17.11 Test curvesBA6AA7598BE3

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A geodesic ray $(\varphi _ t)_{t\in [0,\infty )}$ in $\mathrm{PSH}(X,\theta )$ records the asymptotic behavior of singularities along a path of potentials. In this chapter we encode the same information in the dual variable $\tau $. Instead of recording the potential at time $t$, the Legendre transform records, for each slope $\tau $, the model potential determined by the part of the ray growing at least linearly with slope $\tau $. This produces a concave decreasing family $\tau \mapsto \Gamma _\tau $ of model potentials; more precisely, a test curve $\Gamma $ is a concave decreasing family $(\Gamma _\tau )_{\tau <\Gamma _{\max }}$ of model potentials in $\mathrm{PSH}(X,\theta )$.

The central result is a correspondence initiated by Ross and Witt Nyström¹. It is the pluripotential-theoretic analogue of Legendre duality: a test curve gives a geodesic ray by taking a Legendre transform, and a suitable geodesic ray gives back a test curve by the inverse transform. Thus test curves provide an intrinsic language for the singularity data carried by geodesic rays.

We first develop test curves as self-contained objects. In section 17.12 we introduce their basic properties, including independence of the auxiliary form, behavior under bimeromorphic morphisms, closedness in the parameter $\tau $, log-concavity of the mass function and the $P$-equivalence relation on test curves. The Ross–Witt Nyström correspondence is proved in section 17.13, where the Legendre transform is shown to give a bijection between test curves and geodesic rays.

The later sections adapt this picture to the singularity types needed elsewhere in the book. In section 17.14 we develop the $\mathcal I$-model version of the theory, including the $\mathcal I$-envelope of a test curve, the transition from $P$-model to $\mathcal I$-model test curves, and the corresponding notion of maximal geodesic ray. Finally, section 17.15 studies sums, translations, maxima, suprema, infima and rescalings of test curves. These operations are needed in the non-Archimedean theory of section 22.1, where test curves provide the analytic input for the construction of non-Archimedean potentials.

Throughout the chapter we freely use the convex-analytic results of section 21.11. Those results are stated for convex functions; when they are applied to concave functions here, they are applied to the negatives.