Non-archimedean periods for log Calabi-Yau surfaces

TL;DR

This paper proves that for log Calabi-Yau surfaces, the non-archimedean period map uniquely determines the analytic periods, linking the K-affine structure to the surface's isomorphism class.

Contribution

It confirms a conjecture by Kontsevich-Soibelman, establishing the non-archimedean period map as a complete invariant for log Calabi-Yau surfaces.

Findings

Non-archimedean period map recovers analytic periods.

K-affine structure determines the isomorphism type.

First proof of the conjecture for log Calabi-Yau surfaces.

Abstract

We prove the first instance of a conjecture by Kontsevich-Soibelman that the non-archimedean period map recovers the analytic periods in the case of log Calabi-Yau surfaces. In particular, we show that the K-affine structure, a natural enhancement of the singular integral affine structure on the essential skeleton, determines the isomorphism type of the log Calabi-Yau surface.