On the quiver Grassmannian in the acyclic case
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.
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