Tannaka Duality for Geometric Stacks

Let X and S denote algebraic stacks of finite type over the field C of complex numbers, and let X and S denote their analytifications (which are stacks in the complex analytic setting). Analytification gives a functor φ : HomC(S,X) → Hom(S , X). It is natural to ask for circumstances under which φ is an equivalence. In the case where X and S are projective schemes, a satisfactory answer was obtained long ago. In this case, both algebraic and analytic maps may be classified by their graphs, which are closed in the product X × S. One may then deduce that any analytic map is algebraic by applying Serre’s GAGA theorem (see [6]) to X × S. If S is a projective scheme and X is the classifying stack of the algebraic group GLn, then Hom(S,X) classifies vector bundles on S. If S is a proper scheme, then any analytic vector bundle on S is algebraic (again by Serre’s GAGA theorem), and one may again deduce that φ is an equivalence. By combining the above methods, one can deduce that φ is an equivalence whenever X is given globally as a quotient of a separated algebraic space by the action of a linear algebraic group (and S is proper). The main motivation for this paper was to find a more natural hypothesis on X which forces φ to be an equivalence. We will show that this is the case whenever X is geometric: that is, when X is quasi-compact and the diagonal morphism X → X ×X is affine. More precisely, we have the following: