Cohomology representations of external and symmetric products of varieties

Laurentiu Maxim, Joerg Schuermann

Published 2016-02-21Version 1

We prove refined generating series formulae for characters of (virtual) cohomology representations of external products of suitable coefficients on (possibly singular) complex quasi-projective varieties, e.g., (complexes of) constructible or coherent sheaves, or (complexes of) mixed Hodge modules. These formulae generalize our previous results for symmetric and alternating powers of such coefficients, and apply also to other Schur functors. The proofs of these results are reduced via an equivariant K\"unneth formula to a more general generating series identity for abstract characters of tensor powers $\cV^{\otimes n}$ of an element $\cV$ in a suitable symmetric monoidal category. This abstract approach applies directly also in the equivariant context for varieties with additional symmetries (e.g., finite group actions, finite order automorphisms, resp., endomorphisms).