Deligne cohomology of a complex manifold Y is a hybrid object: it mixes Betti cohomology with the Hodge filtration on de Rham cohomology, producing a complex of locally compact abelian groups. Despite its central role in regulators and special-value conjectures, it lacks a geometric framework that treats the archimedean structure of de Rham cohomology and the discrete structure of Betti cohomology at the same time.
In this talk, we construct an analytic stack \mathcal{X} (in the sense of Clausen-Scholze) that serves as a universal base for such objects. Quasi-coherent sheaves on \mathcal{X} combine archimedean (liquid) modules and non-archimedean (solid) modules. In particular, Deligne cohomology groups appear as quasi-coherent sheaves on \mathcal{X}.
Finally, we present a strategy to associate, to any complex analytic manifold Y, an analytic stack Y^{Del} over \mathcal{X}, whose relative cohomology should recover Deligne cohomology. This is a work in progress. | 
