[Paper Review] Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals (with an Appendix by Jos\'e Ignacio Burgos Gil and Mart\'in Sombra)

Walter Gubler, Philipp Jell+2 more|arXiv (Cornell University)|Dec 4, 2017

Geometry and complex manifolds3 references3 citations

TL;DR

This paper establishes the continuity of plurisubharmonic envelopes in non-archimedean geometry for curves and surfaces over positive characteristic function fields, replacing multiplier ideals with test ideals in the proof strategy. It proves the existence of continuous semipositive metrics solving the non-archimedean Monge-Ampère equation under resolution of singularities, with an appendix providing a counterexample showing that retraction maps in toric models can fail to preserve semipositivity.

ABSTRACT

Let L be an ample line bundle on a smooth projective variety X over a non-archimedean field K. For a continuous metric on L, we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L and that the non-archimedean Monge-Amp\`ere equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that X is a surface defined geometrically over the function field of a curve over a perfect field k of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over k. The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps.

Motivation & Objective

• To establish the continuity of plurisubharmonic envelopes in non-archimedean geometry for curves and surfaces over positive characteristic fields.
• To extend the non-archimedean Calabi–Yau problem to positive characteristic by replacing multiplier ideals with test ideals in the proof framework.
• To investigate the behavior of retraction maps and semipositive metrics in toric models, particularly in relation to the Monge–Ampère equation.
• To provide a counterexample showing that the retraction of a semipositive metric may not remain semipositive in higher-dimensional toric models.
• To generalize the existence result for continuous semipositive solutions to the Monge–Ampère equation beyond the characteristic zero setting.

Proposed method

• Adapt the strategy of Boucksom, Favre, and Jonsson by replacing multiplier ideals with test ideals to handle positive characteristic settings.
• Use the piecewise linear structure on skeletons of SNC-models to define and analyze pluripotential-theoretic objects such as envelopes and Monge–Ampère measures.
• Prove continuity of the semipositive envelope P(∥∥) via asymptotic test ideals and uniform convergence to the zero function in model metrics.
• Employ resolution of singularities and descent techniques for model functions to extend results from curves to surfaces.
• Construct explicit counterexamples in toric geometry using two SNC-models of P²_K with isomorphic skeletons but non-equivalent retraction maps.
• Utilize the identification of toric psh functions with concave functions on the Newton polyhedron to analyze the failure of semipositivity under retraction.

Experimental results

Research questions

• RQ1 Does the semipositive envelope of a continuous metric remain continuous in non-archimedean geometry over positive characteristic fields?
• RQ2 Can the non-archimedean Monge–Ampère equation be solved by a continuous semipositive metric in positive characteristic, assuming resolution of singularities?
• RQ3 Is the retraction map associated to a toric model compatible with the semipositivity of metrics, or can it map semipositive metrics to non-semipositive ones?
• RQ4 Do solutions to the Monge–Ampère equation factor through retraction maps of refined models, even when they do not factor through coarser models?
• RQ5 To what extent can the theory of multiplier ideals be replaced by test ideals in non-archimedean pluripotential theory in positive characteristic?

Key findings

• The semipositive envelope P(∥∥) is continuous for curves over non-archimedean fields, extending Thuillier’s result to the continuous setting.
• For surfaces defined over the function field of a curve over a perfect field of positive characteristic, the semipositive envelope is continuous under the assumption of resolution of singularities.
• The non-archimedean Monge–Ampère equation has a continuous semipositive solution in the curve case and under resolution of singularities in higher dimensions.
• The appendix constructs a toric variety with two SNC-models inducing the same skeleton but different retraction maps, showing that the retraction of a semipositive metric may fail to be semipositive.
• A specific example shows that a solution ϕ′ to the Monge–Ampère equation does not satisfy ϕ′ = ϕ′ ◦ pX, demonstrating that the factorization property fails in dimension ≥2.
• Solutions to the Monge–Ampère equation factor through the retraction of a refined model, suggesting a potential generalization of the factorization property to coarsely refined models.