The Brasselet-Schürmann-Yokura conjecture on $L$-classes of singular varieties

Abstract

In 2010, Brasselet, Schürmann and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky-MacPherson $L$-class $L_{\ast }(X)$ and the Hirzebruch homology class $T_{1\ast }(X)$ for a compact complex algebraic variety $X$ that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions which we find of independent interest.

Exposition improved. Characterization of rational homology manifolds in terms of cubical hyperresolutions added. See arXiv:2103.04836 for another proof using Hodge modules

Cited by in corpus (1)

• Hodge modules and cobordism classes