Rigid rational homotopy types

TL;DR

This paper introduces a new rigid rational homotopy type for varieties over perfect fields of positive characteristic, establishing comparison theorems and exploring Frobenius structures and obstructions to sections.

Contribution

It defines a novel rigid rational homotopy type, proves comparison theorems with existing notions, and applies these to Frobenius structures and homotopy obstructions.

Findings

Frobenius structure on higher rational homotopy groups is mixed over finite fields.

Comparison theorems connect the new type with previous definitions in special cases.

A homotopy obstruction for the existence of sections is established.

Abstract

In this paper we define a rigid rational homotopy type, associated to any variety $X$ over a perfect field $k$ of positive characteristic. We prove comparison theorems with previous definitions in the smooth and proper, and log-smooth and proper case. Using these, we can show that if $k$ is a finite field, then the Frobenius structure on the higher rational homotopy groups is mixed. We also define a relative rigid rational homotopy type, and use it to define a homotopy obstruction for the existence of sections.