Global pluripotential theory on hybrid spaces

TL;DR

This paper develops a pluripotential theory framework on hybrid spaces combining complex and non-Archimedean geometry, establishing extension and continuity results for plurisubharmonic metrics in degenerating families of varieties.

Contribution

It introduces a class of plurisubharmonic metrics on hybrid spaces and proves their extension and continuity properties in degenerations of complex varieties.

Findings

Existence of canonical plurisubharmonic extensions to hybrid spaces.

Continuity of Monge-Ampère measures in hybrid settings.

Explicit description of extensions in canonical degenerations.

Abstract

Let A be an integral Banach ring, and X/A be a projective scheme of finite type, endowed with a semi-ample line bundle L. We define a class PSH(X,L) of plurisubharmonic metrics on L on the Berkovich analytification X^an and prove various basic properties thereof. We focus in particular on the case where A is a hybrid ring of complex power series and X/A is a smooth variety, so that X^an is the hybrid space associated to a degeneration X of complex varieties over the punctured disk. We then prove that when L is ample, any plurisubharmonic metric on L with logarithmic growth at zero admits a canonical plurisubharmonic extension to the hybrid space X^hyb . We also discuss the continuity of the family of Monge-Amp\`ere measures associated to a continuous plurisubharmonic hybrid metric. In the case where X is a degeneration of canonically polarized manifolds, we prove that the canonical psh extension is continuous on Xhyb and describe it explicitly in terms of the canonical model (in the sense of MMP) of the degeneration.