Chern classes of tautological sheaves on Hilbert schemes of points on surfaces
Let \(X\) be a smooth projective surface and \(X^{[n]}\) the Hilbert scheme of generalized \(n\)-tuples on \(X\). The paper deals with the Chern classes of tautological bundles of the form \(p_*({\mathcal O}_\Xi\otimes q^*E)\), where \(E\) is a vector bundle on \(X\) and \((p,q):\Xi\to X^{[n]}\times X\) is the universal family. It gives an algorithmic description of the action of these Chern classes on the cohomology of the Hilbert schemes within the framework of Nakajima's oscillator algebra [\textit{H. Nakajima}, Ann. Math., II. Ser. 145, No. 2, 379-388 (1997; Zbl 0915.14001)]. This method leads to an identification of the cohomology ring of \((\mathbb{A}^2)^n\) with a ring of explicitly given differential operators on a Fock space. The paper ends with the computation of the top Segre classes of tautological bundles associated to line bundles on \(X^{[n]}\) up to \(n=7\), extending computations of Severi, LeBarz, Tikhomirov and Troshina, and gives a conjecture for the generating series.
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