Generalization of Gurjar's Hyperplane section theorem to arbitrary analytic varieties and A$\mathbb{m}$AC classes

TL;DR

This paper extends Gurjar's hyperplane section theorem to all local analytic varieties, including non-isolated intersections, and introduces A$ ext{m}$AC classes for formal varieties, broadening the theorem's applicability.

Contribution

It generalizes Gurjar's theorem to arbitrary analytic varieties and formal varieties with A$ ext{m}$AC classes, allowing for non-isolated intersections and broader function classes.

Findings

Generalization of Gurjar's theorem to arbitrary analytic varieties.

Extension to formal varieties with A$ ext{m}$AC classes.

Applicable to non-isolated hyperplane intersections.

Abstract

The aim of this paper is to generalize the hyperplane section theorem of Gurjar to arbitrary (local) analytic varieties even if the intersection with of hyperplanes is not necessarily isolated. In case of formal varieties, we generalize the statement to work for different classes of functions than just hyperplanes. We call these classes (which are subsets of formal power series ring) to be algebraic $\mathbb{m}$-adicaly closed (A$\mathbb{m}$AC).