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?
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2 Answers
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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.
answered Dec 2, 2013 at 16:10
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$\begingroup$ Ah! That is very nice. $\endgroup$ Commented Dec 2, 2013 at 16:15
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$\begingroup$ Thanks for your answer. This is very inspiring and I look forward to reading it. I am also interested standard bases for the module of Kahler differentials (and exterior powers) themselves. For example, if A is the algebra on the first slide, we have a nice basis for A which consists of elements of the form $x'^n*y^m$ ($n\geq 0$) or $x^n*y^m$. These are for example homogeneous with respect to the action of one of the tori. Similarly for the two forms on A (things like d(x'^n)d(y'^m)). However for the module of Kahler differentials themselves, there doesn't appear to be such a basis $\endgroup$ Commented Dec 2, 2013 at 17:43
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$\begingroup$ (cont) I was wondering whether there is nevertheless some distinguished basis for this module of Kahler differentials that appears in the theory. The main problem in this example is of course that I don't know how to make a precise definition of what I am looking for. Does such a structure arise in your theory? $\endgroup$ Commented Dec 2, 2013 at 17:50
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$\begingroup$ We are taking our cluster varieties to be Spec of cluster algebras with frozen variables inverted. This means they are affine varieties, and hence have an infinite dimensional space of differentials. That doesn't mean that there couldn't be a nice basis anyways, but it's part of why we are looking at bases for H_{DR}, and not for global differentials. $\endgroup$ Commented Dec 2, 2013 at 19:26
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There is not much known in general, as far as I know.
There is a nice 2-form (called the Weil-Petersson 2-form) defined using the cluster algebra structure.
This can be found in article "The Weil-Petersson form on an acyclic cluster variety" by Greg Müller.
Sometimes, the exterior powers of this 2-form provide other algebraic differential forms.
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$\begingroup$ How about "not in general" e.g. under some nice hypotheses? For example even in this $A_1$ example discussed by Muller of the open subvariety of $SL_2(\mathbb{C})$. Of course I can compute the module of Kahler differentials in this setting. But it is clear what a "canonical bases" of these differential forms should be from the point of view of cluster varieties? $\endgroup$ Commented Dec 2, 2013 at 8:26