The motive of a variety with cellular resolution of singularities

TL;DR

This paper explores the motive with compact support of complex varieties that admit cellular resolutions of singularities, providing explicit descriptions and examples including curves and toric varieties.

Contribution

It offers a concrete description of the motive with compact support for varieties with cellular resolutions, linking it to Borel-Moore homology under mild conditions.

Findings

Explicit formulas for motives of rational projective curves.

Descriptions of motives for toric varieties of dimensions two and three.

Examples illustrating the theory with concrete computations.

Abstract

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.