(S.H): Polarized K3 or abelian surfaces defined over \(\mathbb{C}\), the main object in this topic is the moduli spaces of sheaves \(M(r,c_1,c_2)\), (rank, chern classes) space of Gieseker \(H\)-semistable sheaves on \(S\). This is a projective variety with a holomorphic symplectic form.
The next main object is duality involution. When \(c_1=0\) and \(M^{\mu-st}\neq \varnothing\) (\(\mu\)-stable part), we can consider the duality involution by taking the dual of a sheaf
\[D: \mathcal{F}\mapsto \mathcal{F}^\vee \in \mathrm{Aut}(M(r,0,c_2))\]
\(D\) is regular on \(M^{\mu-st}\).
\(D\) is symplectic.
\(D\) is nontrivial if \(r\ge 3\).
\(X\) smooth projective variety with holomorphic symplectic form \(\sigma\). For classification, we may assume that \(X\) is IHS (i.e. \(X\) is simply connected and \(H^0(X,\Omega^2_X)=\mathbb{C}\sigma\)). By B-B decomposition, known IHS manifolds up to deformation:
\(\mathrm{K3}^{[n]}\)
Generalized Kummer \(K_n(A)\)
OG10
OG6
For example Fujiki 1983, Kawatani 2009, \(S^n/G\) and \(K(A)\to K3^{[n]}/G\).
Theorem 1. Assume \(\rho(S)=1\) and \(r\ge 3\). Then \(D\) extends to a regular involution on the whole \(M(r,0,c_2)\). This is equivalent to \(c_2=2r\) if \(S\) is K3, and \(c_2=2\) if \(S\) is abelian.
Remark. Even if these conditions are satisfied, the extended involution is not given by the duality on \(M(r,0,c_2)\).
Theorem 2. Consider
Sis K3
\(M(3,0,6)\): singular of dimension \(20\).