奇点解消的先验障碍
A priori obstructions to resolution of singularities
AI总结：
本文基于Thom的论证构造射影簇奇点嵌入解消的先验上同调障碍并推广至特征p>0的域，证明其对满足特定条件的一般上同调理论成立，给出高Chow群上Steenrod运算的消失结果及Wu公式。
摘要：
我们呈现了Thom提出的一个论证，用于构造射影簇存在奇点嵌入解消的先验上同调障碍，并将该论证推广到特征p>0的域。我们证明，这些障碍对满足局部化、étale切除和$\mathbb{A}^1$-不变性且配备真推进的一般上同调理论均有定义。作为例子，我们表明，通过合适的$E_{\infty}$-代数结构定义的、模$\ell$的高Chow群上的奇数次Steenrod同调运算（$\ell$为任意素数）在光滑簇的基本类上消失。我们将这一一般消失结果推广到任意簇，适用于模$\ell \neq p$定义的运算。在此过程中，我们还得到光滑簇闭嵌入情形下模p Steenrod运算的Wu公式。
英文摘要：
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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