Geometry of Multiplicative Preprojective Algebra

TL;DR

This paper explores the geometric structure of multiplicative preprojective algebras, revealing deep connections with quiver varieties and moduli spaces of local systems, especially for star-shaped quivers.

Contribution

It establishes a complex analytic isomorphism between nilpotent subvarieties of multiplicative and additive quiver varieties and links multiplicative quiver varieties to stable filtered local systems.

Findings

Existence of a complex analytic isomorphism between nilpotent subvarieties.

Symplectomorphism between tubular neighborhoods of these subvarieties.

Parametrization of stable filtered local systems for star-shaped quivers.

Abstract

Crawley-Boevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called the multiplicative preprojective algebra. In this paper we study the moduli space of (semi)stable representations of such an algebra (the multiplicative quiver variety), which in fact has many similarities to the quiver variety. We show that there exists a complex analytic isomorphism between the nilpotent subvariety of the quiver variety and that of the multiplicative quiver variety (which can be extended to a symplectomorphism between these tubular neighborhoods). We also show that when the quiver is star-shaped, the multiplicative quiver variety parametrizes Simpson's (poly)stable filtered local systems on a punctured Riemann sphere with prescribed filtration type, weight and associated graded local system around each puncture.