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

Abstract

In this mostly expository note, I give a 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 d -dimensional definable subset of S ⊆ R n S\subseteq \mathbb {R}^n in an o-minimal expansion of the ordered field of real numbers satisfies the inequality H d ( { x ∈ S : ‖ x ‖ > r } ) ≤ C r d \mathcal {H}^d(\{x\in S:\lVert x\rVert >r\})\leq Cr^d , where H d \mathcal {H}^d denotes the d d -dimensional Hausdorff measure on R n \mathbb {R}^n and C C is a constant depending on S S . 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.