Mixed Hodge Modules

Schürmann

2011

Sel. Math. New Ser.

We give a new proof of formulae for the generating series of (Hodge) genera of symmetric products X (n) with coefficients, which hold for complex quasiprojective varieties X 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 χ y -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. Our proof applies to more general situations and is based on equivariant Künneth formulae and pre-lambda structures on the coefficient theory of a point,K 0 (A( pt)), with A( pt) a Karoubian Q-linear tensor category. Moreover, Atiyah's approach to power operations in K -theory also applies in this context toK 0 (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 (n) .

Section: Corollary 13 Let X Be a Pure Dimensional Complex Quasi-projmentioning

confidence: 99%

Section: Is a Complex Algebraic Variety Or A Compact Complex Analyticmentioning

confidence: 99%

Section: Examples and Homomorphisms Of Pre-lambda Ringsmentioning

confidence: 99%

“…Together with (37), this implies the important Künneth isomorphism (9) from the introduction. (For a more direct approach compare with [34].…”

Section: Remark 23 the Fact That Total Cohomology Functormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Twisted genera of symmetric products

Schürmann

2011

Sel. Math. New Ser.

Hodge genera of algebraic varieties I

Cappell

Shaneson

2007

Comm Pure Appl Math

The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a complex algebraic variety X to such invariants of singularities of proper algebraic maps defined on X . Such formulae severely constrain, both topologically and analytically, the singularities of complex maps, even between smooth varieties. Similar results were announced by the first and third author in [13,32].

Equivariant Characteristic Classes of Singular Complex Algebraic Varieties

Cappell

Schürmann³

et al. 2012

Comm Pure Appl Math

Abstract. Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schürmann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, BaumFulton-MacPherson Todd classes, and Goresky-MacPherson L-classes, respectively). In this note we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms, and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We also compute the new classes in the context of monodromy problems, e.g., for varieties that fiber equivariantly (in the complex topology) over a connected algebraic manifold. As another application, we discuss Atiyah-Meyer type formulae for twisted Hirzebruch classes of global orbifolds.