Some Examples of Family Floer Mirrors

TL;DR

This paper computes explicit examples of family Floer mirrors for non-compact Calabi-Yau surfaces, comparing them with other mirror constructions and exploring their compactifications.

Contribution

It provides explicit calculations of family Floer mirrors and compares them with Gross-Hacking-Keel and cluster variety constructions, highlighting their relationships.

Findings

Explicit family Floer mirror calculations for non-compact Calabi-Yau surfaces

Comparison with Gross-Hacking-Keel mirror construction

Partial compactifications via analytifications of cluster varieties

Abstract

In this article, we give explicit calculations for the family Floer mirrors of some non-compact Calabi-Yau surfaces. We compare it with the mirror construction of Gross-Hacking-Keel for suitably chosen log Calabi-Yau pairs and the rank two cluster varieties of finite type. In particular, the analytifications of the later two give partial compactifications of the family Floer mirrors that we computed.