Complex orthogonal geometric structures of dimension three

TL;DR

This paper explores complex orthogonal geometric structures on three-dimensional complex manifolds, demonstrating the existence of a family of uniformizable structures including Guillot's example, and constructing related projective structures.

Contribution

It introduces new families of uniformizable complex orthogonal and projective structures on three-dimensional compact complex manifolds, expanding understanding of their geometric properties.

Findings

Existence of a family of uniformizable complex orthogonal structures including Guillot's example

Construction of a family of uniformizable complex projective structures

Extension of geometric structures to related compact manifolds

Abstract

A complex orthogonal (geometric) structure on a complex manifold is a geometric structure locally modelled on a non-degenerate quadric. One of the first examples of such a structure on a compact manifold of dimension three was constructed by Guillot. In this paper, we show that the same manifold carries a family of uniformizable complex orthogonal (geometric) structures which includes Guillot's structure; here, a structure is said to be uniformizable if it is a quotient of an invariant open set of a quadric by a Kleinian group. We also construct a family of uniformizable complex (geometric) projective structures on a related compact complex manifold of dimension three.