Curvature and integrability of almost complex structures

TL;DR

This paper explores harmonic complex structures on almost complex manifolds, examining their curvature, integrability, and connections with balanced structures to address fundamental questions in complex geometry.

Contribution

It revisits previous results on harmonic complex structures and investigates their curvature and integrability properties, highlighting their role in understanding complex structures.

Findings

Harmonic complex structures relate to curvature and integrability.

Connections between harmonic structures and balanced structures are discussed.

The work emphasizes the importance of these structures in fundamental geometry problems.

Abstract

This note is concerned in so called harmonic complex structures introduced by the author previously. I will recall some previous results and emphasize the motivation: Provide an attempt to a fundamental problem in geometry--determining the complex structures on an almost complex manifold. I also discuss the almost-Hermitian case of harmonic complex structures and the connections with balanced structures.