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arXiv · 2610.13657

Compact lcK manifolds with Gauduchon parallel vector fields

Andrei Moroianu·Giacomo Perri
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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 $\nabla^t$, for $t\ne \frac{1}{3}$, 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 $\nabla^\frac{1}{3}$-parallel vector field forces the metric to be globally conformally Kähler.

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Andrei Moroianu, Giacomo Perri. 2026-10-09. Compact lcK manifolds with Gauduchon parallel vector fields. https://arxiv.org/abs/2610.13657

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