Volumes of definable sets in o-minimal expansions and affine GAGA theorems

Abstract

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 ${\mathcal{H}}^d(\{x\in S:\Vert x\Vert <r\})\le C{r}^d$, 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.

arXiv commentAdded references to papers of Nguyen and Valette and of Kurdyka and Raby containing proofs of the main volume estimate, Theorem 4, which were pointed out to me after version 1 was posted