Three Descriptions of the Cohomology of Bun G ( X ) ( Lecture 4 ) April 9 , 2013

Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and let G be a smooth affine group scheme over X. We let Bun G (X) denote the moduli stack of G-bundles on X (denoted by Bun G in the previous lecture), and let denote a prime number which is invertible in k. Our main goal in this course is to give a convenient description of the-adic cohomology ring H * (Bun G (X); Q). In the special case where G and X are defined over some finite field F q , this will allow us to compute the trace of ϕ −1 on H * (Bun G (X); Q) (where ϕ denotes the geometric Frobenius morphism from Bun G (X) to itself), which we will use to prove Weil's conjecture following the outline provided in the last lecture. However, the problem of describing H * (Bun G (X); Q) makes sense over an arbitrary algebraically closed field k. In this lecture, we will specialize to the case k = C, and describe several topological approaches to the problem (which we will later adapt to the setting of algebraic geometry). In this case, we do not need to work-adically: the algebraic stack Bun G (X) has a well-defined homotopy type (namely, the homotopy type of the associated analytic stack). Let us henceforth assume that k = C. We will identify X with the corresponding Riemann surface (a smooth manifold of dimension 2). Let us distinguish between two a priori different notions of G-bundle: (a) Since X is an algebraic variety, we can consider the category algebraic G-bundles on X: that is, principal G-bundles P → X in the category of schemes. Algebraic G-bundles are classified by the moduli stack Bun G (X). (b) Regarding X as a smooth manifold, we can consider smooth G-bundles on X. More generally, for any topological space Y , we can consider G-bundles on the product X × Y which are equipped with a smooth structure in the X-direction. By general nonsense, there is a classifying space for such bundles. More precisely, we can choose a topological space M with the property that for any (sufficiently nice) space Y , there is a canonical bijection between the set of isomorphism classes of G-bundles on X × Y with the set of homotopy classes …