Given a Kähler manifold X with an ample line bundle L, we consider the metric space of L 1 geodesic rays associated to the first Chern class c 1 (L). We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems. By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge-Ampère energy of a general L 1 ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations.Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge-Ampère energy in terms of asymptotic expansion, and show the continuity of the projection map from L 1 rays to non-Archimedean rays. We also show that the closure of ample test configurations and filtrations gives the same set of rays.

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Let X be a compact Kähler manifold. Fix a big class α with smooth representative θ and a model potential φ with positive mass. We study the space E p (X, θ; [φ]) of finite energy Kähler potentials with prescribed singularity for each p ≥ 1. We define a metric dp and show that (E p (X, θ; [φ]), dp) is a complete metric space. This construction generalizes the usual dp-metric defined for an ample class.

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Let X be a smooth projective variety. We construct partial Okounkov bodies associated to Hermitian pseudo-effective line bundles (L, φ) on X. We show that partial Okounkov bodies are universal invariants of the singularity of φ. As an application, we generalize the theorem of Boucksom-Chen and construct Duistermaat-Heckman measures associated to finite energy metrics on the Berkovich analytification of an ample line bundle.