Characteristic classes of symmetric products of complex quasi-projective varieties
Let \(X\) be a complex quasi-projective variety and let \(X^{(n)}\) denote its symmetric product. Let \(T_{y*}:K_0(var/X)\rightarrow H_{ev}^{BM}(X)\oplus\mathbb{Q}[y]\) be the un-normalized motivic Hirzebruch class transformation, where \(H_{ev}^{BM}(X)\) denotes the Borel-Moore homology in even degrees and \(K_0(var/X)\) the relative Grothendieck group of complex algebraic varieties over \(X\) (see [\textit{J.-P. Brasselet} et al., J. Topol. Anal. 2, No. 1, 1--55 (2010; Zbl 1190.14009)]). For any positive integer \(r\), let \(d^r:X\rightarrow X^{(r)}\) denote the composition of the diagonal embedding \(X\hookrightarrow X^r\) with the projection \(X^r\rightarrow X^{(r)}\), and let \(\Psi_r\) denote the \(r\)-th homological Adams operation, which on \(H_{2k}^{BM}(X^{(r)},\mathbb{Q})\) is defined by multiplication by \(1/r^k\), together with \(y\mapsto y^r\). The main result of this paper is the folloing generating series formula for the Hirzebruch classes of the symmetric powers \(\mathcal{M}^{(n)}\in D^b\text{MHM}(X^{(n)})\) of a fixed complex of mixed Hodge modules on the variety \(X\): \(\sum_{n\geq 0}T_{(-y)*}(\mathcal{M}^{(n)})\cdot t^n=\text{exp}\left(\sum_{r\geq 1}\Psi_r(d_*^rT_{(-y)*}(\mathcal{M}))\cdot\frac{t^r}{r}\right)\). If one lets \(\mathcal{M}\) be the constant Hodge sheaf \(\mathbb{Q}_X^H\), the above formula becomes the expression \(\sum_{n\geq 0}T_{(-y)*}(X^{(n)})\cdot t^n=\text{exp}\left(\sum_{r\geq 1}\Psi_r(d_*^rT_{(-y)*}(X))\cdot\frac{t^r}{r}\right)\) for the motivic Hirzebruch classes.
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