Let G be a complex affine algebraic group. If C is a smooth algebraic curve and x is a point in C, the affine Grassmannians are algebro-geometric objects that study G-bundles on C together with a trivialization outside x. A particularly smart version is the so-called Beilinson-Drinfeld affine Grassmannian, where the point x is allowed to move, and we can even allow multiple points x. In this talk we present possible analogs for the Beilinson-Drinfeld affine Grassmannian, in the case where the curve is replaced by a smooth projective surface, and the trivialization data are given with respect to flags of closed subschemes. Based on joint work with B. Hennion and G. Vezzosi.
