The motive of a variety with cellular resolution of singularities

A complex variety X admits a cellular resolution of singularities if there exists a resolution of singularities X˜→X such that its exceptional locus as well as X˜ and the singular locus of X admit a cellular decomposition. We give a concrete description of the motive with compact support of X in terms of its Borel–Moore homology, under some mild conditions. We give many examples, including rational projective curves and toric varieties of dimension two and three.