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

TL;DR

This paper provides a quick proof of the definable Chow theorem using volume estimates for definable sets in o-minimal structures, making the concepts accessible to algebraic geometers.

Contribution

It offers a new, simplified proof of the definable Chow theorem leveraging volume estimates, and reviews foundational notions for algebraic geometers unfamiliar with o-minimal structures.

Findings

Volume estimate for definable sets: $ ext{Hausdorff measure} \, ext{bounded by} \, Cr^d$

Simplified proof of the definable Chow theorem

Accessible review of o-minimal structures for algebraic geometers

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 $\mathcal{H}^d(\{x\in S:\lVert x\rVert<r\})\leq Cr^d$, 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.