Comparison of two notions of subharmonicity on non-archimedean curves

TL;DR

This paper proves the equivalence of two notions of subharmonicity on non-archimedean curves, showing stability under pullback and establishing a regularization theorem for certain curves.

Contribution

It establishes the equivalence between Thuillier's and Chambert-Loir and Ducros's definitions of subharmonicity on non-archimedean curves, with implications for stability and regularization.

Findings

Equivalence of subharmonicity notions on non-archimedean curves.

Stability of psh property under morphisms.

Regularization theorem for projective line and Mumford curves.

Abstract

We show that a continuous function on the analytification of a smooth proper algebraic curve over a non-archimedean field is subharmonic in the sense of Thuillier if and only if it is psh, i.e. subharmonic in the sense of Chambert-Loir and Ducros. This equivalence implies that the property psh for continuous functions is stable under pullback with respect to morphisms of curves. Furthermore, we prove an analogue of the monotone regularization theorem on the analytification of the projective line and Mumford curves using this equivalence.