Symmetric products of mixed Hodge modules

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.

Characteristic Classes of Singular Toric Varieties

“…, with the sheaf of Zariski p-forms also holds for algebraic V -manifolds (also called orbifolds), i.e., for spaces locally given as the quotient of a complex manifold Y by the action of a finite group G of automorphisms of Y ; see [43][Rem.2.9]. In particular, for a global orbifold X = Y /G, with Y a projective complex algebraic manifold and G a finite group of algebraic automorphisms of Y , we have Ω …”

“…Note that the equality sign(X) = χ 1 (X) is also a consequence of the Hodge index theorem for a projective orbifold X, which is a special case of the corresponding result in intersection homology for a projective variety X (see [43][Sec. 3.6], and in the smooth case see also [48][Thm.3.12]).…”

Equivariant Characteristic Classes of Singular Complex Algebraic Varieties

“…Such an exterior product with Künneth isomorphisms is readily available in all cases mentioned in Example 6.1 (as explained in more detail in [MS09,MSS]). Similarly, one has…”

Equivariant characteristic classes of external and symmetric products of varieties

“…To a given object F ∈ cat(Z) in a category as above, i.e., coherent or constructible sheaves, or mixed Hodge modules on Z (resp., morphisms f : Y → Z in the motivic context), we attach new objects as follows (see [13,27,28] for details):…”