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Compact lcK manifolds with Gauduchon parallel vector fields
Abstract
We prove that on a compact locally conformally Kähler manifold, the existence of a non-trivial parallel vector field with respect to a Gauduchon connection ∇t, for t≠ 1/3, forces the metric to be either Vaisman (for t=−1), globally conformally Kähler (for t=1), or Kähler for all other values of t. As a consequence, we characterize compact Vaisman manifolds in terms of their Bismut holonomy group. Moreover, we show that in complex dimension two, the existence of a non-trivial ∇/13-parallel vector field forces the metric to be globally conformally Kähler.
Related
- A note on the Chern-Yamabe flow on compact Hermitian manifolds with negative Gauduchon degree
- Deformed Gauduchon-Yamabe problems on compact Hermitian manifolds
- Poincaré compactifications of pullbacks of parallel vector fields under counterexamples to the real Jacobian conjecture
- Generalized Gauduchon metrics on compact complex spaces
- Curvature characterization of Hermitian manifolds with Bismut parallel torsion
BibTeX
@misc{moroianu2026compact,
title = {{Compact lcK manifolds with Gauduchon parallel vector fields}},
author = {Andrei Moroianu and Giacomo Perri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13657},
eprint = {2610.13657},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13657},
}
RIS
TY - UNPB
TI - Compact lcK manifolds with Gauduchon parallel vector fields
AU - Andrei Moroianu
AU - Giacomo Perri
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13657
UR - https://arxiv.org/abs/2610.13657
AB - We prove that on a compact locally conformally Kähler manifold, the existence of a non-trivial parallel vector field with respect to a Gauduchon connection ∇^t, for t≠ 1/3, forces the metric to be either Vaisman (for t=−1), globally conformally Kähler (for t=1), or Kähler for all other values of t. As a consequence, we characterize compact Vaisman manifolds in terms of their Bismut holonomy group. Moreover, we show that in complex dimension two, the existence of a non-trivial ∇^/13-parallel vector field forces the metric to be globally conformally Kähler.
ER -