We show the optimal $C^{1,1}$ regularity of geodesics in nef and big cohomology class on Kähler manifolds away from the non-Kähler locus, assuming sufficiently regular initial data. As a special case, we prove the $C^{1,1}$ regularity of geodesics of Kähler metrics on compact Kähler varieties away from the singular locus. Our main novelty is an improved boundary estimate for the complex Monge–Ampère equation that does not require strict positivity of the reference form near the boundary. We also discuss the case of some special geodesic rays.

中文翻译：

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C1,1 奇异 Kähler 度量测地线的规律

我们展示了最优 $C^{1,1}$假设初始数据足够规则，在远离非 Kähler 轨迹的 Kähler 流形上，nef 和大上同调类中测地线的规律性。作为特例，我们证明$C^{1,1}$远离奇异轨迹的紧凑 Kähler 变体上 Kähler 度量的测地线的规律性。我们的主要新颖之处在于改进了复杂 Monge-Ampère 方程的边界估计，它不需要边界附近参考形式的严格正性。我们还讨论了一些特殊测地线的情况。