Topological and Hodge L-classes of singular covering spaces and varieties with trivial canonical class

Original Paper
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Abstract

The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincaré complexes as examples of C. T. C. Wall show. We establish the multiplicativity of the signature, and more generally, the topological L-class, for closed oriented stratified pseudomanifolds that can be equipped with a middle-perverse Verdier self-dual complex of sheaves, determined by Lagrangian sheaves along strata of odd codimension. This class of spaces, called L-pseudomanifolds, contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We apply this result in proving the Brasselet–Schürmann–Yokura conjecture for normal complex projective 3-folds with at most canonical singularities, trivial canonical class and positive irregularity. The conjecture asserts the equality of topological and Hodge L-class for compact complex algebraic rational homology manifolds.

Keywords

Signature Characteristic classes Pseudomanifolds Stratified spaces Intersection homology Perverse sheaves Hodge theory Canonical singularities Varieties of Kodaira dimension zero Calabi–Yau varieties 

Mathematics Subject Classification (2010)

57R20 55N33 32S60 14J17 14J30 14E20 

Notes

Acknowledgements

We thank Selma Altınok-Bhupal, Gavin Brown, Daniel Greb, Laurentiu Maxim and particularly Jörg Schürmann, who suggested to bring in transfer maps, for helpful communication and discussions.

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Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität HeidelbergHeidelbergGermany

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