The Weil-Petersson form on an acyclic cluster variety (Q2909666)
From MaRDI portal
 | This is the item page for this Wikibase entity, intended for internal use and editing purposes. Please use this page instead for the normal view: The Weil-Petersson form on an acyclic cluster variety |
scientific article; zbMATH DE number 6078258
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The Weil-Petersson form on an acyclic cluster variety |
scientific article; zbMATH DE number 6078258 |
Statements
6 September 2012
0 references
cluster variety
0 references
Weil-Petersson form
0 references
commutative algebra
0 references
Kähler 2-form
0 references
smooth variety
0 references
0.9217292
0 references
0.86686635
0 references
0.86582655
0 references
0.8612577
0 references
0 references
0.8592381
0 references
0.85835826
0 references
0 references
The Weil-Petersson form on an acyclic cluster variety (English)
0 references
This paper deals with the Weil-Petersson form on cluster varieties. Let us recall that cluster algebras are commutative algebras, first introduced by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)]. A natural question about cluster varieties is to know whether the Weil-Petersson form can be extended to the entire cluster variety. In this work, the author proves that the answer is affirmative for a large class of cluster varieties, called the acyclic cluster varieties. More precisely, for these varieties, the form extends to a regular Kähler 2-form. In the case of a smooth variety, it is showed that the form extends to a differential 2-form, defined everywhere. Several acyclic examples as well as interesting counterexamples are given.
0 references
