varieties.Green-Lazarsfeld generic vanishing has fueled a long series of research papers in many different directions.

Extensions, which appear equally fruitful, have been provided recently in positive characteristic and for compact complex manifolds. Differently, no progress has been made in the case of non-compact varieties.

Thus I aim to close this gap, improving our understanding of the geometry of algebraic varieties, both answering classical questions, and opening new investigation frontiers.I will circulate the outcome of the project among the scientific community by means of publications in high level peer reviewed journals.

I seek funding for a PhD student, starting in the fall of 2024.