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 $\widetilde X\to X$ such that its exceptional locus as well as $\widetilde X$ 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.