Frobenius-Ehresmann structures and Cartan geometries in positive characteristic

TL;DR

This paper develops a foundational theory of Frobenius-Ehresmann structures in positive characteristic, relating them to Cartan geometries and indigenous bundles, and explores their deformation and classification on algebraic curves.

Contribution

It introduces Frobenius-Ehresmann structures, generalizing previous Frobenius-projective and affine structures, and establishes their relation to Cartan geometries and indigenous bundles in positive characteristic.

Findings

Established conditions for equivalence of geometric structures

Computed deformation tangent and obstruction spaces

Proved the Ehresmann-Weil-Thurston principle in this setting

Abstract

The aim of the present paper is to lay the foundation for a theory of Ehresmann structures in positive characteristic, generalizing the Frobenius-projective and Frobenius-affine structures defined in the previous work. This theory deals with atlases of \'{e}tale coordinate charts on varieties modeled on homogenous spaces of algebraic groups, which we call Frobenius-Ehresmann structures. These structures are compared with Cartan geometries in positive characteristic, as well as with higher-dimensional generalizations of dormant indigenous bundles. In particular, we investigate the conditions under which these geometric structures are equivalent to each other. Also, we consider the classification problem of Frobenius-Ehresmann structures on algebraic curves. The latter half of the present paper discusses the deformation theory of indigenous bundles in the algebraic setting. The tangent and obstruction spaces of various deformation functors are computed in terms of the hypercohomology groups of certain complexes. As a consequence, we formulate and prove the Ehresmann-Weil-Thurston principle for Frobenius-Ehresmann structures. This fact asserts that deformations of a variety equipped with a Frobenius-Ehresmann structure are completely determined by their monodromy crystals.