Atiyah Classes and Closed Forms on Moduli Spaces of Sheaves

  • Francesco Bottacin

    Università di Padova, Italy


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 kmax{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.

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