On compact affine quaternionic curves and surfaces

TL;DR

This paper investigates the classification of compact affine quaternionic curves and surfaces, establishing uniqueness of affine quaternionic structures and identifying specific types of compact curves and surfaces.

Contribution

It provides a classification of all compact affine quaternionic curves and surfaces, showing that the only such curves are quaternionic tori and Hopf surfaces, and explores the structure of compact surfaces.

Findings

Only quaternionic tori and Hopf surfaces are compact affine quaternionic curves.

Complete compact affine quaternionic surfaces are linked to certain nilpotent hypercomplex Lie groups.

Uniqueness of affine quaternionic structures on affine quaternionic manifolds.

Abstract

This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem that studies compact complex manifolds in complex dimension 2, states that the only compact affine quaternionic curves are the quaternionic tori and the primary Hopf surface S^3 x S^1. As for compact affine quaternionic surfaces, we restrict to the complete ones: the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups, identifies a path towards their classification.