Twisted genera of symmetric products (Q411320)

Let \(X\) be an algebraic variety. The \(n\)-fold symmetric product \(X^{(n)}\) of \(X\) is the quotient of \(X^n\) by the symmetric group on \(n\) elements where the action is given by permuting factors. The authors give a new proof of formulae for the generating series of Hodge genera of symmetric products of a complex quasi-projective variety \(X\) with any kind of singularities. Important specializations include generating series for extensions of Hodge numbers and Hirzebruch's \(\chi_y\)-genus to the singular setting with mixed Hodge module coefficients, and generating series for intersection cohomology Hodge numbers and Goresky--MacPherson intersection cohomology signatures. The proof is based on equivariant Künneth formulae and pre-lambda structures on the coefficient theory of a point. Following the approach of \textit{M. F. Atiyah} [Q. J. Math., Oxf. II. Ser. 17, 165--193 (1966; Zbl 0144.44901)] to power operations in \(K\)-theory, the authors extend operations on the coefficient pre-lambda ring to introduce interesting coefficients on \(X^{(n)}\) related to the corresponding Adams operations.