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We give a proof of the Brasselet-Sch\"urmann-Yokura conjecture for projectivevarieties based on cubical hyperresolutions, the Decomposition Theorem, andclassical Hodge theory. It predicts the equality between the Hodge $L$-classand the Goresky-MacPherson $L$-class for compact complex algebraic varietiesthat are rational homology manifolds.

The Brasselet-Sch\"urmann-Yokura conjecture on $L$-classes of singular varieties