A normal variety of invariant connections on hermitian symmetric spaces

TL;DR

This paper introduces a class of invariant connections on principal bundles over hermitian symmetric spaces, revealing their structure as a normal variety with connections to quiver varieties and integrable complex structures.

Contribution

It defines a new class of invariant connections with a normal variety structure and explores their relation to quiver varieties and integrable complex structures.

Findings

The parameter space of invariant connections forms a normal variety.

The fixed points under an anti-holomorphic involution correspond to integrable complex structures.

Connections to quiver varieties and commuting matrix tuples are established.

Abstract

We introduce a class of $G$-invariant connections on a homogeneous principal bundle $Q$ over a hermitian symmetric space $M=G/K$. The parameter space carries the structure of normal variety and has a canonical anti-holomorphic involution. The fixed points of the anti-holomorphic involution are precisely the integrable invariant complex structures on $Q.$ This normal variety is closely related to quiver varieties and, more generally, to varieties of commuting matrix tuples modulo simultaneous conjugation.