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

for any $i,j=1,\dots ,n$ and any $k\ge \max \{n-i,n-j\}$. 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 $H^i(X,{\Omega }_X^j)$. 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 $X$.