Complete pluriclosed metrics with vanishing Bismut Ricci form on the affine quadric
Abstract
We study a one-parameter family of SO(4)-invariant pluriclosed Hermitian metrics gc with vanishing Bismut Ricci form on the affine quadric Q3≅ TS3, previously described in the physics literature. For every real parameter c, we prove local existence and uniqueness of a real-analytic solution to the defining singular initial-value problem, together with positivity of the associated metric near the singular orbit S3. We then establish global existence and completeness for |c|≤1 and finite-time degeneration for |c|>1. The complete family exhausts all SO(4)-invariant Bismut Hermitian-Einstein metrics on the affine quadric Q3, and joins Stenzel's Kähler Ricci-flat to a metric that was first considered by Chamseddine-Volkov/Maldacena-Nuñez. We also determine the leading asymptotics and derive an explicit scalar curvature formula, proving strict positivity for the non-Kähler members and identifying the change in asymptotic scalar curvature at the endpoint c=1. Finally, we show that all these metrics are not Bismut flat and have full Bismut holonomy SU(3), providing new examples of such manifolds.
Related
- Pluriclosed deformations of Bismut flat metrics
- Pluriclosed manifolds with parallel Bismut torsion
- A gluing construction of Z-extremal Kähler metrics and new examples of Bismut Hermitian-Einstein $3$-folds
- Hodge level and effective non-vanishing for smooth weighted complete intersections
- Nonexistence of complete metrics with uniformly positive scalar curvature
BibTeX
@misc{podesta2026complete,
title = {{Complete pluriclosed metrics with vanishing Bismut Ricci form on the affine quadric}},
author = {Fabio Podestà and Alberto Raffero},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27979},
eprint = {2609.27979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27979},
}
RIS
TY - UNPB
TI - Complete pluriclosed metrics with vanishing Bismut Ricci form on the affine quadric
AU - Fabio Podestà
AU - Alberto Raffero
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27979
UR - https://arxiv.org/abs/2609.27979
AB - We study a one-parameter family of SO(4)-invariant pluriclosed Hermitian metrics g_c with vanishing Bismut Ricci form on the affine quadric Q_3≅ TS^3, previously described in the physics literature. For every real parameter c, we prove local existence and uniqueness of a real-analytic solution to the defining singular initial-value problem, together with positivity of the associated metric near the singular orbit S^3. We then establish global existence and completeness for |c|≤1 and finite-time degeneration for |c|>1. The complete family exhausts all SO(4)-invariant Bismut Hermitian-Einstein metrics on the affine quadric Q_3, and joins Stenzel's Kähler Ricci-flat to a metric that was first considered by Chamseddine-Volkov/Maldacena-Nuñez. We also determine the leading asymptotics and derive an explicit scalar curvature formula, proving strict positivity for the non-Kähler members and identifying the change in asymptotic scalar curvature at the endpoint c=1. Finally, we show that all these metrics are not Bismut flat and have full Bismut holonomy SU(3), providing new examples of such manifolds.
ER -