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From the work of Phong and Sturm in 2007, for a polarised projective manifold and an ample test configuration, one can associate the geodesic ray of plurisubharmonic metrics on the polarising line bundle using the solution of the Monge-Amp\`ere equation on an equivariant resolution of singularities of the test configuration. We prove that the Mabuchi chordal distance between the geodesic rays associated with two ample test configurations coincides with the spectral distance between the associated filtrations on the section ring. This gives an algebraic description of the boundary at the infinity of the space of positive metrics, viewed - as it is usually done for spaces of negative curvature - through geodesic rays.

We study the asymptotics of the $L^2$-optimal holomorphic extensions of holomorphic jets associated with high tensor powers of a positive line bundle along submanifolds. More precisely, for a fixed complex submanifold in a complex manifold, we consider the operator which for a given holomorphic jet along the submanifold of a positive line bundle associates the $L^2$-optimal holomorphic extension of it to the ambient manifold. When the tensor power of the line bundle tends to infinity, we give an explicit asymptotic formula for this extension operator. This is done by a careful study of the Schwartz kernels of the extension operator and related Bergman projectors. It extends our previous results, done for holomorphic sections instead of jets. As an application, we prove the asymptotic isometry between two natural norms on the space of holomorphic jets: one induced from the ambient manifold and another from the submanifold., Comment: 42 pages. This article extends our previous results, done for holomorphic sections instead of jets. The proof of the semiclassical extension theorem for holomorphic sections from arXiv:2109.06851 relied on spectral geometry and localisation techniques. The proof of the corresponding result in the current article, done more generally for holomorphic jets, builds upon Taylor-type expansion of holomorphic differential near submanifold from arXiv:2109.06851, on the study of Toeplitz operators with weak exponential decay from arXiv:2201.04102, and on multiplicative defect operator, introduced in the most simple form in arXiv:2201.04102