Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces
For a smooth algebraic surface \(S\) over the complex numbers let \(S^{[n]}\) be the Hilbert scheme of zero-dimensional subschemes of length \(n\). The authors compute the singular cohomology \(H^*(S^{[n]}, \mathbb{Q})\). For the proof they use perverse sheaves. The result has been shown in special cases by \textit{G. Ellingsrud} and \textit{S. A. Strømme} in Invent. Math. 87, 343-352 (1987; Zbl 0625.14002) and for the case of a projective surface by \textit{L. Göttsche} [Math. Ann. 286, No. 1-3, 193- 207 (1990; Zbl 0686.14022)]. Using the mixed Hodge modules of Saito, the authors also describe the Hodge structure of \(H^*(S^{[n]}, \mathbb{Q})\), proving a conjecture of the first author. With the same methods they also compute the cohomology and its Hodge structure for higher order Kummer varieties, which were defined by \textit{A. Beauville} [J. Differ. Geom. 18, 755-782 (1983; Zbl 0537.53056)].
scientific article; zbMATH DE number 537296