On the Hermitian Geometry of $k$-Gauduchon 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.