Pure Type-One Generalized Hypercomplex Structures in Dimension Four
Abstract
We classify untwisted generalized hypercomplex structures on four-manifolds whose entire sphere has type one. The three Poisson tensors determine a canonical line field, and the local data are Gauduchon--Tod coframes. We determine local equivalence under diffeomorphisms, common closed real B-fields and constant rotations of the triple. On a closed manifold the canonical foliation is automatically Riemannian: a nonzero period obstructs the exact twist required by the non-Riemannian models. The compact structures form four families, distinguished by b1=1,2,3,4: Berger quotients, flat axis quotients, nontrivial circle bundles over flat three-tori, and constant tori. We determine their parameters, equivalences and finite ordered quotients, including nonorientable canonical lines. The underlying manifolds are parallelizable, satisfy b2=2b1−2, and are symplectic exactly when the intrinsic Gauduchon--Tod invariant vanishes. A twisted Hopf example shows why the untwisted hypothesis is essential. The classification is intrinsic and assumes no initial circle action. These results address the four-dimensional untwisted case of the intermediate-type question of Fino and Grantcharov.
Related
- On the Hodge conjecture for hyper-Kähler varieties of generalized Kummer type of dimension divisible by four
- Generalized pseudo-product structures and finite type distributions via abnormal extremals
- Pure dimension and counterexamples to the maximality of purely-primes
- A sound pure type system with ranks as sorts and Type-in-Type
- Formalizing the Intrinsic Dimension of Objective Landscapes by a Generalized Notion of Co-dimension
BibTeX
@misc{ren2026pure,
title = {{Pure Type-One Generalized Hypercomplex Structures in Dimension Four}},
author = {Guangzhen Ren},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35557},
eprint = {2610.35557},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35557},
}
RIS
TY - UNPB
TI - Pure Type-One Generalized Hypercomplex Structures in Dimension Four
AU - Guangzhen Ren
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35557
UR - https://arxiv.org/abs/2610.35557
AB - We classify untwisted generalized hypercomplex structures on four-manifolds whose entire sphere has type one. The three Poisson tensors determine a canonical line field, and the local data are Gauduchon--Tod coframes. We determine local equivalence under diffeomorphisms, common closed real B-fields and constant rotations of the triple. On a closed manifold the canonical foliation is automatically Riemannian: a nonzero period obstructs the exact twist required by the non-Riemannian models. The compact structures form four families, distinguished by b_1=1,2,3,4: Berger quotients, flat axis quotients, nontrivial circle bundles over flat three-tori, and constant tori. We determine their parameters, equivalences and finite ordered quotients, including nonorientable canonical lines. The underlying manifolds are parallelizable, satisfy b_2=2b_1−2, and are symplectic exactly when the intrinsic Gauduchon--Tod invariant vanishes. A twisted Hopf example shows why the untwisted hypothesis is essential. The classification is intrinsic and assumes no initial circle action. These results address the four-dimensional untwisted case of the intermediate-type question of Fino and Grantcharov.
ER -