Twisted Genera of Symmetric Products

We prove very general formulae for the generating series of (Hodge) genera of symmetric products X with coefficients, which hold for complex quasi-projective varietiesX 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’s χy-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 Künneth formulae and pre-lambda structures on the coefficient theory of a point, K̄0(A(pt)), with A(pt) a Karoubian Q-linear tensor category. Moreover, Atiyah’s approach to power operations in K-theory also works in this context for K̄0(A(pt)), 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 X.