Atiyah Classes and Closed Forms on Moduli Spaces of Sheaves

Abstract

Let X be a smooth n-dimensional projective variety, and let Y be a moduli space of stable sheaves on X. By using the local Atiyah class of a universal family of sheaves on Y, which is well defined even when such a universal family does not exist, we are able to construct natural maps f : H\_i\_(X, ΩX\_j\_) → H k+i-n(Y, ΩY\_k\_+j-n), for any i, j = 1,…,n and any k ≥ max{n-i, n-j}. In particular, for k = n-i, the map f associates a closed differential form of degree j-i on the moduli space Y to any element of H\_i\_(X,ΩX\_j\_). This method provides a natural way to construct closed differential forms on moduli spaces of sheaves. We remark that no smoothness hypothesis is made on the moduli space Y. As an application, we describe the construction of closed differential forms on the Hilbert schemes of points of X.