GAGA theorems

Given an algebraic variety X over a topological field (e.g. R, C or Qp), one can often make some sort of analytic space Xan from X. The topology on Xan reflects the topology of the topological field and the functions on Xan should be appropriately holomorphic. The relationship between the vector bundles, subvarieties, cohomology, and coherent sheaves on X and Xan is typically referred to as a “GAGA theorem”. This name goes back to Serre’s paper Géométrie algébrique et géométrie analytique (1956), where the relationship was considered over C. Over the decades, corresponding to different types of analytifications of varieties and schemes, various GAGA theorems have been established (e.g., rigid, adic, and formal). I will discuss a new and unified GAGA theorem. This gives all existing results in the literature and also puts them into a broader context.