Part Iii Seminar (lent Term): Moduli Stacks of Vector Bundles

Let X be a smooth projective curve over C (you can think of X as a Riemann surface in the analytic setting if you wish) throughout. Then there exists a smooth variety PicX over C whose C-points are the isomorphism classes of line bundles on X. More generally, for any scheme (variety) S over C, the set HomC(S,PicX) corresponds to a sheafification of the line bundles on the family of curves X ×S (the Picard group). This is an example of a moduli problem: studying a class of objects (line bundles) on families of schemes (curves). In this case, the Picard scheme PicX is the fine moduli space of line bundles. We will be considering the moduli problem of vector bundles of a fixed rank r on X ×S as S varies. Pullback makes this a “presheaf” on the category of Sch. But what category does this functor go to? Vector bundles have many nontrivial automorphisms (Aut(O) = GLr), so we shouldn’t just ignore these. To account for this, we define a presheaf Bunr : Sch op → Gpd to the 2-category of groupoids. Since you can glue compatible vector bundles together, this presheaf is a sheaf/stack. The goal of this talk will be to outline/convince you that this sheaf has geometric properties which makes it behave almost like a scheme, i.e., Bunr is an algebraic stack [Beh91, Bro10, Ols06, Sor00]. This result is important because once we know Bunr is algebraic, one can define quasi-coherent sheaves and D-modules on it, just as with schemes. The geometric Langlands program studies the correspondence between D-modules on Bunr (↔ automorphic forms) and local systems (rank r vector bundles with flat connections) on X (↔ Galois representations), and this correspondence has deep connections to both number theory (classical Langlands) and quantum physics (cf. [Fre10]).