### Francesco Bottacin

Università di Padova, Italy

## Abstract

Let *X* be a smooth *n*-dimensional projective variety, and let*Y* 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 any*k* ≥ *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 space*Y* 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*.

## Cite this article

Francesco Bottacin, Atiyah Classes and Closed Forms on Moduli Spaces of Sheaves. Rend. Sem. Mat. Univ. Padova 121 (2009), pp. 165–177

DOI 10.4171/RSMUP/121-10