A priori obstructions to resolution of singularities
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916501929328640 |
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| author | Shin, Tobias |
| author_facet | Shin, Tobias |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_05366 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A priori obstructions to resolution of singularities Shin, Tobias Algebraic Geometry Algebraic Topology 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. |
| title | A priori obstructions to resolution of singularities |
| topic | Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2312.05366 |