The space of K\"ahler metrics can, on the one hand, be approximated by subspaces of algebraic metrics, while, on the other hand, can be enlarged to finite-energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of K\"ahler metrics. The former spaces are the finite-dimensional spaces of Fubini--Study metrics of K\"ahler quantization. The goal of this article is to draw a connection between the two. We show that the Finsler structures on the space of K\"ahler potentials can be quantized. More precisely, given a K\"ahler manifold polarized by an ample line bundle we endow the space of Hermitian metrics on powers of that line bundle with Finsler structures and show that the resulting path length metric spaces recover the corresponding metric completions of the Finsler structures on the space of K\"ahler potentials. This has a number of applications, among them a new approach to the rooftop envelopes and Pythagorean formulas of K\"ahler geometry, a new Lidskii type inequality on the space of K\"ahler metrics, and approximation of finite energy potentials, as well as geodesic segments by the corresponding smooth algebraic objects.

中文翻译：

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几何多势论中的量化

K\"ahler 度量的空间一方面可以由代数度量的子空间近似，另一方面可以扩大到多势理论中出现的有限能量空间。后者空间被实现为度量在 K\"ahler 度量空间上完成 Finsler 结构。前面的空间是Fubini的有限维空间--K\"ahler量化的研究度量。本文的目的是在两者之间建立联系。我们证明K\"ahler空间上的Finsler结构电位是可以量化的。更准确地说，给定一个 K\"