Symmetric products of mixed Hodge modules

TL;DR

This paper extends classical theorems to describe the mixed Hodge structure on symmetric products of complexes of mixed Hodge modules and generalizes signature formulas to broader contexts.

Contribution

It provides a formula for the mixed Hodge structure on symmetric products and introduces a generalized signature theorem for symmetric pairings on complexes.

Findings

Derived a formula for the mixed Hodge structure on symmetric products.

Established the existence of symmetric group actions on self-products of mixed Hodge modules.

Generalized the signature theorem to complexes with possibly degenerate pairings.

Abstract

Generalizing a theorem of Macdonald, we show a formula for the mixed Hodge structure on the cohomology of the symmetric products of bounded complexes of mixed Hodge modules by showing the existence of the canonical action of the symmetric group on the multiple external self-products of complexes of mixed Hodge modules. We also generalize a theorem of Hirzebruch and Zagier on the signature of the symmetric products of manifolds to the case of the symmetric products of symmetric parings on bounded complexes with constructible cohomology sheaves where the pairing is not assumed to be non-degenerate.