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On the quiver Grassmannian in the acyclic case. (English) Zbl 1153.14032
Let $$Q$$ be a quiver without oriented cycles, let $$A$$ be its path algebra and let $$M$$ be a finite-dimensional $$A$$-module. The subject of this paper is the geometry of the so-called quiver Grassmannian $$Gr_{\underline e}(M)$$ of $$A$$-submodules of dimension $$\underline e$$, where $$\underline e$$ is a dimension vector; in particular, the Euler characteristics $$\chi_c(Gr_{\underline e}(M))$$ of these Grassmannians. This is related to cluster algebras as follows. If $$M$$ is an exceptional module (that is, without self-extensions), then there is a suitably defined Laurent polynomial $$\chi_M$$ defined in terms of Euler characteristics corresponding to various dimension vectors. It is known that the $$\chi_M$$’s, where $$M$$ runs over the set of exceptional modules, form a set of generators of the cluster algebra, called cluster variables. In this paper, an inductive formula to compute $$\chi_c(Gr_{\underline e}(M))$$ is given, when is $$M$$ is preprojective. This is used to prove a conjecture of Fomin and Zelevinsky for acyclic cluster algebras, namely that $$\chi_c(Gr_{\underline e}(M))$$ is positive when $$M$$ is exceptional. It is also shown that the quiver Grassmannian is smooth when $$M$$ is exceptional.

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14M15 Grassmannians, Schubert varieties, flag manifolds 16G20 Representations of quivers and partially ordered sets
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