A priori obstructions to resolution of singularities

Abstract

We present an argument due to Thom to formulate a priori cohomology obstructions for a projective variety to admit an embedded resolution of singularities, and generalize the argument to a field of characteristic $p>0$. We show that these obstructions are defined for a general cohomology theory that satisfies localization, étale excision, and ${\mathbb{A}}^1$-invariance, and is equipped with a proper pushforward. As examples, we show that odd degree Steenrod homology operations on higher Chow groups mod $\ell$, defined by a suitable $E_{\infty }$-algebra structure, vanish on the fundamental class of a smooth variety, where $\ell$ is any prime. We extend this general vanishing result to arbitrary varieties for operations defined mod $\ell \ne p$. Along the way, we also obtain a Wu formula for the mod $p$ Steenrod operations in the case of closed embeddings of smooth varieties.

arXiv commentRemoved half of exposition, and fixed section which previously relied on embedding bundle trick and now uses stack theory. Comments welcome!