Verdier Duality (Lecture 21)

Let k be an algebraically closed field, ℓ a prime number which is invertible in k, X an algebraic curve over k and G a reductive group scheme over X. The direct image of the constant sheaf along the map RanG(X) → Ran(X) can be regarded as a sheaf A on the Ran space Ran(X), whose stalk at a point µ: S → X(k) having image {x1,..., xm} ⊆ X(k) is given by Aµ ≃ ⊗ C ∗ (GrG,xi; Zℓ). The main result of the first part of this course is that the cochain complex C ∗ (BunG(X); Zℓ) can be identified with the global sections of A. Roughly speaking, our next move is to exploit this fact by analyzing the Verdier dual of A (or, more precisely, the Verdier dual of a “reduced version ” of A which we will discuss in the next two lectures). Since the Ran space Ran(X) is an infinite-dimensional algebro-geometric object, the classical theory of Verdier duality does not apply directly. In this lecture, we will give an overview of Verdier duality in the setting of topology, and describe how it can be adapted to spaces like Ran(X). We begin by reviewing the theory of sheaves. For the remainder of this lecture, let us fix an ∞-category C which admits small limits and colimits. Let Y be a topological space and let U(Y) denote the partially ordered set of all open subsets of Y. Recall that a C-valued sheaf on Y is a functor F: U(Y) op → C with the following property: for each open cover {Uα} of an open set U ⊆ Y, the canonical ma