Twisted genera of symmetric products

TL;DR

This paper develops comprehensive formulas for the generating series of (Hodge) genera of symmetric products of complex varieties, accommodating singularities and extending classical invariants to new settings.

Contribution

It introduces general generating series formulas for (Hodge) genera of symmetric products with coefficients, applicable to singular varieties, unifying and extending classical results.

Findings

Formulas for generating series of Hodge and Hirzebruch genera for singular varieties.

Extension of intersection cohomology invariants to symmetric products.

Application of K-theory power operations to symmetric product invariants.

Abstract

We prove very general formulae for the generating series of (Hodge) genera of symmetric products with coefficients, which hold for complex quasi-projective varieties with any kind of singularities, and which include many of the classical results in the literature as special cases. Important specializations of our results include generating series for extensions of Hodge numbers and Hirzebruch genus to the singular setting and, in particular, generating series for Intersection cohomology Hodge numbers and Goresky-MacPherson Intersection cohomology signatures of symmetric products of complex projective varieties. A very general proof is given based on Kuenneth formulae and pre-lambda structures on the coefficient theory of a point. Moreover, Atiyah's approach to power operations in K-theory also works in this context, giving a nice description of the important related Adams operations. This last approach also allows us to introduce very interesting coefficients on the symmetric products.