Effective local differential topology of algebraic varieties over local fields of positive characteristics
Avraham Aizenbud, Dmitry Gourevitch, David Kazhdan, Eitan Sayag
TL;DR
This work develops an effective local differential-topology framework for algebraic varieties over local fields of positive characteristic, centering on rectification, balls, and fixed measures to achieve uniform control across field extensions. It proves effective, uniform versions of the implicit/inverse function theorems, establishes control over neighborhoods of subvarieties, and provides push-forward continuity/smoothness results for measures under maps. Key contributions include effective surjectivity criteria and quantitative bounds on push-forward measures for submersions, with results that remain robust under base-field extensions. The framework is motivated by, and designed to support, subsequent work (AGKS1, AGKS2) on jet schemes and Harish-Chandra-type integrability in positive characteristic, using Lang-Weil-type uniform point-count bounds.
Abstract
In this paper we provide a framework for quantitative statements on distances and measures when studying algebraic varieties and morphisms of algebraic varieties over local fields. We will concentrate on local fields of the type $\mathbb{F}_\ell((t))$ and work uniformly with respect to finite extensions of $\mathbb{F}_\ell$. In this framework we prove analogues of standard results from local differential topology, including the implicit function theorem and study the behavior of smooth measures under push forward with respect to submersions.