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European Mathematical Society Publishing House
2016-09-19 17:05:48
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
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121
2009
0
Atiyah Classes and Closed Forms on Moduli Spaces of Sheaves
Francesco
Bottacin
Università di Padova, 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 : Hi(X, ΩXj) → H k+i-n(Y, ΩYk+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 Hi(X,ΩXj). 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.
Algebraic geometry
General
165
177
10.4171/RSMUP/121-10
http://www.ems-ph.org/doi/10.4171/RSMUP/121-10