Formal orbifolds and orbifold bundles in positive characteristic

TL;DR

This paper introduces formal orbifolds and orbifold bundles in positive characteristic, defining their fundamental groups and vector bundles, and proves analogues of characteristic zero results in this setting.

Contribution

It extends the theory of orbifolds and vector bundles to positive characteristic, providing new definitions and proving key analogues of characteristic zero theorems.

Findings

Fundamental groups of formal orbifolds approximate those of affine curves.

Defined vector bundles and orbifold bundles in positive characteristic.

Proved analogues of characteristic zero results for vector bundles.

Abstract

We define formal orbifolds over an algebraically closed field of arbitrary characteristic as curves together with some branch data. Their \'etale coverings and their fundamental groups are also defined. These fundamental group approximates the fundamental group of an appropriate affine curve. We also define vector bundles on these objects and the category of orbifold bundles on any smooth projective curve. Analogues of various statement about vector bundles which are true in characteristic zero are proved. Some of these are positive characteristic avatar of notions which appear in the second author's work ([Par]) in characteristic zero.