可定义集在O-极小展开式中的扩张与仿射GAGA定理
Volumes of definable sets in o-minimal expansions and affine GAGA theorems
摘要：
在这个主要是简要说明，我给一个非常快速的证明可定义的周定理的Peterzil和Starchenko使用主教斯托尔定理和体积估计可定义的集由于阮和Valette。体积估计说，在实数的序域的o-最小展开中，$S\subseteq\mathbb{R}^n$的任何$d$维可定义子集满足不等式$\mathcal{H}^d(\{x\in S:\lVert x\rVert<r\})\leq Cr^d$，其中$\mathcal{H}^d$表示$\mathbb{R}^n$上的$d$维Hausdorff测度，并且$C$是依赖于$S$的常数。一个密切相关的亚分析集的体积估计可以追溯到Kurdyka和Raby。由于这说明是为了帮助代数几何不精通O-最小结构和可定义集，我回顾这些概念，也证明了主要的体积估计从头开始。
英文摘要：
In this mostly expository note, I give a very quick proof of the definable Chow theorem of Peterzil and Starchenko using the Bishop-Stoll theorem and a volume estimate for definable sets due to Nguyen and Valette. The volume estimate says that any $d$-dimensional definable subset of $S\subseteq\mathbb{R}^n$ in an o-minimal expansion of the ordered field of real numbers satisfies the inequality $\mathcal{H}^d(\{x\in S:\lVert x\rVert<r\})\leq Cr^d$, where $\mathcal{H}^d$ denotes the $d$-dimensional Hausdorff measure on $\mathbb{R}^n$ and $C$ is a constant depending on $S$. A closely related volume estimate for subanalytic sets goes back to Kurdyka and Raby. Since this note is intended to be helpful to algebraic geometers not versed in o-minimal structures and definable sets, I review these notions and also prove the main volume estimate from scratch.
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