Enumeration of holomorphic cylinders in log Calabi-Yau surfaces. I

TL;DR

This paper introduces a method to count holomorphic cylinders in log Calabi-Yau surfaces using non-archimedean geometry, leading to new invariants and insights into wall-crossing phenomena.

Contribution

It develops a novel approach to enumerate holomorphic cylinders in log Calabi-Yau surfaces, connecting complex geometry with non-archimedean methods and verifying conjectured wall-crossing formulas.

Findings

Defined new geometric invariants from cylinder counting

Verified wall-crossing formula for del Pezzo surfaces

Linked holomorphic cylinders to broken line combinatorics

Abstract

We define the counting of holomorphic cylinders in log Calabi-Yau surfaces. Although we start with a complex log Calabi-Yau surface, the counting is achieved by applying methods from non-archimedean geometry. This gives rise to new geometric invariants. Moreover, we prove that the counting satisfies a property of symmetry. Explicit calculations are given for a del Pezzo surface in detail, which verify the conjectured wall-crossing formula for the focus-focus singularity. Our holomorphic cylinders are expected to give a geometric understanding of the combinatorial notion of broken line by Gross, Hacking, Keel and Siebert. Our tools include Berkovich spaces, tropical geometry, Gromov-Witten theory and the GAGA theorem for non-archimedean analytic stacks.