Holomorphic aspects of moduli of representations of quivers

TL;DR

This paper explores the complex-analytic structure of moduli spaces of stable quiver representations, constructing a complex manifold with a Kähler metric and a Hermitian line bundle whose curvature relates to the Kähler form.

Contribution

It introduces a natural complex manifold structure on the moduli space and defines a Hermitian line bundle with curvature proportional to the Kähler form.

Findings

Constructed a complex manifold structure on the moduli space.

Defined a Kähler metric on the moduli space.

Established a Hermitian line bundle with curvature related to the Kähler form.

Abstract

This article describes some complex-analytic aspects of the moduli space of the finite-dimensional complex representations of a finite quiver, which are stable with respect to a fixed rational weight. We construct a natural structure of a complex manifold on this moduli space, and a K\"ahler metric on the complex manifold. We then define a Hermitian holomorphic line bundle on the moduli space, and show that its curvature is a rational multiple of the K\"ahler form.