Computing cup products in integral cohomology of Hilbert schemes of points on \(K3\) surfaces (Q2804224)

The Hilbert schemes of points on a \(K3\) surface are irreducible holomorphic symplectic manifolds. The cup products in the integral cohomology are studied. A computer algebra program is used to compute the cup products. The source code and an explanation how to use it is given in an appendix.NEWLINENEWLINELet \(S^{[3]}\) be the Hilbert scheme of three points on a projective \(K3\) surface. \(\mathrm{Sym}^kH^2(S^{[3]}, \mathbb{Z})\) can be identified with its image in \(H^{2k}(S^{[3]}, \mathbb{Z})\) under the cup product mapping. As a result of the computations the following theorem is obtained:NEWLINENEWLINE\(H^4(S^{[3]}, \mathbb{Z})/\mathrm{Sym}^2H^2(S^{[3]}, \mathbb{Z})\simeq\mathbb{Z}/3\mathbb{Z}\oplus \mathbb{Z}^{23}\)NEWLINENEWLINE\(H^6(S^{[3]}, \mathbb{Z})/H^2 (S^{[3]}, \mathbb{Z})\cup H^4(S^{[3]}, \mathbb{Z})\simeq(\mathbb{Z}/3\mathbb{Z})^{23}\).