The Weil-Petersson form on an acyclic cluster variety
  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. 
Greg Muller
math.RA math.AG