The moduli space of Higgs pairs

TL;DR

This paper investigates the structure of the moduli space of Higgs pairs, establishing stability conditions, the Kobayashi-Hitchin correspondence, and analyzing its geometric and topological properties on Riemann surfaces.

Contribution

It introduces $ au$-stability for Higgs pairs, proves the moduli space is a smooth complex manifold under certain conditions, and computes its Poincaré polynomial for rank 2 bundles.

Findings

Moduli space is a non-singular complex manifold for suitable $ au$.

Poincaré polynomial of the moduli space for rank 2 bundles is determined.

Constructs a fibration from the moduli space of Higgs pairs to Higgs bundles.

Abstract

In this paper, we study the moduli space of Higgs pairs, which can be considered as a generalization of holomorphic pairs. Higgs pairs are an example of quiver bundles. We introduce the notion of $\tau$-stability of Higgs pairs for $\tau\in\mathbb{R}$ and establish the Kobayashi-Hitchin correspondence for Higgs pairs. The differential-geometric objects corresponding to stable Higgs pairs is called the vortex equations for Higgs bundles. We analyze the moduli space of stable Higgs pairs when the base space of vector bundle is a compact Riemann surface and obtaine the following results. Firstly, we prove that the moduli space is non-singular complex manifold for a suitable choice of $\tau$. Secondly, we determine the Poincar\'{e} polynomial of the moduli space for rank 2 bundle. Lastly, we construct a map from the moduli space of stable Higgs pairs to the moduli space of stable Higgs bundles and proved that the map is a fibration under suitable assumptions.