The symplectic structure on the moduli space of line bundles on a noncommutative Azumaya surface

TL;DR

This paper proves that the moduli space of certain modules over an Azumaya algebra on a K3-surface forms an irreducible symplectic variety, similar to a Hilbert scheme of points on the surface.

Contribution

It establishes the symplectic structure and deformation equivalence of the moduli space to a Hilbert scheme on a K3-surface, extending geometric understanding of noncommutative surfaces.

Findings

Moduli space is an irreducible symplectic variety.

Deformation equivalent to a Hilbert scheme of points.

Provides new insights into noncommutative algebraic geometry.

Abstract

In this note we prove that the moduli space of torsion-free modules of rank one over an Azumaya algebra on a K3-surface is an irreducible symplectic variety deformation equivalent to a Hilbert scheme of points on the K3-surface.