A Natural Hermitian Line Bundle on the Moduli Space of Semistable Representations of a Quiver
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DOI: 10.1007/s13226-020-0446-0
A NATURAL HERMITIAN LINE BUNDLE ON THE MODULI SPACE OF SEMISTABLE REPRESENTATIONS OF A QUIVER Pradeep Das Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India e-mail: [email protected] (Received 1 May 2019; accepted 20 May 2019) This paper describes the construction of a natural Hermitian holomorphic line bundle on the stratified moduli space of complex representations of a finite quiver, which are semistable with respect to a fixed rational weight and have a fixed type. It is shown that the curvature of this Hermitian line bundle on each stratum of the moduli space is essentially the K¨ahler form of that stratum. Key words : Representations of quivers; moduli spaces; moment maps; Hermitian line bundles. 2010 Mathematics Subject Classification : 16G20, 53D20.
1. I NTRODUCTION Moduli spaces of semistable representations of a finite quiver over an algebraically closed field were studied by King in [5]. A consequence of his construction of these moduli spaces is that they are quasiprojective varieties. In particular, when the base field is that of the complex numbers, there exists a positive Hermitian holomorphic line bundle on each such moduli space. For a general survey about the moduli spaces of representations of quivers see [6]. The aim of this paper is to construct a natural and explicit such line bundle. It extends the results of [1], where such a line bundle was constructed on the smooth open subset of the moduli space whose points correspond to stable representations of the quiver. The second section contains definitions about quivers and their representations, and describes the moduli spaces in question. In Section 3, we construct a natural algebraic line bundle on the moduli space of semistable representations of the quiver. This section follows the methods of Drezet
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and Narasimhan [2] which dealt with the descent of line bundles to certain moduli spaces of vector bundles over curves. Section 4 recalls some facts about complex spaces with K¨ahler stratifications, as developed by Sjamaar [7]. Finally, Section 5 contains the derivations of the main results of the paper. 2. P RELIMINARIES This section describes some standard notions about quivers and their representations, semistability and stability of representations. The reason for including this material is that there are several conventions and notations about these objects, and it is necessary to state the conventions and notations that are used in this article. 2a. Quivers and their representations A quiver Q is a quadruple (Q0 , Q1 , s, t), where Q0 and Q1 are finite sets, Q0 6= ∅, and s : Q1 → Q0 and t : Q1 → Q0 are functions. The elements of Q0 (respectively, Q1 ) are called the vertices (respectively, arrows) of Q. For any arrow α of Q, the vertex s(α) (respectively, t(α)) is called the source (respectively, target) of α. If s(α) = a and t(α) = b, then we say that α is an arrow from a to b, and write α : a → b. A representation of Q is a pair (V, ρ), where V
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