We give a complete description of all locally conformally K\"ahler structures with holomorphic Lee vector field on a compact complex manifold of Vaisman type. This provides in particular examples of such structures whose Lee vector field is not homothetic to the Lee vector field of a Vaisman structure. More generally, dropping the condition of being of Vaisman type, we show that on a compact complex manifold, any lcK metric with potential and with holomorphic Lee vector field admits a potential which is positive and invariant along the anti-Lee vector field.

中文翻译：

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Vaisman 型流形上具有全纯 Lee 向量场的 LcK 结构

我们在 Vaisman 类型的紧复流形上给出了所有具有全纯 Lee 向量场的局部共形 K\"ahler 结构的完整描述。这提供了这样的结构的特定例子，其 Lee 向量场与 a 的 Lee 向量场不相似Vaisman 结构. 更一般地，放弃 Vaisman 类型的条件，我们证明在紧复流形上，任何具有势和全纯 Lee 向量场的 lcK 度量都承认一个势，该势沿反李向量场为正且不变.