Euler characteristics and geometric properties of quiver Grassmannians

Abstract

The quiver Grassmannian is the projective variety of subrepresentations of a finite-dimensional representation of a quiver with a fixed dimension vector. <br /> Quiver Grassmannians occur naturally in different contexts. Fomin and Zelevinsky introduced cluster algebras in 2000. Caldero and Keller used Euler characteristics of quiver Grassmannians for the categorification of acyclic cluster algebras. This was generalized to arbitrary antisymmetric cluster algebras by Derksen, Weyman and Zelevinsky. The quiver Grassmannians play a crucial role in the construction of Ringel-Hall algebras. Moreover, they arise in the study of general representations of quivers by Schofield and in the theory of local models of Shimura varieties. Motivated by this, we study the geometric properties of quiver Grassmannians, their Euler characteristics and Ringel-Hall algebras. This work is divided into three parts. <br /> In the first part of this thesis, we study geometric properties of quiver Grassmannians. In some cases we compute the dimension of this variety, we detect smooth points and we prove semicontinuity of the rank functions and of the dimensions of homomorphism spaces. Moreover, we compare the geometry of quiver Grassmannians with the geometry of the module varieties and we develop tools to decompose the quiver Grassmannian into irreducible components. <br /> In the following we consider some special classes of quiver representations, called string, tree and band modules. There is an important family of finite-dimensional algebras, called string algebras, such that each indecomposable module is either a string or a band module. <br /> In the second part, for the complex field we compute the Euler characteristics of quiver Grassmannians and of quiver flag varieties in the case that the quiver representation is a direct sum of string, tree and band modules. We prove that these Euler characteristics are positive if the corresponding variety is non-empty. This generalizes some results of Cerulli Irelli. <br /> In the third part, we consider the Ringel-Hall algebra of a string algebra. We give a complete combinatorial description of the product of an important subalgebra of the Ringel-Hall algebra. <br /> In covering theory we resemble the results of the last two parts.

<strong>Euler Charakteristiken und geometrische Eigenschaften von Köcher Grassmannschen </strong><br /> The quiver Grassmannian is the projective variety of subrepresentations of a finite-dimensional representation of a quiver with a fixed dimension vector. <br /> Quiver Grassmannians occur naturally in different contexts. Fomin and Zelevinsky introduced cluster algebras in 2000. Caldero and Keller used Euler characteristics of quiver Grassmannians for the categorification of acyclic cluster algebras. This was generalized to arbitrary antisymmetric cluster algebras by Derksen, Weyman and Zelevinsky. The quiver Grassmannians play a crucial role in the construction of Ringel-Hall algebras. Moreover, they arise in the study of general representations of quivers by Schofield and in the theory of local models of Shimura varieties. Motivated by this, we study the geometric properties of quiver Grassmannians, their Euler characteristics and Ringel-Hall algebras. This work is divided into three parts. <br /> In the first part of this thesis, we study geometric properties of quiver Grassmannians. In some cases we compute the dimension of this variety, we detect smooth points and we prove semicontinuity of the rank functions and of the dimensions of homomorphism spaces. Moreover, we compare the geometry of quiver Grassmannians with the geometry of the module varieties and we develop tools to decompose the quiver Grassmannian into irreducible components. <br /> In the following we consider some special classes of quiver representations, called string, tree and band modules. There is an important family of finite-dimensional algebras, called string algebras, such that each indecomposable module is either a string or a band module. <br /> In the second part, for the complex field we compute the Euler characteristics of quiver Grassmannians and of quiver flag varieties in the case that the quiver representation is a direct sum of string, tree and band modules. We prove that these Euler characteristics are positive if the corresponding variety is non-empty. This generalizes some results of Cerulli Irelli. <br /> In the third part, we consider the Ringel-Hall algebra of a string algebra. We give a complete combinatorial description of the product of an important subalgebra of the Ringel-Hall algebra. <br /> In covering theory we resemble the results of the last two parts.