Higher-dimensional classification of minimal Gauduchon-parallel vector fields

Characterize the existence of a nontrivial vector field parallel with respect to the minimal Gauduchon connection \(\nabla^{1/3}\) on compact locally conformally Kähler manifolds of complex dimension greater than two.

Background

The paper studies compact locally conformally Kähler manifolds admitting a nontrivial vector field parallel with respect to a Gauduchon connection $\nabla^t$. The main theorem gives a complete description for generic parameters and for the Bismut and Chern connections. For the exceptional minimal Gauduchon parameter $t=\frac{1}{3}$, the authors prove that in complex dimension two the Lee form is exact, so the metric is globally conformally Kähler, and they construct a non-Kähler globally conformally Kähler example.

The authors explicitly state that the problem remains open in higher dimensions. Although they subsequently construct examples in every complex dimension of compact globally conformally Kähler non-Kähler manifolds admitting a nontrivial $\nabla^{1/3}$-parallel vector field, these examples do not resolve the broader higher-dimensional classification problem.