Topological and Hodge L-Classes of Singular Covering Spaces and Varieties with Trivial Canonical Class

TL;DR

This paper proves the multiplicativity of topological L-classes for a broad class of singular spaces, including algebraic varieties, and confirms a conjecture relating topological and Hodge L-classes for certain complex projective 3-folds.

Contribution

It establishes the multiplicativity of the topological L-class for L-pseudomanifolds and proves the Brasselet-Schürmann-Yokura conjecture for specific complex algebraic threefolds.

Findings

Proved multiplicativity of the topological L-class for L-pseudomanifolds.

Confirmed the Brasselet-Schürmann-Yokura conjecture for certain complex projective 3-folds.

Unified topological and Hodge L-classes in the specified setting.

Abstract

The signature of closed oriented manifolds is well-known to be multiplicative under finite covers. This fails for Poincar\'e 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 (so-called L-pseudomanifolds). This class of spaces contains all Witt spaces and thus all pure-dimensional complex algebraic varieties. We apply this result in proving the Brasselet-Sch\"urmann-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.