Ramification theory and formal orbifolds in arbitrary dimension

TL;DR

This paper extends ramification theory and formal orbifolds to higher dimensions, defining their fundamental groups and étale sites, and demonstrates their use in studying wild ramification and related algebraic structures.

Contribution

It introduces formal orbifolds in arbitrary dimensions, defines their étale fundamental groups and sites, and applies these concepts to analyze wild ramification and related sheaf theories.

Findings

Fundamental groups of formal orbifolds have finiteness properties.

Formal orbifolds can approximate étale fundamental groups of normal varieties.

Framework enables organized study of wild ramification.

Abstract

Formal orbifolds are defined in higher dimension. Their \'etale fundamental groups are also defined. It is shown that the fundamental groups of formal orbifolds have certain finiteness property and it is also shown that they can be used to approximate the \'etale fundamental groups of normal varieties. Etale site on formal orbifolds are also defined. This framework allows one to study wild ramification in an organised way. Brylinski-Kato filtration, Lefschetz theorem for fundamental groups and $l$-adic sheaves in these contexts are also studied.