The Weil-Petersson form on an acyclic cluster variety

TL;DR

This paper proves that for acyclic cluster varieties, the Weil-Petersson 2-form, originally defined on a subset, extends smoothly to the entire variety, enhancing understanding of its geometric structure.

Contribution

It demonstrates that the Weil-Petersson form extends to a regular Kähler form on all acyclic cluster varieties, a significant extension of previous local definitions.

Findings

Weil-Petersson form extends to entire acyclic cluster varieties.

The extended form is a regular Kähler 2-form.

This extension provides new insights into the geometry of cluster varieties.

Abstract

The Weil-Petersson form on a cluster variety is a 2-form on a certain open smooth subvariety; the union of the cluster tori. We show that for acyclic cluster varieties, the Weil-Petersson 2-form extends to a regular K\"ahler 2-form on the entire cluster variety.