Atiyah Classes and Closed Forms on Moduli Spaces of Sheaves

Abstract

Let X be a smooth n-dimensional projective variety, and letY be a moduli space of stable sheaves on X. By using thelocal Atiyah class of a universal family of sheaves on Y,which is well defined even when such a universal family doesnot 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 anyk ≥ max{n-i, n-j}.In particular, for k = n-i, the map f associates aclosed differential form of degree j-i on the moduli spaceY to any element of H_i_(X,ΩX_j_). This methodprovides a natural way to construct closed differential formson moduli spaces of sheaves. We remark that no smoothnesshypothesis is made on the moduli space Y. As an application,we describe the construction of closed differential forms onthe Hilbert schemes of points of X.