On the Hermitian Geometry of $k$-Gauduchon Orthogonal Complex Structures

TL;DR

This paper investigates the geometry of complex structures orthogonal to a Riemannian metric, focusing on torsion and showing pre-compactness of certain $k$-Gauduchon structures within the moduli space.

Contribution

It introduces new results on the pre-compactness of $k$-Gauduchon orthogonal complex structures for specific $k$ ranges, emphasizing torsion over curvature.

Findings

Spaces of $k$-Gauduchon orthogonal structures are pre-compact for certain $k$.

Provides a preliminary understanding of how special complex structures are situated within the orthogonal complex structures.

Highlights the role of torsion in the geometry of orthogonal complex structures.

Abstract

The purpose of this note is to study the complex structures orthogonal to a given Riemannian metric. For another paper on this topic, we highly recommend the work of Salamon. His work describes in great detail the role that curvature plays in this question. We instead focus on torsion, which lends itself to somewhat different analysis of the problem. In terms of novel results, we show that for a certain range of $k$, the spaces of $k$-Gauduchon orthogonal complex structures are pre-compact within the moduli space of complex structures and give a preliminary picture of how certain special complex structures fit into the space of orthogonal complex structures.