AI 中文总结
本文研究本质p-维数与陈数，证明有限p群在特定光滑射影簇上的作用可分解，推导对基域算术敏感的不动点定理，得到Cremona群循环p子群阶的界，关键利用Karpenko与Merkurjev的p群本质p-维数计算。
AI 中文摘要
设X是定义在域k上的光滑、射影、几何连通的簇，且k包含p次本原单位根。若X存在与p互素的陈数，我们证明有限p群在X上的每个作用都可分解为GLₙ(k)的子群作用，其中n=dim X。这使得有限p群的表示性质可转移到其在X上的作用。我们推导出一个不动点定理，与此前同类已知结果不同，该定理对基域的算术性质敏感。我们还得到了Cremona群中循环p子群阶的界：例如当p≥n+2时，Crₙ(ℚ)不含p²阶元素。该方法基于一个具有独立意义的观察：对于特征零域上的仿射代数群G，edₚ(G)+dim G是具有G的一般自由作用且次数映射CH_G(Y)→𝔽ₚ非零的光滑射影簇Y的最小维数。我们结果的关键输入是Karpenko与Merkurjev对p群本质p-维数的计算。
英文摘要
Let $X$ be a smooth, projective, geometrically connected variety over a field $k$ containing a root of unity of order $p$. If $X$ has a Chern number prime to $p$, we show that every action of a finite $p$-group on $X$ factors through a subgroup of $\operatorname{GL}_n(k)$, where $n=\dim X$. This allows one to transfer properties of representations of finite $p$-groups to their actions on $X$. We deduce a fixed-point theorem which, unlike previously known results of this kind, is sensitive to the arithmetic of the base field. We also obtain a bound on the orders of cyclic $p$-subgroups of the Cremona groups: for instance $\operatorname{Cr}_n(\mathbb{Q})$ contains no element of order $p^2$ when $p \ge n+2$. The method is based on the following observation, of independent interest. For an affine algebraic group $G$ over a field of characteristic zero, $\operatorname{ed}_p(G) + \dim G$ is the least dimension of a smooth projective variety $Y$ with a generically free $G$-action such that the degree map $\operatorname{CH}_G(Y) \to \mathbb{F}_p$ is nonzero. A key input for our result is Karpenko and Merkurjev's computation of the essential $p$-dimension of $p$-groups.
