Non-Archimedean Calabi-Yau Potentials on Certain Affine Varieties

TL;DR

This paper develops a non-Archimedean analog of Ricci-flat metric potentials on certain affine varieties by solving a Monge-Ampère equation on Berkovich spaces, bridging complex and non-Archimedean geometry.

Contribution

It provides the first solution to a non-Archimedean Monge-Ampère equation on specific affine varieties, extending complex geometric results to the non-Archimedean setting.

Findings

Solved a non-Archimedean Monge-Ampère equation on Berkovich analytification.

Established limits of complex potentials coincide with non-Archimedean counterparts.

Connected complex Ricci-flat metrics with their non-Archimedean analogs.

Abstract

We solve a non-Archimedean Monge-Amp\`ere equation on the Berkovich analytification of a complex log Calabi-Yau pair whose dual complex is a standard simplex, answering a question of Collins-Li and offering a non-Archimedean analog of Ricci-flat metric potentials on complex affine varieties. This work builds on the solution to a complex Monge-Amp\`ere equation obtained by Collins-Li and Collins-Tong-Yau. We also show the suitably rescaled limits of the complex potentials coincide with their non-Archimedean counterparts in some situations, strengthening their connections.