Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

TL;DR

This paper proves that compact complex 3-manifolds with holomorphic Riemannian metrics are, up to a finite cover, modeled on spaces of constant sectional curvature, establishing a uniformization result.

Contribution

It establishes a uniformization theorem for compact complex 3-manifolds with holomorphic Riemannian metrics, showing they admit metrics of constant sectional curvature after a finite cover.

Findings

Such manifolds are, up to a finite cover, of constant sectional curvature

The result generalizes classical uniformization to complex 3-manifolds with holomorphic metrics

Provides a classification framework for these geometric structures

Abstract

We study compact complex 3-manifolds admitting holomorphic Riemannian metrics. We prove a uniformization result: up to a finite unramified cover, such a manifold admits a holomorphic Riemannian metric of constant sectionnal curvature.