A priori obstructions to resolution of singularities

Tobias Shin

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arXiv · 2312.05366

A priori obstructions to resolution of singularities

Tobias Shin

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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 $\ell \neq p$. Along the way, we also obtain a Wu formula for the mod $p$ Steenrod operations in the case of closed embeddings of smooth varieties.

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BibTeX RIS

Tobias Shin. 2024-12-02. A priori obstructions to resolution of singularities. https://arxiv.org/abs/2312.05366

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