Closed-String Mirror Symmetry for Log Calabi-Yau Surfaces

TL;DR

This paper proves closed-string mirror symmetry for log Calabi-Yau surfaces with generic parameters, showing how blowdowns affect the potential and ensuring semi-simplicity of quantum cohomology.

Contribution

It establishes mirror symmetry for all such surfaces and analyzes the effects of blowdowns on the potential and critical points.

Findings

Blowing down a (-1)-divisor removes one critical point.

The potential remains a Morse function after blowdown.

Quantum cohomology is semi-simple due to distinct critical values.

Abstract

This paper establishes closed-string mirror symmetry for all log Calabi-Yau surfaces with generic parameters, where the exceptional divisor are sufficiently small. We demonstrate that blowing down a $(-1)$-divisor removes a single geometric critical point, ensuring that the resulting potential remains a Morse function. Additionally, we show that the critical values are distinct, which implies that the quantum cohomology $QH^{\ast}(X)$ is semi-simple.