On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. More explicitly, he shows that one can decompose this vector space in terms of weight decompositions and obtain a sort of canonical basis for these differential forms. Is there some literature which does the same for cluster algebras or varieties?

Thomas Lam and I are working on a paper about this, which we hope to get out by next summer. Here 1 2 are some slides describing what we knew last Spring when I talked about these results in Austin. We know more now, but it doesn't look like there is going to be a clean complete answer. 


There is not much known in general, as far as I know. There is a nice 2form (called the WeilPetersson 2form) defined using the cluster algebra structure. This can be found in article "The WeilPetersson form on an acyclic cluster variety" by Greg Müller. Sometimes, the exterior powers of this 2form provide other algebraic differential forms. 

