Let X be a compact Kähler manifold. Given a big cohomology class $\left\{\theta \right\}$, there is a natural equivalence relation on the space of θ-psh functions giving rise to $𝒮(X,\theta )$, the space of singularity types of potentials. We introduce a natural pseudo-metric $d_{𝒮}$ on $𝒮(X,\theta )$ that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge–Ampère equations with varying singularity type converge as governed by the $d_{𝒮}$-topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.

中文翻译：

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奇异类型的度量几何

令X为紧凑的Kähler流形。给定同调类$\left\{\theta \right\}$，在θ-psh函数空间上存在自然等价关系 $𝒮（X，\theta ）$，即奇点类型的空间。我们介绍一个自然的伪度量$d_{𝒮}$ 上 $𝒮（X，\theta ）$在模型奇异类型的空间上是非退化的，并且其原子恰好是相对完整质量类别。在存在正质量的情况下，我们表明该度量空间是完整的。作为应用，我们证明了对于具有不同奇异类型的一组复杂Monge-Ampère方程的解收敛，由$d_{𝒮}$-拓扑，我们获得了与奇异类型相关的乘子理想滑轮的半连续结果，从而从局部上下文扩展了先前结果的范围。