Tempered subanalytic topology on algebraic varieties

TL;DR

This paper develops new sheaf-theoretic frameworks on algebraic varieties over complex numbers, introducing tempered subanalytic sites and analyzing their relations with classical analytification and algebraization.

Contribution

It constructs tempered subanalytic and Stein tempered subanalytic sites, along with associated sheaves and functors, linking analytic and algebraic geometry in novel ways.

Findings

Sheaves of tempered holomorphic functions are isomorphic to regular functions on Zariski open sets.

Defined tempered and Stein tempered analytifications and studied their relations with classical analytification.

Obtained algebraization and flatness results in the non-proper case.

Abstract

On a smooth algebraic variety over $\mathbb{C}$, we build the tempered subanalytic and Stein tempered subanalytic sites. We construct the sheaf of holomorphic functions tempered at infinity over these sites and study their relations with the sheaf of regular functions, proving in particular that these sheaves are isomorphic on Zariski open subsets. We show that these data allow to define the functors of tempered and Stein tempered analytifications. We study the relations between these two functors and the usual analytification functor. We also obtain algebraization results in the non-proper case and flatness results.