Learning Almost Complex and Complex Structures
Abstract
We study the numerical constructions of almost complex and complex structures on spheres and products of spheres up to real dimension six. Our general set-up parametrises almost complex structures via a global matrix field in an ambient Euclidean space, compressed by the tangent projector. The projector's algebraic residual measures the almost complex condition, while its Nijenhuis tensor is evaluated in the intrinsic norm of the round product metric. Polynomial features and neural networks are trained without prescribed targets to perform general searches for these structures. The almost complex search passes numerical admissibility checks on all nine examples that admit such structures and rejects S4, where a topological obstruction is known. A matrix correction then enforces the almost complex identity, subject to a novel uniform admissibility condition. For integrability, a free polynomial representation with Levenberg-Marquardt optimisation learns the complex structures on S2× S2, S3× S1, S3× S3 and S5× S1, with the largest Nijenhuis norms found by adversarial search all below 10−9. These general searches were unsuccessful on S6, S4× S2 and S4× T2. Therefore, for these three cases direct numerical reconstructions were built in torus-fibred coordinates. They are integrable by construction, satisfy their compatibility conditions to better than 10−11, and corroborate the recent constructions on S6 and S4× S2.
Related
- On the integrability of generalized almost complex structures on 𝕊6
- Invariant almost Hermitian structures on the maximal real flag manifold of type 𝒜3
- Harmonicity of normal almost contact metric structures
- Four-dimensional almost-Einstein manifolds with circulant structures
- Weakly Einstein and weakly η-Einstein structures on almost contact metric three-manifolds
BibTeX
@misc{hirst2026learning,
title = {{Learning Almost Complex and Complex Structures}},
author = {Edward Hirst and Tancredi Schettini Gherardini and Tomás S. R. Silva and Jakob Stein},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03872},
eprint = {2611.03872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03872},
}
RIS
TY - UNPB
TI - Learning Almost Complex and Complex Structures
AU - Edward Hirst
AU - Tancredi Schettini Gherardini
AU - Tomás S. R. Silva
AU - Jakob Stein
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03872
UR - https://arxiv.org/abs/2611.03872
AB - We study the numerical constructions of almost complex and complex structures on spheres and products of spheres up to real dimension six. Our general set-up parametrises almost complex structures via a global matrix field in an ambient Euclidean space, compressed by the tangent projector. The projector's algebraic residual measures the almost complex condition, while its Nijenhuis tensor is evaluated in the intrinsic norm of the round product metric. Polynomial features and neural networks are trained without prescribed targets to perform general searches for these structures. The almost complex search passes numerical admissibility checks on all nine examples that admit such structures and rejects S^4, where a topological obstruction is known. A matrix correction then enforces the almost complex identity, subject to a novel uniform admissibility condition. For integrability, a free polynomial representation with Levenberg-Marquardt optimisation learns the complex structures on S^2× S^2, S^3× S^1, S^3× S^3 and S^5× S^1, with the largest Nijenhuis norms found by adversarial search all below 10^−9. These general searches were unsuccessful on S^6, S^4× S^2 and S^4× T^2. Therefore, for these three cases direct numerical reconstructions were built in torus-fibred coordinates. They are integrable by construction, satisfy their compatibility conditions to better than 10^−11, and corroborate the recent constructions on S^6 and S^4× S^2.
ER -