Cohomogeneity one solutions of the IIB system
Abstract
The IIB system is a system of PDEs for an SU(3)--structure and a positive function on a 6-manifold. Its solutions describe conformally balanced pluriclosed Hemitian metrics on complex 3-folds with holomorphically trivial canonical line bundle. In particular, solutions of the IIB system are steady solitons for the generalized Ricci-flow and Bismut--Hermitian--Einstein metrics. In this paper we study cohomogeneity one solutions of the IIB system. We construct 1-parameter families (up to scaling symmetries) of complete non-compact non-Kähler solutions on the smoothing of the conifold with controlled geometry at infinity. The generic member of the family is asymptotically conical with tangent cone at infinity the Calabi--Yau cone metric on the conifold. As a limit of the family we recover an explicit solutions known in the physics literature as the Chamseddine--Volkov/Maldacena--Nuñez solution, which has an exotic asymptotic geometry. We show that there are no complete cohomogeneity one solutions on crepant resolutions of the conifold (or its quotient), but we also establish the existence of an analogous 1-parameter family of forward complete solutions defined on exterior domains of the conifold with prescribed incomplete behaviour along the interior boundary and similar asymptotic behaviour at infinity. As a byproduct of these existence results, we produce infinitely many complete non-compact non-Kähler Bismut--Hermitian--Einstein metrics in complex dimension 3 with full holonomy of the Bismut connection, in contrast to the holonomy reduction forced upon compact examples. We also find infinitely many such complete non-compact full-holonomy non-Kähler examples (with at least two ends) in complex dimension 2 by revisiting a construction due to Callan--Harvey--Strominger in the physics literature.
Related
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- Cohomogeneity one Hermann actions and backward mean curvature flows
- Cohomogeneity one Einstein metrics on complex projective spaces
- Gauge theory on Cohomogeneity One Spin(7)-Manifolds from Aloff--Wallach Spaces
- Cohomogeneity-One Ruled Hypersurfaces in ℂℙ2 and ℂH2
BibTeX
@misc{foscolo2026cohomogeneity,
title = {{Cohomogeneity one solutions of the IIB system}},
author = {Lorenzo Foscolo and Mario Garcia-Fernandez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28280},
eprint = {2609.28280},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28280},
}
RIS
TY - UNPB
TI - Cohomogeneity one solutions of the IIB system
AU - Lorenzo Foscolo
AU - Mario Garcia-Fernandez
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28280
UR - https://arxiv.org/abs/2609.28280
AB - The IIB system is a system of PDEs for an SU(3)--structure and a positive function on a 6-manifold. Its solutions describe conformally balanced pluriclosed Hemitian metrics on complex 3-folds with holomorphically trivial canonical line bundle. In particular, solutions of the IIB system are steady solitons for the generalized Ricci-flow and Bismut--Hermitian--Einstein metrics. In this paper we study cohomogeneity one solutions of the IIB system. We construct 1-parameter families (up to scaling symmetries) of complete non-compact non-Kähler solutions on the smoothing of the conifold with controlled geometry at infinity. The generic member of the family is asymptotically conical with tangent cone at infinity the Calabi--Yau cone metric on the conifold. As a limit of the family we recover an explicit solutions known in the physics literature as the Chamseddine--Volkov/Maldacena--Nuñez solution, which has an exotic asymptotic geometry. We show that there are no complete cohomogeneity one solutions on crepant resolutions of the conifold (or its quotient), but we also establish the existence of an analogous 1-parameter family of forward complete solutions defined on exterior domains of the conifold with prescribed incomplete behaviour along the interior boundary and similar asymptotic behaviour at infinity. As a byproduct of these existence results, we produce infinitely many complete non-compact non-Kähler Bismut--Hermitian--Einstein metrics in complex dimension 3 with full holonomy of the Bismut connection, in contrast to the holonomy reduction forced upon compact examples. We also find infinitely many such complete non-compact full-holonomy non-Kähler examples (with at least two ends) in complex dimension 2 by revisiting a construction due to Callan--Harvey--Strominger in the physics literature.
ER -