We prove a dichotomy theorem about compact Kähler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields, which has the following consequence: either such a manifold satisfies an additional integrability condition, or through every zero of the real-holomorphic geodesic gradient there passes an uncountable family of totally geodesic, holomorphically immersed complex projective spaces, each carrying a fixed multiple of the Fubini–Study metric. We also obtain a classification result for the case where the integrability condition holds, implying that the manifold must then be biholomorphically isometric to a bundle of complex projective spaces with a bundle-like metric.

中文翻译：

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具有测地线全同梯度的Kähler流形

我们证明了一个紧致的Kähler流形的二分法定理，它允许非平凡的实全地测地梯度矢量场，其结果如下：要么该流形满足一个附加的可积性条件，要么通过实全地的全地测地梯度的每个零，通过一个不可数完全测地的，全纯的，沉浸式的复杂投影空间族，每个空间都承载着Fubini-Study度量的固定倍数。对于可积性条件成立的情况，我们还获得了分类结果，这意味着流形必须然后对于具有束状度量的一束复杂的射影空间是双全等距的。