projects:clusterhms:start
Table of Contents
HMS for cluster $A_n$
Overview
There are three approaches,
- Daping's canonical skeleton using $\omega = Im(\Omega)$.
- Constructing Kahler potential by extending (after modification) standard KP on $(\C^*)^n$.
- Semi-Tropicalize the cluster variety, put the main torus on some flat region. Let trickle down from
They each need to solve some technical problems.
Skeleton for framed $A_1$ quiver
The equation for $X_1$ is $x x' = 1 + q$, $q \in \C^*, x,x' \in \C$. We have several approaches to build a Liouville structure on this space. They are related here but have different abilities to generalize to higher dimension.
- Build a Kahler potential $\varphi$.
- Use symplectic SYZ fibration, singular affine structure in the base.
- Non-toric blow-up for $\C \times \C^*$
- Use the holomorphic symplectic structure.
projects/clusterhms/start.txt · Last modified: 2026/03/06 17:44 by pzhou